Download Functions and Graphs

Solving Linear Equations by
Graphing (3-2)
Objective: Solve equations by graphing.
Estimate solutions to an equation by graphing.
Solve by Graphing
• A linear function is a function for
which the graph is a line.
• The simplest linear function is
f(x) = x and is called the parent
function of the family of linear
functions.
• A family of graphs is a group of
graphs with one or more similar
characteristics.
Linear Function
•
•
•
•
Parent function: f(x) = x
Type of graph: line
Domain: all real numbers
Range: all real numbers
y
4
3
2
1
x
-4
-3
-2
-1
-1
-2
-3
-4
-5
1
2
3
4
5
Linear Function
• The solution or root of an equation
is any value that makes the
equation true.
• A linear equation has at most one
root.
• You can find the root of an
equation by graphing its related
function.
• To write the related function for an
equation, replace 0 with f(x).
– Linear Equation: 2x – 8 = 0
– Related Function: f(x) = 2x – 8 or
y = 2x – 8
Linear Function
• Values of x for which f(x) = 0 are
called zeros of the function f.
• The zero of a function is located at
the x-intercepts of the function.
• The root of an equation is the value
of the x-intercept.
Linear Function
• Example:




4 is the x-intercept of 2x – 8 = 0.
4 is the solution of 2x – 8 = 0.
4 is the root of 2x – 8 = 0.
4 is the zero of f(x) = 2x – 8.
y
y2= 2x – 8
x
-2
2
-2
-4
-6
-8
-10
4
6
8
10
Example 1
• Solve each equation algebraically
and graphically.
Graph y = ½ x + 3
a. 0 = ½ x + 3
-3
-3
2 • -3 = ½ x • 2
-6 = x
x = -6
10
X
Y
0
3
2
4
4
5
y
x-int = -6
x = -6
8
6
4
2
x
-10 -8 -6 -4 -2
-2
-4
-6
-8
-10
2
4
6
8
10
Example 1
• Solve each equation algebraically
and graphically.
2 = 1/3 x + 3
-2
-2
0 = 1/3 x + 1
Graph y = 1/3 x + 1
b. 2 = 1/3 x + 3
-3
-3
3 • -1 = 1/3 x • 3
-3 = x
x = -3
X
Y
8
0
1
6
3
2
6
3
10
y
4
2
x
-10 -8 -6 -4 -2
-2
-4
-6
-8
-10
2
4
6
8
10
x-int = -3
x = -3
Linear Function
• For equations with the same
variable on each side of the
equation, use addition or
subtraction to get the terms with
variables on one side.
• Then solve.
Example 2
• Solve each equation algebraically
and graphically.
2x + 5 = 2x + 3
-2x
-2x
5=3
-3 -3
2=0
Graph y = 2
a. 2x + 5 = 2x + 3
-2x
-2x
5=3
No Solution
10
y
8
6
4
2
x
-10 -8 -6 -4 -2
-2
-4
-6
-8
-10
2
4
6
8
10
Since there is no xintercept, there is no
solution.
Example 2
• Solve each equation algebraically
and graphically.
5x – 7 = 5x + 2
-5x
-5x
-7 = 2
-2 -2
-9 = 0
Graph y = -9
b. 5x – 7 = 5x + 2
-5x
-5x
-7 = 2
No Solution
10
y
8
6
4
2
x
-10 -8 -6 -4 -2
-2
-4
-6
-8
-10
2
4
6
8
10
Since there is no xintercept, there is no
solution.
Estimate Solutions by Graphing
• Graphing may provide only an
estimate.
• In these cases, solve algebraically
to find the exact solution.
Example 3
•
Kendra’s class is selling greeting cards to raise money
for new soccer equipment. They paid $115 for the
cards, and they are selling each card for $1.75. The
function y = 1.75x – 115 represents their profit y for
selling x greeting cards. Find the zero of this function.
Describe what this value means in this context.
X
80
100
Y
25
60
X-int is
around 65?
0 = 1.75x – 115
+115
+115
115 = 1.75x
1.75
1.75
65.7 = x
The class must sell at least 66
cards to make any profit.
y
80
60
40
20
x
20
40
60
80
100
Check Your Progress
• Choose the best answer for the
following.
A. Solve -2 = 2/3 x + 4 algebraically.
A.
B.
C.
D.
x = -4
x = -9
x=4
x=9
-2 = 2/3 x + 4
-4
-4
3/ • -6 = 2/ x • 3/
2
3
2
Check Your Progress
• Choose the best answer for the
following.
B. Solve 6 = -¾ x + 9 by graphing.
A. x = 4;
6 = -¾ x + 9
-6
-6
0 = -¾ x + 3
C. x = -3;
y
y
4
4
2
2
x
-4
-2
2
4
6
x
-4
-2
2
-2
-2
-4
-4
-6
-6
4
D. x = 3;
B. x = -4;
y
y
4
4
2
2
-2
2
4
6
x
-4
-2
2
-2
-2
-4
-4
-6
-6
Graph
y = -¾ x + 3
X
Y
0
3
4
0
-4 6
x
-4
6
4
6
Check Your Progress
• Choose the best answer for the
following.
A. Solve -3x + 6 = 7 – 3x algebraically.
A.
B.
C.
D.
x=0
x=1
x = -1
no solution
-3x + 6 = 7 – 3x
+3x
+3x
6=7
Check Your Progress
• Choose the best answer for the
following.
B. Solve 4 – 6x = -6x + 3 by graphing.
A. x = -1;
C. x = 1;
y
y
4
4
2
2
x
-4
-2
2
4
6
x
-4
-2
2
-2
-2
-4
-4
-6
-6
4
6
D. no solution
B. x = 1;
y
y
4
4
2
2
x
x
-4
-2
2
-2
-4
-6
4
6
-4
-2
2
-2
-4
-6
4
6
4 – 6x = -6x + 3
+6x +6x
4=3
-3 -3
1=0
Graph y = 1
Check Your Progress
• Choose the best answer for the
following.
0 = 150 – 45t
-150 -150
-150 = -45t
-45
-45
– On a trip to his friend’s house, Raphael’s
average speed was 45 miles per hour. The
distance that Raphael is from his friend’s
house at a certain moment in the trip can
be represented by d = 150 – 45t, where d
represents the distance in miles and t is
the time in hours. Find the zero of this
function.
A. 3; Raphael will arrive at his friend’s house
in 3 hours.
B. 3 1/3; Raphael will arrive at his friend’s
house in 3 hours and 20 minutes.
C. 3 1/3; Raphael will arrive at his friend’s
house in 3 hours 30 minutes.
D. 4; Raphael will arrive at his friend’s house
in 4 hours.