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Interactive Study Guide for Students
Chapter 8: Functions and Graphing
Section 1: Functions
Relations and Functions
Examples
A relation is a set of ________ _______.
Is this relation a Function?
A ___________ is a special relation in which each member of the
domain(x) is paired with ___________ one member in the range(y).
1. {(-10, -34), (0, -22), (10, -9),
(20,3)}
A function ________________ the relationship between ______
quantities such as time and distance.
Since functions are _____________, they can be represented using
____________ _______, ___________, and __________.
Ordered Pairs and Tables
2. {(-10, -34), (-10, -22), (10, -9),
(20,3)}
Determine whether each relation is a function. Explain
{(-3,1), (-2, 4), (-1, 7), (0, 10), (1, 13)}
X
y
Is it a function? Explain__________________________________
________________________________________________________
Graphs
Another way to tell whether a relation is a function is to use the
____________ line test.
Place a pencil on the graph if for every value of x in the domain, it passes
through no more than _______ point on the graph, then the graph
represents a _____________.
This graph is a function!
Is this a function?
Chapter 8: Functions and Graphing
Section 2: Linear Equations in two variables
Solutions of Equations
Functions can be represented in ________, in a table, as ordered pairs,
with a __________, and with an ______________.
Examples
1. Find four solutions of
y = 2x – 1
___________ ____________: an equation in which the variables appear
in separate terms and neither variable contains an exponent other than
1. ex: y= 1.5x
Solutions to linear equations are __________ ________ that make the
equation true. One way to do this is a table.
x
y = -x + 8
y
(x,y)
-1
y = -(-1) + 8
9
(-1 , 9)
0
y = -0 + 8
8
(0, 8)
1
y = -1 + 8
7
(1, 7)
2
y = -2 + 8
6
(2, 6)
2.
x
y = 2x + 3
y
(x,y)
Solve an equation for y
0
y = 2( ) + 3
(
,
)
Sometimes it is necessary to first rewrite an equation by __________ for
y.
1
y = 2( ) + 3
(
,
)
2
y = 2( ) + 3
(
,
)
Example: Find four solutions of the equation: y – 2x = 3
3
y = 2( ) + 3
(
,
)
1st: solve for y:
y = 2x + 3
nd
2 : choose 4 x values and substitute them into y = 2x + 3 to find
four solutions.
Graph a Linear Equation
3. Graph y = -x + 8
First find ____________ _________ solutions. Then plot these points and
draw a _________ through them.
Any ordered pair on the line is a ____________.
Graph y = 2x +3
4. What is the x-intercept of
y = 2x +3?
x-intercept: Where a ___________ crosses the x-axis
- To find it let y=0 and solve for x or find it on the graph
y-intercept: Where a graph crosses the __________.
5. What is the y-intercept of
y=2x + 3?
- To find it let x=0 and solve for y or find it on the graph
Chapter 8: Functions and Graphing
Section 4 & 5: Slope and Rate of Change
Slope
Examples
Slope describes the ____________ of a line. It is the ratio of the rise, or
___________ change, to the run, or ____________ change.
m = Slope =
1. Graph the line that goes
through the points (-2, 1) and
(0,-3).
rise
vertical _ change

run
horizontal _ change
It is the ______ for any two points on the same line.
Example:
Find the slope of a road that rises 25 feet for every horizontal change of
80 feet.
Slope =
What is the slope of the line?
25 ft 5
=
= 0.3125
80 ft 16
Using two points to find the slope
The ________ (m) of a line passing through points at (x1, y1) and (x2, y2) is
the _________ of the difference in the y-coordinates to the
_____________ in the x-coordinates.
2. Find the slope of the line that
goes through the points (-1,1)
and (3,1).
m = Slope = y1-y2 , where x1 ≠ x2
x1 – x2
Example: Find the slope of the line between the points (2,2) and (5,3)
3. Graph the line that goes
through the points (-5, 6) and
(-5, 0).
32 1
m = Slope =
=
52 3
Rate of Change
A change in one quantity with respect to another quantity is called the
Rates of Change. Rates of change can be described using slope.
Compare the rates of change of the perimeter of the triangle and square
give in the table.
Triangle =
change _ in _ y 6
= =3
Change _ in _ x 2
change _ in _ y 8
Square=
= =4
Change _ in _ x 2
Side
x
0
2
4
Perimeter
y
Triangle
Square
0
6
12
0
8
18
What is the slope of the line?
The perimeter of the square increases at a faster rate than the perimeter
of the triangle.
Chapter 8: Functions and Graphing
Slope and y-Intercept
Section 6: Slope-Intercept Form
Examples
All the equations here are written in the
form y=mx + b, where m is the
____________ and b is the ___-intercept.
This is called slope-intercept form.
y = mx + b
slope 
 y-intercept
1. State the slope and y-intercept:
y = x +8
m=
b=
Example: State the slope and the y-intercept
of the graph of y = 3/5x – 7:
1st: Write the equation in slope-intercept form
y = mx + b
3
y = x + (-7)
5
2. State the slope and y-intercept:
x+3y = 6
nd
2 : State the slope (m) and the y-intercept (b)
m=
Slope-intercept form:
3
5
m=
b = -7
b=
Sometimes you must first write an _______________ in slope-intercept
form before finding the __________ and y-_______________.
State the slope and the y-intercept of the graph of 5x + y = 3.
1st: write the original equation
5x + y = 3
2nd: solve for y to get it in slope-intercept form y= -5x + 3
3rd: State the slope (m) and the y-intercept (b)
m = -5
b=3
3. Graph the equation:
y=x+5
Graph Equations
Step 1: Find the slope and y-intercept.
Step 2: Graph the y-intercept point
Step 3: Use the slope to find another point
Step 4: Use the two points to draw a line
Graph y=
1
x -4
2
Slope=m =
Intercept=b=
Chapter 8: Functions and Graphing
Solve Systems by Graphing
Section 9: Solving Systems of Equations
Examples
The equations y=10x + 50 and y=15x together are called a ___________
of _____________. The solutions of this system is the ___________
_______ that is a solution of both equations, (10, 150)
1. Solve by graphing:
Y = 2x + 1
Y = -x + 1
y = 10x + 50
y = 15x
150 = 10(10) + 50
150=15(10)
150 = 150
150=150
One method for solving is to graph both ____________ on the same
coordinate plane. The solution is where they ______________.
Find the solution:
2. Solve by graphing:
Y = 2x + 4
Y = 2x - 1
y=-x
y=x+2
To check it substitute the coordinates
Into each equation.
y = -x
y=x+2
1 = -(-1)
1=(-1)+2
1=1
1=1
One Solution
No Solutions
Infinitely Many
3. Solve by graphing:
2y = x + 6
Y=
1
x+3
2
Solve Systems by Substitution
A more accurate way to solve a system of _____________ is by using a
method called _________________.
Solve by substitution:
y=x+5
y=3
Since y must be equal in both equations, replace y with 3 in the
first equation.
3=x+5
 then solve for x
x=-2
The solution is an ordered pair of (-2,3). You can check it by
graphing it.
Chapter 8: Functions and Graphing
4. Solve by substitution:
y=x+2
y=0
Section 10: Graphing Inequalities
Examples