5-1
Identifying Linear Functions
Warm Up
1.Solve 2x – 3y = 12 for y.
2. Graph
for D: {–10, –5, 0, 5, 10}.
3.The ratio of red hair to brown hair in a class is
2:7. If one student is randomly chosen, what
is the probability that the student has red hair?
4.
Suppose a letter of the alphabet is randomly
chosen. What is the probability that it will be a
letter that is found in the word Tennessee?
Holt Algebra 1
5-1
Identifying Linear Functions
Objectives
Identify linear functions and linear equations.
Graph linear functions that represent realworld situations and give their domain and
range.
Holt Algebra 1
5-1
Identifying Linear Functions
The graph represents a
function because each
domain value (x-value) is
paired with exactly one
range value (y-value).
Notice that the graph is a
straight line. A function
whose graph forms a
straight line is called a
linear function.
Holt Algebra 1
5-1
Identifying Linear Functions
Example 1A: Identifying a Linear Function by Its Graph
Identify whether the graph represents a function.
Explain. If the graph does represent a function, is
the function linear?
Each domain value is
paired with exactly
one range value. The
graph forms a line.
linear function
Holt Algebra 1
5-1
Identifying Linear Functions
Example 1B: Identifying a Linear Function by Its Graph
Identify whether the graph represents a function.
Explain. If the graph does represent a function, is
the function linear?
Each domain value is
paired with exactly one
range value. The graph
is not a line.
not a linear function
Holt Algebra 1
5-1
Identifying Linear Functions
Example 1C: Identifying a Linear Function by Its Graph
Identify whether the graph represents a function.
Explain. If the graph does represent a function, is
the function linear?
The only domain value,
–2, is paired with many
different range values.
not a function
Holt Algebra 1
5-1
Identifying Linear Functions
Check It Out! Example 1b
Identify whether the graph represents a function.
Explain. If the graph does represent a function, is
the function linear?
Each domain value is
paired with exactly one
range value. The graph
forms a line.
linear function
Holt Algebra 1
5-1
Identifying Linear Functions
Check It Out! Example 1c
Identify whether the graph represents a function.
Explain. If the graph does represent a function, is
the function linear?
Each domain value is
not paired with exactly
one range value.
not a function
Holt Algebra 1
5-1
Identifying Linear Functions
You can sometimes identify a linear function
by looking a table or a list of ordered pairs. In
a linear function, a constant change in x
corresponds to a constant change in y.
Holt Algebra 1
5-1
Identifying Linear Functions
The points from
this table lie on a
line.
In this table, a constant
change of +1 in x
corresponds to constant
change of –3 in y. These
points satisfy a linear
function.
Holt Algebra 1
5-1
Identifying Linear Functions
The points from
this table do not lie
on a line.
In this table, a constant
change of +1 in x does not
correspond to a constant
change in y. These points do
not satisfy a linear function.
Holt Algebra 1
5-1
Identifying Linear Functions
Example 2A: Identifying a Linear Function by Using
Ordered Pairs
Tell whether the set of ordered pairs satisfies a
linear function. Explain.
{(0, –3), (4, 0), (8, 3), (12, 6), (16, 9)}
x
+4
+4
+4
+4
Holt Algebra 1
y
0
–3
4
0
8
3
12
6
16
9
+3
+3
+3
+3
Write the ordered pairs in a table.
Look for a pattern.
A constant change of +4 in x
corresponds to a constant
change of +3 in y.
These points satisfy a linear
function.
5-1
Identifying Linear Functions
Example 2B: Identifying a Linear Function by Using
Ordered Pairs
Tell whether the set of ordered pairs satisfies a
linear function. Explain.
{(–4, 13), (–2, 1), (0, –3), (2, 1), (4, 13)}
+2
+2
+2
+2
Holt Algebra 1
x
y
–4
13
–2
1
0
–3
2
1
4
13
–12
–4
+4
+12
Write the ordered pairs in a table.
Look for a pattern.
A constant change of 2 in x
corresponds to different
changes in y.
These points do not satisfy
a linear function.
5-1
Identifying Linear Functions
Another way to determine whether a function is
linear is to look at its equation. A function is linear
if it is described by a linear equation. A linear
equation is any equation that can be written in the
standard form shown below.
Holt Algebra 1
5-1
Identifying Linear Functions
Notice that when a linear equation is written in
standard form
• x and y both have exponents of 1.
• x and y are not multiplied together.
• x and y do not appear in denominators,
exponents, or radical signs.
Holt Algebra 1
5-1
Identifying Linear Functions
Holt Algebra 1
5-1
Identifying Linear Functions
For any two points, there is exactly one line that
contains them both. This means you need only
two ordered pairs to graph a line.
Holt Algebra 1
5-1
Identifying Linear Functions
Example 3A: Graphing Linear Functions
Tell whether the function is linear. If so,
graph the function.
x = 2y + 4
x = 2y + 4
–2y –2y
x – 2y =
4
Write the equation in standard form.
Try to get both variables on the
same side. Subtract 2y from both
sides.
The equation is in standard form
(A = 1, B = –2, C = 4).
The equation can be written in standard form, so
the function is linear.
Holt Algebra 1
5-1
Identifying Linear Functions
Example 3A Continued
x = 2y + 4
To graph, choose three values
of y, and use them to
generate ordered pairs. (You
only need two, but graphing
three points is a good check.)
y
0
x = 2y + 4
x = 2(0) + 4 = 4
(x, y)
(4, 0)
–1
–2
x = 2(–1) + 4 = 2
x = 2(–2) + 4 = 0
(2, –1)
(0, –2)
Holt Algebra 1
Plot the points and
connect them with a
straight line.
•
•
•
5-1
Identifying Linear Functions
Example 3B: Graphing Linear Functions
Tell whether the function is linear. If so,
graph the function.
xy = 4
This is not linear, because x and y are
multiplied. It is not in standard form.
Holt Algebra 1
5-1
Identifying Linear Functions
Check It Out! Example 3a
Tell whether the function is linear. If so,
graph the function.
y = 5x – 9
y = 5x – 9
–5x –5x
–5x + y =
–9
Write the equation in standard form.
Try to get both variables on the
same side. Subtract 5x from both
sides.
The equation is in standard form
(A = –5, B = 1, C = –9).
The equation can be written in standard form, so
the function is linear.
Holt Algebra 1
5-1
Identifying Linear Functions
Check It Out! Example 3a Continued
y = 5x – 9
To graph, choose three values
of x, and use them to
generate ordered pairs. (You
only need two, but graphing
three points is a good check.)
x
0
y = 5x – 9
y = 5(0) – 9 = –9
(x, y)
(0, –9)
1
y = 5(1) – 9 = –4
(1, –4)
2
y = 5(2) – 9 = 1
(2, 1)
Holt Algebra 1
Plot the points and
connect them with a
straight line.
•
•
•
5-1
Identifying Linear Functions
Check It Out! Example 3b
Tell whether the function is linear. If so,
graph the function.
y = 12
The equation is in standard form
(A = 0, B = 1, C = 12).
The equation can be written in standard form,
so the function is linear.
Holt Algebra 1
5-1
Identifying Linear Functions
Check It Out! Example 3b Continued
y = 12
y
Holt Algebra 1
5-1
Identifying Linear Functions
Check It Out! Example 3c
Tell whether the function is linear. If so,
graph the function.
y = 2x
This is not linear, because x is an exponent.
Holt Algebra 1
5-1
Identifying Linear Functions
For linear functions whose graphs are not
horizontal, the domain and range are all real
numbers. However, in many real-world
situations, the domain and range must be
restricted. For example, some quantities cannot
be negative, such as time.
Holt Algebra 1
5-1
Identifying Linear Functions
Sometimes domain and range are restricted
even further to a set of points. For example, a
quantity such as number of people can only be
whole numbers. When this happens, the graph
is not actually connected because every point on
the line is not a solution. However, you may see
these graphs shown connected to indicate that
the linear pattern, or trend, continues.
Holt Algebra 1
5-1
Identifying Linear Functions
Check It Out! Example 4
What if…? At a salon, Sue can rent a station for
$10.00 per day plus $3.00 per manicure. The
amount she would pay each day is given by f(x)
= 3x + 10, where x is the number of manicures.
Graph this function and give its domain and
range.
Holt Algebra 1
5-1
Identifying Linear Functions
Check It Out! Example 4 Continued
Choose several values of x and make a table of ordered
pairs.
x
f(x) = 3x + 10
0
f(0) = 3(0) + 10 = 10
1
f(1) = 3(1) + 10 = 13
2
f(2) = 3(2) + 10 = 16
3
f(3) = 3(3) + 10 = 19
4
f(4) = 3(4) + 10 = 22
5
f(5) = 3(5) + 10 = 25
Holt Algebra 1
The number of manicures
must be a whole number, so
the domain is {0, 1, 2, 3, …}.
The range is {10.00, 13.00,
16.00, 19.00, …}.
5-1
Identifying Linear Functions
Check It Out! Example 4 Continued
Graph the ordered pairs.
The individual
points are
solutions in this
situation. The
line shows that
the trend
continues.
Holt Algebra 1
5-2 Using Intercepts
Warm-Up
Tell whether each set of ordered pairs
satisfies a linear function. Explain.
1. {(–3, 10), (–1, 9), (1, 7), (3, 4), (5, 0)}
2. {(3, 4), (5, 7), (7, 10), (9, 13), (11, 16)}
Tell whether each function is linear. If so,
graph the function.
3. y = 3 – 2x
4. 3y = 12
5. The cost of a can of iced-tea mix at Save More
Grocery is $4.75. The function f(x) = 4.75x
gives the cost of x cans of iced-tea mix. Graph
this function and give its domain and range.
Holt Algebra 1
5-2 Using Intercepts
Objectives
Find x- and y-intercepts and interpret their
meanings in real-world situations.
Use x- and y-intercepts to graph lines.
Holt Algebra 1
5-2 Using Intercepts
The y-intercept is the ycoordinate of the point
where the graph intersects
the y-axis. The x-coordinate
of this point is always 0.
The x-intercept is the xcoordinate of the point
where the graph intersects
the x-axis. The y-coordinate
of this point is always 0.
Holt Algebra 1
5-2 Using Intercepts
Example 1A: Finding Intercepts
Find the x- and y-intercepts.
The graph intersects the
y-axis at (0, 1).
The y-intercept is 1.
The graph intersects the
x-axis at (–2, 0).
The x-intercept is –2.
Holt Algebra 1
5-2 Using Intercepts
Example 1B: Finding Intercepts
Find the x- and y-intercepts.
5x – 2y = 10
To find the x-intercept,
To find the y-intercept,
replace y with 0 and solve
replace x with 0 and solve
for x.
for y. 5x – 2y = 10
5x – 2y = 10
5x – 2(0) = 10
5x – 0 = 10
5x = 10
x=2
The x-intercept is 2.
Holt Algebra 1
5(0) – 2y = 10
0 – 2y = 10
– 2y = 10
y = –5
The y-intercept is –5.
5-2 Using Intercepts
Example 2: Sports Application
Trish can run the 200 m dash in 25 s. The
function f(x) = 200 – 8x gives the distance
remaining to be run after x seconds. Graph
this function and find the intercepts. What
does each intercept represent?
Neither time nor distance can be negative, so choose
several nonnegative values for x. Use the function to
generate ordered pairs.
x
f(x) = 200 – 8x
Holt Algebra 1
0
5
10
20
25
200
160
120
40
0
5-2 Using Intercepts
Example 2 Continued
Graph the ordered pairs. Connect
the points with a line.
y-intercept: 200. This is the
number of meters Trish has to
run at the start of the race.
x-intercept: 25. This is the time it
takes Trish to finish the race, or
when the distance remaining is 0.
Holt Algebra 1
5-2 Using Intercepts
Check It Out! Example 2a
The school sells pens for $2.00 and notebooks
for $3.00. The equation 2x + 3y = 60 describes
the number of pens x and notebooks y that you
can buy for $60.
Graph the function and find its intercepts.
Neither pens nor notebooks can be negative, so choose
several nonnegative values for x. Use the function to
generate ordered pairs.
x
Holt Algebra 1
0
15
30
20
10
0
5-2 Using Intercepts
Check It Out! Example 2a Continued
The school sells pens for $2.00 and notebooks
for $3.00. The equation 2x + 3y = 60 describes
the number of pens x and notebooks y that you
can buy for $60.
Graph the function and find its intercepts.
x-intercept: 30; y-intercept: 20
Holt Algebra 1
5-2 Using Intercepts
Check It Out! Example 2b
The school sells pens for $2.00 and notebooks
for $3.00. The equation 2x + 3y = 60 describes
the number of pens x and notebooks y that you
can buy for $60.
What does each intercept represent?
x-intercept: 30. This is the
number of pens that can be
purchased if no notebooks are
purchased.
y-intercept: 20. This is the
number of notebooks that can
be purchased if no pens are
purchased.
Holt Algebra 1
5-2 Using Intercepts
Remember, to graph a linear function, you need
to plot only two ordered pairs. It is often
simplest to find the ordered pairs that contain
the intercepts.
Helpful Hint
You can use a third point to check your line. Either
choose a point from your graph and check it in the
equation, or use the equation to generate a point
and check that it is on your graph.
Holt Algebra 1
5-2 Using Intercepts
Example 3A: Graphing Linear Equations by Using
Intercepts
Use intercepts to graph the line described by
the equation.
3x – 7y = 21
Step 1 Find the intercepts.
x-intercept:
y-intercept:
3x – 7y = 21
3x – 7y = 21
3x – 7(0) = 21
3(0) – 7y = 21
3x = 21
–7y = 21
x=7
Holt Algebra 1
y = –3
5-2 Using Intercepts
Example 3A Continued
Use intercepts to graph the line described by
the equation.
3x – 7y = 21
Step 2 Graph the line.
Plot (7, 0) and (0, –3).
x
Holt Algebra 1
Connect with a straight line.
5-2 Using Intercepts
Check It Out! Example 3b
Use intercepts to graph the line described by
the equation.
Step 1 Write the equation in standard form.
Multiply both sides 3, to
clear the fraction.
3y = x – 6
–x + 3y = –6
Holt Algebra 1
Write the equation in
standard form.
5-2 Using Intercepts
Check It Out! Example 3b Continued
Use intercepts to graph the line described by
the equation.
–x + 3y = –6
Step 2 Find the intercepts.
x-intercept:
y-intercept:
–x + 3y = –6
–x + 3y = –6
–(0) + 3y = –6
3y = –6
–x + 3(0) = –6
–x = –6
x=6
y = –2
Holt Algebra 1
5-2 Using Intercepts
Check It Out! Example 3b Continued
Use intercepts to graph the line described by
the equation.
–x + 3y = –6
Step 3 Graph the line.
Plot (6, 0) and (0, –2).
Connect with a straight
line.
Holt Algebra 1
5-3
Rate of Change and Slope
Warm-Up
1. An amateur filmmaker has $6000 to make a film
that costs $75/h to produce. The function f(x) =
6000 – 75x gives the amount of money left to
make the film after x hours of production. Graph
this function and find the intercepts. What does
each intercept represent?
2. Use intercepts to graph the line described by
Holt Algebra 1
5-3
Rate of Change and Slope
Objectives
Find rates of change and slopes.
Relate a constant rate of change to the slope
of a line.
Holt Algebra 1
5-3
Rate of Change and Slope
A rate of change is a ratio that compares the
amount of change in a dependent variable to
the amount of change in an independent
variable.
Holt Algebra 1
5-3
Rate of Change and Slope
Example 1: Application
The table shows the average temperature (°F)
for five months in a certain city. Find the rate of
change for each time period. During which time
period did the temperature increase at the
fastest rate?
Step 1 Identify the dependent and independent
variables.
dependent: temperature
Holt Algebra 1
independent: month
5-3
Rate of Change and Slope
Example 1 Continued
Step 2 Find the rates of change.
2 to 3
3 to 5
5 to 7
7 to 8
The temperature increased at the greatest rate
from month 5 to month 7.
Holt Algebra 1
5-3
Rate of Change and Slope
Example 2: Finding Rates of Change from a Graph
Graph the data from Example 1 and show the
rates of change.
Graph the ordered pairs. The
vertical segments show the
changes in the dependent
variable, and the horizontal
segments show the changes in
the independent variable.
Notice that the greatest rate of
change is represented by the
steepest of the red line
segments.
Holt Algebra 1
5-3
Rate of Change and Slope
Example 2 Continued
Graph the data from Example 1 and show the
rates of change.
Also notice that between months
2 to 3, when the balance did
not change, the line segment is
horizontal.
Holt Algebra 1
5-3
Rate of Change and Slope
If all of the connected segments have the same
rate of change, then they all have the same
steepness and together form a straight line. The
constant rate of change of a line is called the
slope of the line.
Holt Algebra 1
5-3
Rate of Change and Slope
Holt Algebra 1
Rate of Change and Slope
5-3
Example 3: Finding Slope
Find the slope of the line.
(–6, 5)
Run –9
•
Rise 3
Rise –3
•
Run 9
Holt Algebra 1
(3, 2)
Begin at one point and count
vertically to fine the rise.
Then count horizontally to the
second point to find the run.
It does not matter which point
you start with. The slope is
the same.
5-3
Rate of Change and Slope
Example 4: Finding Slopes of Horizontal and Vertical
Lines
Find the slope of each line.
A.
B.
You cannot
divide by 0
The slope is undefined.
Holt Algebra 1
The slope is 0.
5-3
Rate of Change and Slope
As shown in the previous examples, slope can be
positive, negative, zero or undefined. You can tell
which of these is the case by looking at a graph of
a line–you do not need to calculate the slope.
Holt Algebra 1
5-3
Rate of Change and Slope
Check It Out! Example 5
Tell whether the slope of each line is positive,
negative, zero or undefined.
a.
The line is vertical.
The slope is undefined.
Holt Algebra 1
b.
The line rises from left to right.
The slope is positive.
5-3
Rate of Change and Slope
Holt Algebra 1
5-4
The Slope Formula
Warm-Up
Name each of the following.
1. The table shows the number of bikes made by a
company for certain years. Find the rate of change
for each time period. During which time period did
the number of bikes increase at the fastest rate?
Find the slope of each line.
2.
Holt Algebra 1
3.
5-4
The Slope Formula
Objective
Find slope by using the slope formula.
Holt Algebra 1
5-4
The Slope Formula
Holt Algebra 1
5-4
The Slope Formula
Example 1: Finding Slope by Using the Slope Formula
Find the slope of the line that contains (2, 5)
and (8, 1).
The slope of the line that contains (2, 5) and (8, 1)
is
.
Holt Algebra 1
5-4
The Slope Formula
Check It Out! Example 1a
Find the slope of the line that contains (–2, –2)
and (7, –2).
=0
The slope of the line that contains (–2, –2) and
(7, –2) is 0.
Holt Algebra 1
5-4
The Slope Formula
Check It Out! Example 1c
Find the slope of the line that contains
and
The slope of the line that contains
is 2.
Holt Algebra 1
and
5-4
The Slope Formula
Sometimes you are not given two points
to use in the formula. You might have to
choose two points from a graph or a
table.
Holt Algebra 1
5-4
The Slope Formula
Example 2A: Finding Slope from Graphs and Tables
The graph shows a linear relationship.
Find the slope.
Let (0, 2) be (x1, y1) and (–2, –2) be (x2, y2).
Holt Algebra 1
5-4
The Slope Formula
Example 2B: Finding Slope from Graphs and Tables
The table shows a linear relationship.
Find the slope.
Step 1 Choose any two points from the table. Let
(0, 1) be (x1, y1) and (–2, 5) be (x2, y2).
Step 2 Use the slope formula.
The slope equals −2
Holt Algebra 1
5-4
The Slope Formula
Remember that slope is a rate of change.
In real-world problems, finding the slope
can give you information about how a
quantity is changing.
Holt Algebra 1
5-4
The Slope Formula
Example 3: Application
The graph shows the
average electricity costs
(in dollars) for operating a
refrigerator for several
months. Find the slope of
the line. Then tell what the
slope represents.
Step 1 Use the slope formula.
Holt Algebra 1
5-4
The Slope Formula
Example 3 Continued
Step 2 Tell what the slope represents.
In this situation y represents the cost of electricity
and x represents time.
So slope represents
in units of
.
A slope of 6 mean the cost of running the
refrigerator is a rate of 6 dollars per month.
Holt Algebra 1
5-4
The Slope Formula
If you know the equation that describes
a line, you can find its slope by using
any two ordered-pair solutions. It is
often easiest to use the ordered pairs
that contain the intercepts.
Holt Algebra 1
5-4
The Slope Formula
Example 4: Finding Slope from an Equation
Find the slope of the line described by 4x – 2y = 16.
Step 1 Find the x-intercept. Step 2 Find the y-intercept.
4x – 2y = 16
4x – 2(0) = 16
4x = 16
Let y = 0.
4x – 2y = 16
4(0) – 2y = 16
Let x = 0.
–2y = 16
y = –8
x=4
Step 3 The line contains (4, 0) and (0, –8). Use the
slope formula.
Holt Algebra 1
1-6
Midpoint and Distance
in the Coordinate Plane
5 Minute Warm-Up
Find the area and perimeter/circumference of
each figure.
1.
2.
3.
4. circle with radius 2 cm
5. circle with diameter 12 ft
Holt Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
Objectives
Develop and apply the formula for midpoint.
Use the Distance Formula and the
Pythagorean Theorem to find the distance
between two points.
Holt Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
A coordinate plane is a plane that is
divided into four regions by a horizontal
line (x-axis) and a vertical line (y-axis) .
The location, or coordinates, of a point are
given by an ordered pair (x, y).
Holt Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
You can find the midpoint of a segment by
using the coordinates of its endpoints.
Calculate the average of the x-coordinates
and the average of the y-coordinates of the
endpoints.
Holt Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
Holt Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
Example 1: Finding the Coordinates of a Midpoint
Find the coordinates of the midpoint of PQ
with endpoints P(–8, 3) and Q(–2, 7).
= (–5, 5)
Holt Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
Check It Out! Example 1
Find the coordinates of the midpoint of EF
with endpoints E(–2, 3) and F(5, –3).
Holt Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
Example 2: Finding the Coordinates of an Endpoint
M is the midpoint of XY. X has coordinates
(2, 7) and M has coordinates (6, 1). Find
the coordinates of Y.
Step 1 Let the coordinates of Y equal (x, y).
Step 2 Use the Midpoint Formula:
Holt Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
Example 2 Continued
Step 3 Find the x-coordinate.
Set the coordinates equal.
Multiply both sides by 2.
12 = 2 + x
– 2 –2
Simplify.
Subtract.
10 = x
Simplify.
The coordinates of Y are (10, –5).
Holt Geometry
2=7+y
– 7 –7
–5 = y
1-6
Midpoint and Distance
in the Coordinate Plane
Check It Out! Example 2 Continued
Step 3 Find the x-coordinate.
Set the coordinates equal.
Multiply both sides by 2.
–2 = –6 + x
+ 6 +6
4=x
Simplify.
Add.
2 = –1 + y
+1 +1
Simplify.
3=y
The coordinates of T are (4, 3).
Holt Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
The Ruler Postulate can be used to find the distance
between two points on a number line. The Distance
Formula is used to calculate the distance between
two points in a coordinate plane.
Holt Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
Example 3: Using the Distance Formula
Find FG and JK.
Then determine whether FG  JK.
Step 1 Find the
coordinates of each point.
F(1, 2), G(5, 5), J(–4, 0),
K(–1, –3)
Holt Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
Example 3 Continued
Step 2 Use the Distance Formula.
Holt Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
Check It Out! Example 3 Continued
Step 2 Use the Distance Formula.
Holt Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
You can also use the Pythagorean Theorem to
find the distance between two points in a
coordinate plane. You will learn more about the
Pythagorean Theorem in Chapter 5.
In a right triangle, the two sides that form the
right angle are the legs. The side across from the
right angle that stretches from one leg to the
other is the hypotenuse. In the diagram, a and b
are the lengths of the shorter sides, or legs, of the
right triangle. The longest side is called the
hypotenuse and has length c.
Holt Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
Holt Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
Example 4: Finding Distances in the Coordinate Plane
Use the Distance Formula and the
Pythagorean Theorem to find the distance, to
the nearest tenth, from D(3, 4) to E(–2, –5).
Holt Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
Example 4 Continued
Method 1
Use the Distance Formula. Substitute the
values for the coordinates of D and E into the
Distance Formula.
Holt Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
Example 4 Continued
Method 2
Use the Pythagorean Theorem. Count the units for
sides a and b.
a = 5 and b = 9.
c2 = a2 + b2
= 52 + 92
= 25 + 81
= 106
c = 10.3
Holt Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
Example 5: Sports Application
A player throws the ball
from first base to a point
located between third
base and home plate and
10 feet from third base.
What is the distance of
the throw, to the nearest
tenth?
Holt Geometry
1-6
Midpoint and Distance
in the Coordinate Plane
Example 5 Continued
Set up the field on a coordinate plane so that home
plate H is at the origin, first base F has coordinates
(90, 0), second base S has coordinates (90, 90), and
third base T has coordinates (0, 90).
The target point P of the throw has coordinates (0, 80).
The distance of the throw is FP.
Holt Geometry
5-5
Direct Variation
Warm-Up
1. Find the slope of the line that contains (5, 3)
and (–1, 4).
2. Find the slope of the line. Then tell what the
slope represents.
50; speed of bus is 50 mi/h
3. Find the slope of the line described by x + 2y = 8.
Holt Algebra 1
5-5
Direct Variation
Objective
Identify, write, and graph direct variation.
Holt Algebra 1
5-5
Direct Variation
A recipe for paella calls for 1 cup of rice to make 5
servings. In other words, a chef needs 1 cup of
rice for every 5 servings.
The equation y = 5x describes this relationship. In
this relationship, the number of servings varies
directly with the number of cups of rice.
Holt Algebra 1
5-5
Direct Variation
A direct variation is a special type of linear
relationship that can be written in the form
y = kx, where k is a nonzero constant called
the constant of variation.
Holt Algebra 1
5-5
Direct Variation
Example 1A: Identifying Direct Variations from
Equations
Tell whether the equation represents a direct
variation. If so, identify the constant of variation.
y = 3x
This equation represents a direct variation because
it is in the form of y = kx. The constant of
variation is 3.
Holt Algebra 1
5-5
Direct Variation
Example 1B: Identifying Direct Variations from
Equations
Tell whether the equation represents a direct
variation. If so, identify the constant of variation.
3x + y = 8
–3x
–3x
y = –3x + 8
Solve the equation for y.
Since 3x is added to y, subtract 3x
from both sides.
This equation is not a direct variation because it
cannot be written in the form y = kx.
Holt Algebra 1
5-5
Direct Variation
Example 1C: Identifying Direct Variations from
Equations
Tell whether the equation represents a direct
variation. If so, identify the constant of variation.
–4x + 3y = 0
Solve the equation for y.
+4x
+4x
Since –4x is added to 3y, add 4x
3y = 4x
to both sides.
Since y is multiplied by 3, divide
both sides by 3.
This equation represents a direct variation because
it is in the form of y = kx. The constant of
variation is .
Holt Algebra 1
5-5
Direct Variation
What happens if you solve y = kx for k?
y = kx
Divide both sides by x (x ≠ 0).
So, in a direct variation, the ratio is equal to
the constant of variation. Another way to identify
a direct variation is to check whether
is the
same for each ordered pair (except where x = 0).
Holt Algebra 1
5-5
Direct Variation
Example 2A Continued
Tell whether the relationship
is a direct variation. Explain.
Method 2 Find
for each ordered pair.
This is a direct variation because
each ordered pair.
Holt Algebra 1
is the same for
5-5
Direct Variation
Example 2B Continued
Tell whether the relationship
is a direct variation. Explain.
Method 2 Find
for each ordered pair.
…
This is not direct variation because
same for all ordered pairs.
Holt Algebra 1
is the not the
5-5
Direct Variation
Example 3: Writing and Solving Direct Variation
Equations
The value of y varies directly with x, and y = 3,
when x = 9. Find y when x = 21.
Method 1 Find the value of k and then write the
equation.
y = kx
Write the equation for a direct variation.
3 = k(9)
Substitute 3 for y and 9 for x. Solve for k.
Since k is multiplied by 9, divide both sides
by 9.
The equation is y =
Holt Algebra 1
x. When x = 21, y =
(21) = 7.
5-5
Direct Variation
Check It Out! Example 3
The value of y varies directly with x, and y = 4.5
when x = 0.5. Find y when x = 10.
Method 1 Find the value of k and then write the
equation.
y = kx
4.5 = k(0.5)
9=k
Write the equation for a direct variation.
Substitute 4.5 for y and 0.5 for x. Solve
for k.
Since k is multiplied by 0.5, divide both
sides by 0.5.
The equation is y = 9x. When x = 10, y = 9(10) = 90.
Holt Algebra 1
5-5
Direct Variation
Lesson Quiz: Part I
Tell whether each equation represents a
direct variation. If so, identify the constant
of variation.
1. 2y = 6x
yes; 3
no
2. 3x = 4y – 7
Tell whether each relationship is a direct
variation. Explain.
3.
Holt Algebra 1
4.
5-5
Direct Variation
Lesson Quiz: Part II
5. The value of y varies directly with x, and
y = –8 when x = 20. Find y when x = –4. 1.6
6. Apples cost $0.80 per pound. The equation
y = 0.8x describes the cost y of x pounds
of apples. Graph this direct variation.
6
4
2
Holt Algebra 1
5-6
Slope-Intercept Form
Objectives
Write a linear equation in slope-intercept form.
Graph a line using slope-intercept form.
Holt Algebra 1
5-6
Slope-Intercept Form
Example 1A: Graphing by Using Slope and y-intercept
Graph the line given the slope and y-intercept.
y intercept = 4
Rise = –2
Step 1 The y-intercept is 4, so the
line contains (0, 4). Plot (0, 4).
Step 2 Slope =
y
•
•
•
Run = 5
Count 2 units down and 5 units
right from (0, 4) and plot another
point.
Step 3 Draw the line through the two points.
Holt Algebra 1
•
5-6
Slope-Intercept Form
Check It Out! Example 1a
Graph the line given the slope and y-intercept.
slope = 2, y-intercept = –3
Step 1 The y-intercept is –3, so the
line contains (0, –3). Plot (0, –3).
Step 2 Slope =
Count 2 units up and 1 unit right
from (0, –3) and plot another
point.
Step 3 Draw the line through the
two points.
Holt Algebra 1
Run = 1
•
Rise = 2
•
5-6
Slope-Intercept Form
If you know the slope of a line and the y-intercept,
you can write an equation that describes the line.
Step 1 If a line has a slope of 2 and the y-intercept
is 3, then m = 2 and (0, 3) is on the line. Substitute
these values into the slope formula.
Holt Algebra 1
5-6
Slope-Intercept Form
Step 2 Solve for y:
Simplify the denominator.
•
•
2x = y – 3
+3
+3
2x + 3 = y, or y = 2x + 3
Holt Algebra 1
Multiply both sides by x.
Add 3 to both sides.
5-6
Slope-Intercept Form
Any linear equation can be written in slope-intercept
form by solving for y and simplifying. In this form,
you can immediately see the slope and y-intercept.
Also, you can quickly graph a line when the equation
is written in slope-intercept form.
Holt Algebra 1
5-6
Slope-Intercept Form
Example 2A: Writing linear Equations in SlopeIntercept Form
Write the equation that describes the line in
slope-intercept form.
slope =
; y-intercept = 4
y = mx + b
y=
Holt Algebra 1
x+4
Substitute the given values for
m and b.
Simplify if necessary.
5-6
Slope-Intercept Form
Example 2B: Writing linear Equations in SlopeIntercept Form
Write the equation that describes the line in
slope-intercept form.
slope = –9; y-intercept =
Holt Algebra 1
y = mx + b
Substitute the given values for
m and b.
y = –9x +
Simplify if necessary.
5-6
Slope-Intercept Form
Example 2E: Writing linear Equations in SlopeIntercept Form
Write the equation that describes the line in
slope-intercept form.
slope = 2; (3, 4) is on the line
Step 1 Find the y-intercept.
y = mx + b
Write the slope-intercept form.
4 = 2(3) + b
Substitute 2 for m, 3 for x, and
4 for y.
4=6+b
–6 –6
–2 = b
Holt Algebra 1
Solve for b. Since 6 is added to
b, subtract 6 from both sides
to undo the addition.
5-6
Slope-Intercept Form
Example 2E Continued
Write the equation that describes the line in
slope-intercept form.
slope = 2; (3, 4) is on the line
Step 2 Write the equation.
y = mx + b
Write the slope-intercept form.
y = 2x + (–2)
Substitute 2 for m, and –2 for b.
y = 2x – 2
Holt Algebra 1
5-6
Slope-Intercept Form
Example 3B: Using Slope-Intercept Form to Graph
Write the equation in slope-intercept form.
Then graph the line described by the equation.
2y + 3x = 6
Step 1 Write the equation in slope-intercept form
by solving for y.
2y + 3x = 6
–3x –3x
2y = –3x + 6
Subtract 3x from both sides.
Since y is multiplied by 2,
divide both sides by 2.
Holt Algebra 1
5-6
Slope-Intercept Form
Example 3B Continued
Write the equation in slope-intercept form.
Then graph the line described by the equation.
Step 2 Graph the line.
is in the form
y = mx + b.
slope: m =
y-intercept: b = 3
Plot (0, 3).
•
•
• Count 3 units down and 2 units right and plot
another point.
• Draw the line connecting the two points.
Holt Algebra 1
5-6
Slope-Intercept Form
Check It Out! Example 3a
Write the equation in slope-intercept form.
Then graph the line described by the equation.
is in the form y = mx + b.
Holt Algebra 1
5-6
Slope-Intercept Form
Check It Out! Example 3a Continued
Write the equation in slope-intercept form.
Then graph the line described by the equation.
Step 2 Graph the line.
y=
x + 0 is in the form
y = mx + b.
slope:
•
•
y-intercept: b = 0
Step 1 Plot (0, 0).
Step 2 Count 2 units up and 3 units right and
plot another point.
Step 3 Draw the line connecting the two points.
Holt Algebra 1
5-6
Slope-Intercept Form
Check It Out! Example 3c
Write the equation in slope-intercept form.
Then graph the line described by the equation.
y = –4
y = –4 is in the form y = mx + b.
slope: m = 0 = = 0
y-intercept: b = –4
Step 1 Plot (0, –4).
Since the slope is 0, the line
will be a horizontal at y = –4.
Holt Algebra 1
•
5-6
Slope-Intercept Form
Example 4: Application
A closet organizer charges a $100 initial
consultation fee plus $30 per hour. The cost
as a function of the number of hours worked
is graphed below.
Holt Algebra 1
5-6
Slope-Intercept Form
Example 4: Application
A closet organizer charges $100 initial
consultation fee plus $30 per hour. The cost
as a function of the number of hours worked
is graphed below.
a. Write an equation that represents the cost as a
function of the number of hours.
Cost
is
$30
y
=
30
for each
hour
•x
An equation is y = 30x + 100.
Holt Algebra 1
plus
$100
+
100
5-6
Slope-Intercept Form
Example 4 Continued
A closet organizer charges $100 initial
consultation fee plus $30 per hour. The cost
as a function of the number of hours worked
is graphed below.
b. Identify the slope and y-intercept and describe
their meanings.
The y-intercept is 100. This is the cost for 0 hours,
or the initial fee of $100. The slope is 30. This is the
rate of change of the cost: $30 per hour.
c. Find the cost if the organizer works 12 hrs.
y = 30x + 100
Substitute 12 for x in the
= 30(12) + 100 = 460 equation
The cost of the organizer for 12 hours is $460.
Holt Algebra 1
5-7
Point-Slope Form
Warm-Up
Write the equation that describes each line in
the slope-intercept form.
1. slope = 3, y-intercept = –2
2. slope = 0, y-intercept =
3. slope =
, (2, 7) is on the line
Write each equation in slope-intercept form.
Then graph the line described by the equation.
4. 6x + 2y = 10
5. x – y = 6
Holt Algebra 1
5-7
Point-Slope Form
Objectives
Graph a line and write a linear equation using
point-slope form.
Write a linear equation given two points.
Holt Algebra 1
5-7
Point-Slope Form
In lesson 5-6 you saw that if you know the
slope of a line and the y-intercept, you can
graph the line. You can also graph a line if you
know its slope and any point on the line.
Holt Algebra 1
5-7
Point-Slope Form
Example 1A: Using Slope and a Point to Graph
Graph the line with the given slope that contains
the given point.
slope = 2; (3, 1)
Step 1 Plot (3, 1).
Step 2 Use the slope to move from
(3, 1) to another point.
1
2
•
Move 2 units up and 1 unit
right and plot another point.
Step 3 Draw the line connecting the two points.
Holt Algebra 1
•
(3, 1)
5-7
Point-Slope Form
Example 1C: Using Slope and a Point to Graph
Graph the line with the given slope that contains
the given point.
slope = 0; (4, –3)
A line with a slope of 0 is
horizontal. Draw the horizontal
line through (4, –3).
•
(4, –3)
Holt Algebra 1
5-7
Point-Slope Form
If you know the slope and any point on the
line, you can write an equation of the line by
using the slope formula. For example, suppose
a line has a slope of 3 and contains (2, 1). Let
(x, y) be any other point on the line.
Substitute into the
slope formula.
Slope formula
Multiply both sides by
(x – 2).
3(x – 2) = y – 1
y – 1 = 3(x – 2)
Holt Algebra 1
Simplify.
5-7
Point-Slope Form
Holt Algebra 1
5-7
Point-Slope Form
Example 2: Writing Linear Equations in Point-Slope
Form
Write an equation in point-slope form for the line
with the given slope that contains the given point.
C.
A.
B.
Holt Algebra 1
5-7
Point-Slope Form
Example 3: Writing Linear Equations in SlopeIntercept Form
Write an equation in slope-intercept form for
the line with slope 3 that contains (–1, 4).
Step 1 Write the equation in point-slope form:
y – y1 = m(x – x1)
y – 4 = 3[x – (–1)]
Step 2 Write the equation in slope-intercept form by
solving for y.
Rewrite subtraction of negative
y – 4 = 3(x + 1)
numbers as addition.
y – 4 = 3x + 3 Distribute 3 on the right side.
+4
+ 4 Add 4 to both sides.
y = 3x + 7
Holt Algebra 1
5-7
Point-Slope Form
Example 4A: Using Two Points to Write an Equation
Write an equation in slope-intercept form for
the line through the two points.
(2, –3) and (4, 1)
Step 1 Find the slope.
Step 2 Substitute the slope and one of the points
into the point-slope form.
y – y1 = m(x – x1)
y – (–3) = 2(x – 2)
Holt Algebra 1
Choose (2, –3).
5-7
Point-Slope Form
Example 4A Continued
Write an equation in slope-intercept form for
the line through the two points.
(2, –3) and (4, 1)
Step 3 Write the equation in slope-intercept form.
y + 3 = 2(x – 2)
y + 3 = 2x – 4
–3
–3
y = 2x – 7
Holt Algebra 1
5-7
Point-Slope Form
Example 5: Problem-Solving Application
The cost to stain a deck is a linear function
of the deck’s area. The cost to stain 100,
250, 400 square feet are shown in the
table. Write an equation in slope-intercept
form that represents the function. Then
find the cost to stain a deck whose area is
75 square feet.
Holt Algebra 1
Point-Slope Form
5-7
Example 5 Continued
1
Understand the Problem
• The answer will have two parts—an equation
in slope-intercept form and the cost to stain
an area of 75 square feet.
• The ordered pairs given in the table—(100,
150), (250, 337.50), (400, 525)—satisfy the
equation.
Holt Algebra 1
5-7
Point-Slope Form
Example 5 Continued
2
Make a Plan
You can use two of the ordered pairs to find
the slope. Then use point-slope form to write
the equation. Finally, write the equation in
slope-intercept form.
Holt Algebra 1
5-7
Point-Slope Form
Example 5 Continued
3
Solve
Step 1 Choose any two ordered pairs from
the table to find the slope.
Use (100, 150)
and (400, 525).
Step 2 Substitute the slope and any ordered
pair from the table into the point-slope
form.
y – y1 = m(x – x1)
y – 150 = 1.25(x – 100)
Holt Algebra 1
Use (100, 150).
5-7
Point-Slope Form
Example 5 Continued
Step 3 Write the equation in slope-intercept
form by solving for y.
y – 150 = 1.25(x – 100)
y – 150 = 1.25x – 125
Distribute 1.25.
y = 1.25x + 25
Add 150 to both
sides.
Step 4 Find the cost to stain an area of 75 sq. ft.
y = 1.25x + 25
y = 1.25(75) + 25 = 118.75
The cost of staining 75 sq. ft. is $118.75.
Holt Algebra 1
5-8
Slopes of Parallel and
Perpendicular Lines
Warm-Up
Write an equation in point-slope form and
slope-intercept form for the line through the
two points.
1. (–1, 7) and (2, 1)
2. The cost to take a taxi from
the airport is a linear function of
the distance driven. The cost for
5, 10, and 20 miles are shown in the
table. Write an equation in
slope-intercept form that represents the function.
Holt Algebra 1
5-8
Slopes of Parallel and
Perpendicular Lines
Objectives
Identify and graph parallel and perpendicular
lines.
Write equations to describe lines parallel or
perpendicular to a given line.
Holt Algebra 1
5-8
Slopes of Parallel and
Perpendicular Lines
To sell at a particular farmers’ market for a year,
there is a $100 membership fee. Then you pay $3
for each hour that you sell at the market. However,
if you were a member the previous year, the
membership fee is reduced to $50.
• The red line shows the
total cost if you are a
new member.
• The blue line shows the
total cost if you are a
returning member.
Holt Algebra 1
5-8
Slopes of Parallel and
Perpendicular Lines
These two lines are parallel.
Parallel lines are lines in
the same plane that have
no points in common. In
other words, they do not
intersect.
Holt Algebra 1
5-8
Slopes of Parallel and
Perpendicular Lines
Holt Algebra 1
5-8
Slopes of Parallel and
Perpendicular Lines
Example 1A: Identifying Parallel Lines
Identify which lines are parallel.
The lines described by
and
both have slope
.
These lines are parallel. The lines
described by y = x and y = x + 1
both have slope 1. These lines
are parallel.
Holt Algebra 1
5-8
Slopes of Parallel and
Perpendicular Lines
Check It Out! Example 1b
Identify which lines are parallel.
Write all equations in slope-intercept form to
determine the slope.
y = 3x
Slope-intercept form
Holt Algebra 1
Slope-intercept form
5-8
Slopes of Parallel and
Perpendicular Lines
Check It Out! Example 1b Continued
Identify which lines are parallel.
Write all equations in slope-intercept form to
determine the slope.
–3x + 4y = 32
+3x
+3x
4y = 3x + 32
y – 1 = 3(x + 2)
y – 1 = 3x + 6
+1
+1
y = 3x + 7
Holt Algebra 1
5-8
Slopes of Parallel and
Perpendicular Lines
Check It Out! Example 1b Continued
The lines described by
–3x + 4y = 32 and y =
–3x + 4y = 32
+8
have the same slope, but they
y = 3x
are not parallel lines. They are
the same line.
The lines described by
y = 3x and y – 1 = 3(x + 2)
represent parallel lines. They
each have slope 3.
Holt Algebra 1
y – 1 = 3(x + 2)
5-8
Slopes of Parallel and
Perpendicular Lines
Perpendicular lines are lines that intersect to
form right angles (90°).
Holt Algebra 1
5-8
Slopes of Parallel and
Perpendicular Lines
Holt Algebra 1
5-8
Slopes of Parallel and
Perpendicular Lines
Example 3: Identifying Perpendicular Lines
Identify which lines are perpendicular: y = 3;
x = –2; y = 3x;
.
The graph described by y = 3
is a horizontal line, and the
graph described by x = –2 is
a vertical line. These lines are
perpendicular.
The slope of the line described
by y = 3x is 3. The slope of the
line described by
is
Holt Algebra 1
.
x = –2
y =3x
y=3
5-8
Slopes of Parallel and
Perpendicular Lines
Example 3 Continued
Identify which lines are perpendicular: y = 3;
x = –2; y = 3x;
.
x = –2
These lines are
perpendicular because
the product of their
slopes is –1.
Holt Algebra 1
y =3x
y=3
5-8
Slopes of Parallel and
Perpendicular Lines
Example 4: Geometry Application
Show that ABC is a right triangle.
If ABC is a right triangle, AB
will be perpendicular to AC.
slope of
slope of
AB is perpendicular to AC
because
Therefore, ABC is a right triangle because it
contains a right angle.
Holt Algebra 1
5-8
Slopes of Parallel and
Perpendicular Lines
Example 5A: Writing Equations of Parallel and
Perpendicular Lines
Write an equation in slope-intercept form for
the line that passes through (4, 10) and is
parallel to the line described by y = 3x + 8.
Step 1 Find the slope of the line.
The slope is 3.
y = 3x + 8
The parallel line also has a slope of 3.
Step 2 Write the equation in point-slope form.
y – y1 = m(x – x1)
Use the point-slope form.
y – 10 = 3(x – 4)
Substitute 3 for m, 4
for x1, and 10 for y1.
Holt Algebra 1
5-8
Slopes of Parallel and
Perpendicular Lines
Example 5A Continued
Write an equation in slope-intercept form for
the line that passes through (4, 10) and is
parallel to the line described by y = 3x + 8.
Step 3 Write the equation in slope-intercept form.
y – 10 = 3(x – 4)
y – 10 = 3x – 12
y = 3x – 2
Holt Algebra 1
Distribute 3 on the right side.
Add 10 to both sides.
5-8
Slopes of Parallel and
Perpendicular Lines
Helpful Hint
If you know the slope of a line, the
slope of a perpendicular line will be
the "opposite reciprocal.”
Holt Algebra 1
5-8
Slopes of Parallel and
Perpendicular Lines
Check It Out! Example 5a
Write an equation in slope-intercept form for
the line that passes through (5, 7) and is
parallel to the line described by y = x – 6.
Step 1 Find the slope of the line.
y=
x –6
The slope is
The parallel line also has a slope of
.
.
Step 2 Write the equation in point-slope form.
y – y1 = m(x – x1)
Holt Algebra 1
Use the point-slope form.
5-8
Slopes of Parallel and
Perpendicular Lines
Check It Out! Example 5a Continued
Write an equation in slope-intercept form for
the line that passes through (5, 7) and is
parallel to the line described by y = x – 6.
Step 3 Write the equation in slope-intercept form.
Distribute
on the right side.
Add 7 to both sides.
Holt Algebra 1
5-10 Transforming Linear Functions
Warm-Up
Write an equation in slope-intercept form
for the line described.
1. contains the point (8, –12) and is parallel to
2. contains the point (4, –3) and is perpendicular
to y = 4x + 5
3. Show that WXYZ is a rectangle.
Holt McDougal Algebra 1
5-10 Transforming Linear Functions
Objective
Describe how changing slope and y-intercept
affect the graph of a linear function.
Holt McDougal Algebra 1
5-10 Transforming Linear Functions
A family of functions is a set of functions whose
graphs have basic characteristics in common. For
example, all linear functions form a family because
all of their graphs are the same basic shape.
A parent function is the most basic function in a
family. For linear functions, the parent function is
f(x) = x.
The graphs of all other linear functions are
transformations of the graph of the parent
function, f(x) = x. A transformation is a change
in position or size of a figure.
Holt McDougal Algebra 1
5-10 Transforming Linear Functions
There are three types of transformations–
translations, rotations, and reflections.
Look at the four functions and their graphs below.
Holt McDougal Algebra 1
5-10 Transforming Linear Functions
Notice that all of the lines are parallel. The
slopes are the same but the y-intercepts are
different.
Holt McDougal Algebra 1
5-10 Transforming Linear Functions
The graphs of g(x) = x + 3, h(x) = x – 2, and
k(x) = x – 4, are vertical translations of the graph
of the parent function, f(x) = x. A translation is a
type of transformation that moves every point the
same distance in the same direction. You can think
of a translation as a “slide.”
Holt McDougal Algebra 1
5-10 Transforming Linear Functions
Holt McDougal Algebra 1
5-10 Transforming Linear Functions
Example 1: Translating Linear Functions
Graph f(x) = 2x and g(x) = 2x – 6. Then
describe the transformation from the graph of
f(x) to the graph of g(x).
f(x) = 2x
g(x) = 2x – 6
The graph of g(x) = 2x – 6 is the result of
translating the graph of f(x) = 2x 6 units down.
Holt McDougal Algebra 1
5-10 Transforming Linear Functions
Check It Out! Example 1
Graph f(x) = x + 4 and g(x) = x – 2. Then
describe the transformation from the graph of
f(x) to the graph of g(x).
f(x) = x + 4
g(x) = x – 2
The graph of g(x) = x – 2 is the result of translating
the graph of f(x) = x + 4 6 units down.
Holt McDougal Algebra 1
5-10 Transforming Linear Functions
The graphs of g(x) = 3x,
h(x) = 5x, and k(x) =
are rotations of the graph
f(x) = x. A rotation is a
transformation about a
point. You can think of a
rotation as a “turn.” The
y-intercepts are the
same, but the slopes are
different.
Holt McDougal Algebra 1
5-10 Transforming Linear Functions
Holt McDougal Algebra 1
5-10 Transforming Linear Functions
Example 2: Rotating Linear Functions
Graph f(x) = x and g(x) = 5x. Then describe the
transformation from the graph of f(x) to the
graph of g(x).
f(x) = x
g(x) = 5x
The graph of g(x) = 5x is the result of rotating the
graph of f(x) = x about (0, 0). The graph of g(x) is
steeper than the graph of f(x).
Holt McDougal Algebra 1
5-10 Transforming Linear Functions
Check It Out! Example 2
Graph f(x) = 3x – 1 and g(x) = x – 1. Then
describe the transformation from the graph of
f(x) to the graph of g(x).
f(x) = 3x – 1
g(x) =
x–1
The graph of g(x) is the result of rotating the graph
of f(x) about (0, –1). The graph of g(x) is less steep
than the graph of f(x).
Holt McDougal Algebra 1
5-10 Transforming Linear Functions
The diagram shows the
reflection of the graph of
f(x) = 2x across the y-axis,
producing the graph of
g(x) = –2x. A reflection is
a transformation across a
line that produces a mirror
image. You can think of a
reflection as a “flip” over a
line.
Holt McDougal Algebra 1
5-10 Transforming Linear Functions
Holt McDougal Algebra 1
5-10 Transforming Linear Functions
Example 3: Reflecting Linear Functions
Graph f(x) = 2x + 2. Then reflect the graph of
f(x) across the y-axis. Write a function g(x) to
describe the new graph.
f(x) = 2x + 2
g(x)
f(x)
To find g(x), multiply the value of m by –1.
In f(x) = 2x + 2, m = 2.
2(–1) = –2
This is the value of m for g(x).
g(x) = –2x + 2
Holt McDougal Algebra 1
5-10 Transforming Linear Functions
Example 5: Business Application
A florist charges $25 for a vase plus $4.50 for
each flower. The total charge for the vase and
flowers is given by the function f(x) = 4.50x +
25. How will the graph change if the vase’s cost
is raised to $35? if the charge per flower is
Total Cost
lowered to $3.00?
f(x) = 4.50x + 25 is graphed
in blue.
If the vase’s price is raised to
$35, the new function is
f(g) = 4.50x + 35. The
original graph will be
translated 10 units up.
Holt McDougal Algebra 1
5-10 Transforming Linear Functions
Example 5 Continued
A florist charges $25 for a vase plus $4.50 for
each flower. The total charge for the vase and
flowers is given by the function f(x) = 4.50x +
25. How will the graph change if the vase’s cost
is raised to $35? If the charge per flower is
Total Cost
lowered to $3.00?
If the charge per flower is
lowered to $3.00. The new
function is h(x) = 3.00x + 25.
The original graph will be
rotated clockwise about
(0, 25) and become less
steep.
Holt McDougal Algebra 1
5-10 Transforming Linear Functions
5 Minute Warm-Up
Directions: Write an equation in slope intercept
form of the line that passes through the points.
1. (5, 32) and (7, 16)
Directions: Which of the lines are perpendicular?
2. line p: y = 1 x + 2 line q: y = 5x + ½ line r: y = -5x +3
5
Directions: Find the distance and midpoint of the two
points.
3. (-3, 8) and (-10, 5)
Holt McDougal Algebra 1
5-10 Transforming Linear Functions
Lesson Quiz: Part I
Describe the transformation from the graph
of f(x) to the graph of g(x).
1. f(x) = 4x, g(x) = x
rotated about (0, 0) (less steep)
2. f(x) = x – 1, g(x) = x + 6
translated 7 units up
3. f(x) = x, g(x) = 2x
rotated about (0, 0) (steeper)
4. f(x) = 5x, g(x) = –5x
reflected across the y-axis
Holt McDougal Algebra 1
5-10 Transforming Linear Functions
Lesson Quiz: Part II
5. f(x) = x, g(x) = x – 4
translated 4 units down
6. f(x) = –3x, g(x) = –x + 1
rotated about (0, 0) (less steep),
translated 1 unit up
7. A cashier gets a $50 bonus for working on a
holiday plus $9/h. The total holiday salary is given
by the function f(x) = 9x + 50. How will the graph
change if the bonus is raised to $75? if the hourly
rate is raised to $12/h?
translate 25 units up; rotated about (0, 50)
(steeper)
Holt McDougal Algebra 1