Chapter 5 - Linear Functions
Algebra I
Table of Contents
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•
•
•
•
5.1 - Identifying Linear Functions
5.2 - Using Intercepts
5.3 - Rate of Change and Slope
5.4 - The Slope Formula
5.5 - The Distance and Midpoint Formulas
5.1 - Identifying Linear Functions
Algebra I
5.1
Algebra 1 (bell work)
The graph below 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.
5.1
Example 1
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?
linear function
not a function
not a linear function
5.1
Example 2
Identifying a Linear Function by Using Ordered Pairs
Does this group of points represent a linear equation?
Don’t Copy
5.1
Tell whether the set of ordered pairs satisfies a linear function. Explain.
{(0, –3), (4, 0), (8, 3), (12, 6), (16, 9)}
Copy in Notes
+4
+4
+4
+4
x
y
0
–3
4
0
8
3
12
6
16
9
+3
+3
+3
+3
These points satisfy a linear function.
5.1 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.
5.1
Math Joke
• Q: How are linear functions similar to Cupid?
• A: They have a partner for every number
5.1
Example 3
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
•
x = 2y + 4
The equation can be written in standard
form, so the function is linear.
y
0
–1
x = 2y + 4
x = 2(0) + 4 = 4
x = 2(–1) + 4 = 2
(x, y)
(4, 0)
(2, –1)
–2
x = 2(–2) + 4 = 0
(0, –2)
5.1
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.
y = 5x – 9
y = 5x – 9
–5x –5x
–5x + y = – 9
y = 5x – 9
The equation can be written in standard form, so
the function is linear.
x
0
y = 5x – 9
(x, y)
y = 5(0) – 9 = –9 (0, –9)
1
2
y = 5(1) – 9 = –4 (1, –4)
y = 5(2) – 9 = 1 (2, 1)
•
•
•
5.1
Example 4
Career Application
The relationship between human years and dog years is given by the function y = 7x,
where x is the number of human years.
Graph this function and give its domain and range.
x
f(x) = 7x
1
f(1) = 7(1) = 7
2
f(2) = 7(2) = 14
3
f(3) = 7(3) = 21
The number of human years must be positive,
so the domain is {x ≥ 0} and the range is {y ≥ 0}.
(3,21) •
• (2,14)
• (1,7)
HW pg. 304
• 5.1– 15-24, 13, 14, 30, 32, 42, 53, 64-71
– Ch: 50, 54, 55, 56, 60
Section 5.2 - Using Intercepts
Algebra I
5.2
Algebra 1 (bell work)
The y-intercept is the y-coordinate of the point where
the graph intersects the y-axis.
The x-coordinate of this point is always 0.
The x-intercept is the x-coordinate of the point where
the graph intersects the x-axis.
The y-coordinate of this point is always 0.
5.2
Example 1
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.
5.2
Find the x- and y-intercepts.
5x – 2y = 10
To find the x-intercept, replace y with 0 and
solve for x.
5x – 2y = 10
5x – 2(0) = 10
5x – 0 = 10
5x = 10
x=2
The x-intercept is 2.
To find the y-intercept, replace x with 0 and
solve for y.
5x – 2y = 10
5(0) – 2y = 10
0 – 2y = 10
– 2y = 10
y = –5
The y-intercept is –5.
5.2
Find the x- and y-intercepts.
–3x + 5y = 30
To find the x-intercept, replace y with 0 and
solve for x.
–3x + 5y = 30
To find the y-intercept, replace x with 0 and
solve for y.
–3x + 5y = 30
–3x + 5(0) = 30
–3(0) + 5y = 30
–3x – 0 = 30
0 + 5y = 30
–3x = 30
5y = 30
x = –10
y=6
The x-intercept is –10.
The y-intercept is 6.
5.2
Example 2
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?
x
f(x) = 200 – 8x
0
200
5
160
10
120
20
40
25
0
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.
Math Joke
• Teacher: Where’s the graph of your function?
• Student: It was intercepted on the way to
school
5.2
Example 3
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
y = –3
5.2
Use intercepts to graph the line described by the equation.
3x – 7y = 21
Step 2 Graph the line.
x
5.2
Use intercepts to graph the line described by the equation.
y = –x + 4
Step 1 Write the equation in standard form.
y = –x + 4
+x +x
x+y=4
x+y=4
Step 2 Find the intercepts.
x-intercept:
y-intercept:
x+y=4
x+y=4
x+0=4
0+y=4
x=4
y=4
5.2
Use intercepts to graph the line described by the equation.
x+y=4
Step 3 Graph the line.
HW pg. 310
• 5.2– 13-23, 30, 33, 48-53
– Ch: 32, 38, 39
5.3 - Rate of Change and Slope
Algebra I
5.3
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.
5.3 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
independent: month
5.3
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.
5.3 Graph the data from Example 1 and show the rates of change.
5.3
Math Joke
• Q: Why were the math students getting up
and sprinting around the classroom?
• A: Their teacher kept saying rise and run.
5.3
Find the slope of the line.
Run –9
(–6, 5)
•
Rise 3
Rise –3
•
Run 9
(3, 2)
5.3
Find the slope of the line that contains (0, –3) and (5, –5).
Run –5
•
Rise –2
Rise 2
•
Run 5
5.3
Find the slope of each line.
A.
B.
You cannot
divide by 0
The slope is undefined.
The slope is 0.
5.3
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.
5.3
Tell whether the slope of each line is positive, negative, zero or undefined.
A.
B.
The line rises from left to right.
The line falls from left to right.
The slope is positive.
The slope is negative.
5.3
Tell whether the slope of each line is positive, negative, zero or undefined.
a.
b.
The line is vertical.
The line rises from left to right.
The slope is undefined.
The slope is positive.
5.3
HW pg. 318
• 5-3
– 2-11, 14-18, 21-23, 25, 37, 38
– E.C. pg 282 Multi Step Test Prep
– (Due Friday Nov 15th) *Must Show Work*
5.4- The Slope Formula
Algebra I
5.4
5.4
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
.
5.4
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.
5.4
The graph shows a linear relationship. Find the slope.
Let (0, 2) be (x1, y1) and (–2, –2) be (x2, y2).
5.4
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
5.4
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.
5.4
Math Joke
• A baker told his assistant to halve the filling
recipe for a cake.
• When the baker asked for the filing, the
assistant replied, “But I ate it!”
• “why did you do that?” demanded the baker
• “You told me to have it!”
5.4
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.
A slope of 6 mean the cost of
running the refrigerator is a rate
of 6 dollars per month.
So slope represents
in units of
5.4
The graph shows the height of a plant over a
period of days.
Find the slope of the line. Then tell what the
slope represents.
Step 1 Use the slope formula.
A slope of
mean the plant grows at rate of 1
centimeter every two days.
So slope represents
.
in units of
HW pg. 327
• 1-10, 16, 17, 21, 22, 24, 25, 38, 42
5.5 – The Midpoint and Distance
Formula
Algebra I
5.5
Midpoint formula pg. 330
5.5
Finding the coordinates of a Midpoint
5.5
Finding the coordinates of an Endpoint
Math Joke
• Q: What’s the midpoint of an algebra test?
• A: The point where fear of not having enough
time to finish overcomes the fear of not
having studied enough
5.5
The Distance Formula pg. 331
5.5
Finding Distance in a coordinate plane
5.5
Geography Application
HW pg. 333
• 5.5– 2-10, 20-22, 28, 44-47 B: 34