Ch1: Graphs
y axis
Quadrant II (-, +) Quadrant I (+, +)
y - $$ in thousands
Origin (0, 0)
-6
-2
(-6,-3)
Quadrant III (-, -)
(6,0)
x axis
2 4 6
(5,-2)
y
intercept
(0,-3)
x
Yrs
x
intercept
Quadrant IV (+, -)
When distinct points are plotted as above
the graph is called a scatter plot – ‘points
that are scattered about’
A point in the x/y coordinate plane is
described by an ordered pair of
coordinates (x, y)
Graphs represent trends in data.
For example:
x – number of years in business
y – thousands of dollars of profit
Equation : y = ½ x – 3
1.1 Distance & Midpoint
y
Things to know:
1. Find distance or midpoint given 2 points
2. Given midpoint and 1 point, find the other point
Origin (0, 0)
-6
A point in the x/y coordinate plane is
described by an ordered pair of
coordinates (x, y)
The Distance Formula
To find the distance between 2 points
(x1, y1) and (x2, y2)
d =  (x2 – x1)2 + (y2 – y1)2
-2
x
2 4 6
(-6,-3)
(5,-2)
The Midpoint Formula
To find the coordinates of the midpoint (M)
of a segment given segment endpoints of
(x1, y1) and (x2, y2)
M
x 1 + x2, y 1 + y2
2
2
1.1 & 1.2 Linear Equations
The graph of a linear equation is a line.
A linear function is of the form y = mx + b, where m and b are constants.
y
y = 3x + 2
y = 3x + 5x
y = -2x –3
x
y = (2/3)x -1
y=4
6x + 3y = 12
All of these equations are linear.
Three of them are graphed above.
x y=3x+2
x y=2/3x –1
0 2
0
-1
1 5
3
1
X and Y intercepts
Equation: y = ½ x – 3
y
(6,0)
y
intercept
(0,-3)
x
x
intercept
-3
The y intercept happens where y is something & x = 0: (0, ____)
Let x = 0 and solve for y: y = ½ (0) – 3 = -3
6 0)
The x intercept happens where x is something & y = 0: (____,
Let y = 0 and solve for x: 0 = ½ x – 3 => 3 = ½ x
=> x = 6
Slope
Slope is the ratio of
RISE (How High)
=
RUN (How Far)
y
y 2 – y1
y (Change in y)
x2 – x1
x (Change in x)
Slope = 5 – 2 = 3
1-0
Slope = 1 – (-1) = 2
3–0
3
Things to know:
1. Find slope from graph
2. Find a point using slope
3. Find slope using 2 points
4. Understand slope between
2 points is always the same
on the same line
x y = mx + b
m = slope
b = y intercept
y=2/3x –1
x y=3x+2
x
0 2
0
-1
1 5
3
1
The Possibilities for a Line’s Slope (m)
Negative Slope
Positive Slope
y
Zero Slope
y
m>0
Line rises from left to right.
Example:
y=½x+2
y
m<0
x
Undefined Slope
y
m is
undefined
m=0
x
x
Line falls from left to right.
Line is horizontal.
Example:
y = -½ x + 1
Example:
y=2
x
Line is vertical.
Example:
x=3
Question: If 2 lines are parallel do you know anything about their slopes?
Things to know:
1. Identify the type of slope given a graph.
2. Given a slope, understand what the graph would look like and draw it.
3. Find the equation of a horizontal or vertical line given a graph.
4. Graph a horizontal or vertical line given an equation
5. Estimate the point of the y-intercept or x-intercept from a graph.
Linear Equation Forms (2 Vars)
Standard Form
Example: 6x + 3y = 12
Slope Intercept Form
Example: y = - ½ x - 2
Point Slope Form
Ax + By = C
A, B, C are real numbers.
A & B are not both 0.
Things to know:
1. Graph using x/y chart
2. Know this makes a line graph.
y = mx + b
m is the slope
b is the y intercept
Things to know:
1. Find Slope & y-intercept
2. Graph using slope & y-intercept
3. Application meaning of of slope & intercepts
y – y1 = m(x – x1)
Example: Write the linear equation through point P(-1, 4) with slope 3
y – y1 = m(x – x1)
Things to know:
y – 4 = 3(x - - 1)
1. Change from point slope to/from other forms.
y – 4 = 3(x + 1)
2. Find the x or y-intercept of any linear equation
Parallel and Perpendicular
Lines & Slopes
Things to know:
1. Identify parallel/non-parallel lines.
PARALLEL
• Vertical lines are parallel
• Non-vertical lines are parallel if and only if they have the same slope
y=¾x+2
y = ¾ x -8
PERPENDICULAR
Same Slope
Things to know:
1. Identify (non) perpendicular lines.
2. Find the equation of a line parallel or
perpendicular to another line through
a point or through a y-intercept.
•Any horizontal line and vertical line are perpendicular
• If the slopes of 2 lines have a product of –1 and/or
are negative reciprocals of each other then the lines are perpendicular.
y= ¾ x+2
Negative reciprocal slopes
y = - 4/3 x - 5
3 • -4 = -12 = -1
4 3
12
Product
is -1
Practice Problems
1.
Find the slope of a line passing through (-1, 2) and (3, 8)
2.
Graph the line passing through (1, 2) with slope of - ½
3.
Is the point (2, -1) on the line specified by: y = -2(x-1) + 3 ?
4.
Parallel, Perpendicular or Neither? 3y = 9x + 3 and 6y + 2x = 6
5.
Find the equation of a line parallel to y = 4x + 2 through the point (-1,5)
6.
Find the equation of a line perpendicular to y = - ¾ x –8 through point (2, 7)
7.
Find the equation of a line passing through the points (-2, 1) and (3, 7)
8. Graph (using an x/y chart – plotting points) and find intercepts of
any equation such as: y = 2x + 5 or y = x2 – 4
Symmetry and Odd/Even Functions
Y-Axis Symmetry  even functions f (-x) = f (x)
For every point (x,y), the point (-x, y) is also on the graph.
Test for symmetry: Replace x by –x in equation. Check for equivalent equation.
Origin Symmetry  odd functions f (-x) = -f (x)
For every point (x, y), the point (-x, -y) is also on the graph.
Test for symmetry: Replace x by –x , y by –y in equation. Check for equivalent equation.
X-Axis Symmetry
(For every point (x, y), the point (x, -y) is also on the graph.)
Test for symmetry: Replace y by –y in equation. Check for equivalent equation.
y = x3
Origin
Symmetry
Symmetry
Test
-y = (-x)3
-y = -x3
y = x3
y = x2
ODD)
Y-axis
Symmetry
(EVEN)
Symmetry
Test
y = (-x)2
y = x2
x = y2
X-axis
Symmetry
Symmetry
Test
x = (-y)2
X = y2
Try these without
Using a graph:
y = 3x2 – 2
y = x2 + 2x + 1
A Rational Function Graph & Symmetry
y= 1
x
x
-2
-1
-1/2
0
½
1
2
y
-1/2
-1
-2
Undefined
2
1
½
Intercepts:
No intercepts exist
If y = 0, there is no solution for x.
If x = 0, y is undefined
The line x = 0 is called a vertical asymptote.
The line y = 0 is called a horizontal asymptote.
Symmetry:
y = 1/-x => No y-axis symmetry
-y = 1/-x => y = 1/x => origin symmetry
-y = 1/x => y = -1/x => no x-axis symmetry
1.3 Functions and Graphs
Year
1997
1998 1999 2000
$3111 $3247 $3356 $3510
Cost
The cost depends on the year.
independent variable (x)
dependent variable (y)
The table above establishes a relation between the year and the cost of tuition
at a public college. For each year there is a cost, forming a set of ordered pairs.
A relation is a set of ordered pairs (x, y). The relation above can be written
as 4 ordered pairs as follows:
S = {(1997, 3111), (1998, 3247), (1999, 3356), (2000, 3510)}
x
y
x
y
x
y
x
y
Domain – the set of all x-values. D = {1977, 1998, 1999, 2000}
Range – the set of all y-values. R = {3111, 3247, 3356, 3510}
Thinking Exercise: Draw a ‘line’ in the x/y axes.
What is the Domain & Range?
Year(x)
1997
1998
1999
2000
Cost(y)
3111
3247
3356
3510
Input
x
Functions & Linear Data Modeling
y – Profit in thousands of $$
(Dependent Var)
Function
f
(6,0)
Output
y=f(x)
y
intercept
(0,-3)
x - Years in business
(Independent Var)
x
intercept
Equation: y = ½ x – 3
Function: f(x) = ½ x – 3
A function has exactly one output value (y)
for each valid input (x).
x
0
2
6
8
Use the vertical line test to see if an equation is
a function.
•If it touches 1 point at a time then FUNCTION
•If it touches more than 1 point at a time then
NOT A FUNCTION.
y = f(x)
-3 f(0) = ½(0)-3=-3
-2 f(2) = ½(2)-3=-2
0 f(6) = ½(6)-3=0
1 f(8) = ½(8)-3=1
Diagrams of Functions
A function is a correspondence fro the domain to the range such that each element
in the domain corresponds to exactly one element in the range.
f
Function: f(x) = ½ x – 3
x
0
2
6
8
y = f(x)
-3 f(0) = ½(0)-3=-3
-2 f(2) = ½(2)-3=-2
0 f(6) = ½(6)-3=0
1 f(8) = ½(8)-3=1
0
2
-3
-2
0
1
6
8
f
1
2
4
4
5
5
6
3
A function
NOT a function
How to Determine if an equation is a
function
Graphically: Use the vertical line test
Symbolically/Algebraically: Solve for y to see if there is only 1 y-value.
Example 1: x2 + y = 4
Example 2: x2 + y2 = 4
y = 4 – x2
y2 = 4 – x2
For every value of x there
Is exactly 1 value for y, so
This equation IS A FUNCTION.
y = 4 – x2 or y = -
4 – x2
For every value of x there
are 2 possible values for y, so
This equation IS NOT A
FUNCTION.
Are these graphs functions?
Use the vertical line test to tell if the following are functions:
y = x2
y = x3
Origin
Symmetry
Y-axis
Symmetry
x = y2
X-axis
Symmetry
More on Evaluation of Functions
f(x) = x2 + 3x + 5
Evaluate: f(2)
f(2) = (2)2 + 3(2) + 5
f(2) = 4 + 6 + 5
f(2) = 15
Evaluate: f(x + 3)
f(x + 3) = (x + 3)2 + 3 (x + 3) + 5
f(x + 3) = (x + 3)(x + 3) + 3x + 9 + 5
f(x + 3) = (x2 + 3x + 3x + 9) + 3x + 14
f(x + 3) = (x2 + 6x + 9) + 3x + 14
f(x + 3) = x2 + 9x + 23
Evaluate: f(-x)
f (-x) = ( -x)2 + 3( -x) + 5
f (-x) = x2 - 3x + 5
More on Domain of Functions
A function’s domain is the largest set of real numbers for which the value f(x)
is a real number. So, a function’s domain is the set of all real numbers
MINUS the following conditions:
• specific conditions/restrictions placed on the function
• Bounds relating to real-life data modeling
(Example: y = 7x, where y is dog years and x is dog’s age)
• values that cause division by zero
• values that result in an even root of a negative number
What is the domain the following functions:
1.
f(x) = 6x
x2 – 9
2. g(x) =
3x + 12
3. h(x) = 2x + 1
Slope & Average Rate of Change
y - $$ in thousands
y=½x–3
(6,0)
y = x2 - 4x + 4
x
Yrs
(0,-3)
The slope of a line may be
interpreted as the rate of change.
The rate of change for a line is
constant (the same for any 2 points)
y2 – y1
x2 – x1
Non-linear equations do not have a
constant rate of change. But you can
Find the average rate of change from
x1 to x2 along a secant to the graph.
f(x2) – f(x1)
x2 – x1
See Page 38-39 for more examples.
Definition of a Difference Quotient
The average rate of change for f(x) is called the “difference quotient”
and is defined below. (This is an important concept in calculus – it
becomes the mathematical definition of the derivative you will learn
about next semester. f (x + h) - f (x)
h
Example: Find the difference quotient for : f(x) = 2x2 -3
f(x + h) = 2(x + h )2 - 3
= 2(x + h)(x + h) -3
= 2(x2 + 2xh + h2 ) -3
= 2x2 + 4xh + 2h2 -3
=
=
=
So, the difference quotient is:
2x2 + 4xh + 2h2 -3 – (2x2 -3)
h
2x2 + 4xh + 2h2 -3 – 2x2 + 3
h
4xh + 2h2
h
4x + 2h
1.4 Increasing, Decreasing, and
Constant Functions
A function is increasing on an interval if for any x1, and x2 in the
interval, where x1 < x2, then f (x1) < f (x2).
A function is decreasing on an interval if for any x1, and x2 in the
interval, where x1 < x2, then f (x1) > f (x2).
A function is constant on an interval if for any x1, and x2 in the interval,
where x1 < x2, then f (x1) = f (x2).
(x2, f (x2))
(x1, f (x1))
(x2, f (x2))
(x1, f (x1))
(x1, f (x1))
(x2, f (x2))
Increasing
f (x1) < f (x2)
Decreasing
f (x1) > f (x2)
Constant
f (x1) = f (x2)
More Examples
a.
b.
5
5
4
4
3
3
2
1
1
-5 -4 -3 -2
-1
-1
-2
1
2
3 4
5
-5 -4 -3 -2
-1
-1
-2
-3
-3
-4
-5
-4
-5
Observations
• Decreasing on the
interval (-oo, 0)
1
2
3 4
5
Observations
a. Two pieces (a piecewise function)
b. Constant on the interval (-oo, 0).
•
•
Increasing on the
interval (0, 2)
Decreasing on the
interval (2, oo).
c. Increasing on the interval (0, oo).
Challenge Yourself: What might be
the definition of the piecewise function
for this graph? (You will learn about these
Later. Can you guess what it might be?)
f(x) = sin (x) x
0
/2
The point at which a function changes its increasing or decreasing

behavior is called a relative minimum or relative maximum.
3/2
y
2
2
Relative (local) Min & Max
(90, f(90))
y
0
1
0
-1
0
f(90), or 1, is a local max
1
x
0
90
180
270
360
-1
-2
A function value f(a) is a relative
maximum of f if there exists an open
interval about a such that f(a) > f(x) for
all x in the open interval.
(270, f(270))
f(270), or -1, is a local min
A function value f(b) is a relative
minimum of f if there exists an open
interval about b such that f(b) < f(x) for
all x in the open interval.
1.4 Library of Functions/Common Graphs
y=c
y=x
y = x2
x
x
y = x3
y=
x
y = |x|
x
y = 1/x
x
x
y = x1/3
x
x
Step Function Application Example
y = int(x) or
f(x) = int(x)
y = [[x]]
(Greatest Integer Function)
y – Tax (+) or Refund (-) in thousands of $$
x – Income in $10,000’s
Find:
1) f (1.06)
2) f (1/3)
3) f (-2.3)
•
What other applications of the
step function can you think of?
Piecewise Functions
A function that is defined by two (or more) equations over
a specified domain is called a piecewise function.
f(x) =
x2 + 3
5x + 3
if x < 0
if x>=0
f(-5) = (-5)2 + 3 = 25 + 3 = 28
f(6) = 5(6) + 3 = 33
See Page 247 for more examples
1.5 Transformation of Functions
A transformation of a graph is a change in its position, shape or size.
Example function: y = x2
For a given function, y = f(x)
10
8
6
y = f(x) +c [shift up c]
y = f(x) – c [shift down c]
4
2
-10 -8
y = f(x + c) [shift left c]
y = f(x – c) [shift right c]
-6
-4
-2
2
4
6
8 10
-2
-4
-6
-8
y = -f(x) [flip over x-axis]
y = f(-x) [flip over y-axis]
y = cf(x) [multiply y value by c]
[if c > 1, stretch vertically]
[if 0 < c < 1, shrink vertically]
-10
Graph: y = x2 + 4
y = x2 - 4
y = (x+4)2
y = (x – 4)2
y = -x2
y = (-x)2
y = ½ x2
Can you graph : y = ½ (x + 2)3 + 2
More Transformation Practice
Suppose that the x-intercepts of the graph of y = f(x) are -5 and 3
(a)What are the x-intercepts of the graph of y = f(x + 2)
(b)What are the x-intercepts of the graph of y = f(x – 2)
(c)What are the x-intercepts of the graph y = 4f(x)
(d)What are the x-intercepts of the graph of y = f(-x)
1.6 Sum, Difference, Product, and
Quotient of Functions
Let f and g be two functions. The sum of f + g, the difference f – g,
the product fg, and the quotient f /g are functions whose domains are the set of
all real numbers common to the domains of f and g, defined as follows:
Sum:
Difference:
Product:
Quotient:
(f + g)(x) = f (x)+g(x)
(f – g)(x) = f (x) – g(x)
(f • g)(x) = f (x) • g(x)
(f / g)(x) = f (x)/g(x), provided g(x) does not equal 0
Example: Let f(x) = 2x+1 and g(x) = x2-2.
f+g = 2x+1 + x2-2 = x2+2x-1
f-g = (2x+1) - (x2-2)= -x2+2x+3
fg = (2x+1)(x2-2) = 2x3+x2-4x-2
f/g = (2x+1)/(x2-2)
Adding & Subtracting Functions
If f(x) and g(x) are functions, then:
(f + g)(x) = f(x) + g(x)
(f – g)(x) = f(x) – g(x)
Examples: f(x) = 2x + 1 and g(x) = -3x – 7
Method1
Method1
(f + g)(4) = 2(4) + 1 + -3(4) – 7
(f – g)(6) = 2(6) + 1 – [-3(6) – 7]
= 8+1
+ -12 – 7
= 12 + 1 - [-18 – 7]
= 9
+ -19
= 13 - [-25]
=
-10
= 13 + 25
Method2
Method2 =
38
(f + g)(4) = 2x + 1 + -3x – 7
(f - g)(6) = 2x + 1 - [-3x – 7]
=
-x – 6
= 2x + 1 + 3x + 7
=
-4 – 6
= 5x + 8
= - 10
= 5(6) + 8
= 30 + 8
=
38
Adding/subtracting also extends to non-linear
functions you will see in a subsequent chapter.
The Composition of Functions
f o g - composition of the function f with g is is defined by the equation
(f o g)(x) = f (g(x)).
f (x) = 3x – 4
and
g(x) = x2 + 6
(f o g)(x) = f (g(x)) = 3g(x) – 4
= 3(x2 + 6) – 4
= 3x2 + 18 – 4
= 3x2 + 14
(g o f)(x) = g(f (x)) = (f (x))2 + 6
= (3x – 4)2 + 6
= 9x2 – 24x + 16 + 6
= 9x2 – 24x + 22
1.7 Inverse Function
If f (g(x)) = x for every x in the domain of g
g(f (x)) = x for every x in the domain of f.
and
Then the function g is the inverse of the function f denoted by f -1 and
the function f is the inverse of the function g denoted by g -1
The domain of f is equal to the range of f -1, and vice versa. (x,y) in f => (y, x) in f--1
Examples: Verifying inverses (Are f & g inverses?)
f (x) = 5x and
g(x) = x/5.
f(x)= 3x + 2 g(x) = x - 2
3
f (g(x)) = 5(g(x)) = 5 x = x
f (g(x)) = 3(g(x))+2
5
= 3 x-2 + 2
g( f (x)) = f(x) = 5x = x
3
5
5
=x–2+2=x
f (g(x)) = x and g( f (x)) = x
Thus they are inverses.
f(x) = 5x f-1(x)=x/5
g(f(x) = f(x) – 2 = 3x + 2 – 2 = x
3
3
Thus they are inverses
f(x ) = 3x + 2 f-1(x) = (x-2)/3
How to Find the Inverse of a Function
Example: Find the inverse of f (x) = 7x – 5.
Step 1: Replace f (x) by y
: y = 7x – 5
Step 2: Interchange x and y
: x = 7y – 5
.
: x + 5 = 7y
Step 3: Solve for y.
x+5=y
7
Step 4
Replace y by f -1(x).
f -1(x) =
x+5
7
The Horizontal Line Test For
Inverse Functions
A function f has an inverse that is a function, f –1, if there is no horizontal line
that intersects the graph of the function f at more than one point
10
y
8
6
4
2
-10 -8 -6 -4 -2
2
4
6
8 10
-2
x
-4
-6
-8
-10
f(x) = x2+3x-1
NO Inverse Function
f(x) = x + 4
YES this has an inverse Function
Graphing the Inverse of a Function
y
x
Create your own Example:
1. Draw any function (f) that passes the horizontal line test.
2. To graph the inverse (f--1) reverse each (x, y) point on f,
graphing (y, x).