13-4 Linear functions
13-4 Linear functions MATH 8 MR. DIXON
Warm Up Determine if each relationship represents a function. yes
1. 2. y = 3x2 – 1 yes
3. For the function f(x) = x2 + 2, find f(0), f(3), and f(–2).
2, 11, 6
A linear function is a function that can be described by a linear equation. You can use function notation to show that the output value of the function f, written f(x), corresponds to the input value x. Important: Any linear function can be written in slope-intercept form f(x) = mx +b. f(x) just replaces the y.
Example 1: Identifying Linear Functions Determine whether the function f(x) = –2x3 is linear. If so, give the slope and y-intercept of the function’s graph. The function is not linear because x has an exponent other than 1. The function cannot be written in the form f(x) = mx + b.
Check It Out: Example 2
Determine whether the function f(x) = –2x + 4 + x is linear. If so, give the slope and y-intercept of the function’s graph. f(x) = –2x + 4 + x f(x) = –x + 4 The function is linear because it can be written in the form f(x) = mx + b. The slope is –1 and the y-intercept is 4.
Additional Example 3: Writing the Equation for a Linear Function
Write a rule for the linear function. Step 1 Identify the y-intercept b from the graph. b=2 Step 2 Locate another point on the graph, such as (1, 4). Step 3 Substitute the x- and yvalues of the point into the equation, f(x) = mx + b, and solve for m.
Additional Example 3 Continued
f(x) = mx + b 4 = m(1) + 2
(x, y) = (1, 4)
4=m+2 –2 –2 2=m The rule is f(x) = 2x + 2.
Check It Out: Example 4
Write a rule for the linear function. y
4
2
x -4
-2
2
-4
4
Step 1 Identify the y-intercept b from the graph. b=1 Step 2 Locate another point on the graph, such as (5, 2). Step 3 Substitute the x- and yvalues of the point into the equation, f(x) = mx + b, and solve for m.
Check It Out: Example 4 Continued f(x) = mx + b 2 = m(5) + 1
(x, y) = (5, 2)
2 = 5m + 1 –1 –1 1 = 5m
m= 1 5 1 The rule is f(x) = x + 1. 5
Additional Example 5: Writing the Equation for a Linear Function
Write a rule for the linear function. x
y
Step 1 Locate two points.
–3 –8
(1, 4) and (3, 10)
–1 –2
Step 2 Find the slope m.
1
4
3
10
y2 – y1 m= x2 – x1
10 – 4 = = 3–1
6=3 2
Step 3 Substitute the x- and y-values of the point into the equation, f(x) = mx + b, and solve for b.
Additional Example 5 Continued f(x) = mx + b
4 = 3(1) + b
(x, y) = (1, 4)
4= 3+b –3 –3 1= b
The rule is f(x) = 3x + 1.
Check It Out: Example 6
Write a rule for the linear function. x
y
Step 1 Locate two points.
0
5
(0, 5) and (1, 6)
1
6
Step 2 Find the slope m.
2
7
–1
4
y2 – y1 6–5 1 m= x –x = = 1 =1 1 – 0 2 1 Step 3 Substitute the x- and y-values of the point into the equation, f(x) = mx + b, and solve for b.
Check It Out: Example 6 Continued f(x) = mx + b 5 = 1(0) + b (x, y) = (0, 5) 5= b The rule is f(x) = x + 5.
ASSIGNMENT PGS. 702 – 703 (1 – 12) ALL Extra Credit: (22 – 29) All Show all work to receive full credit. Calculators may be used. Use grid paper to make neat graphs. Use a ruler to draw straight lines.
Warm Up Determine if each relationship represents a function. yes
1. 2. y = 3x2 – 1 yes
3. For the function f(x) = x2 + 2, find f(0), f(3), and f(–2).
2, 11, 6
A linear function is a function that can be described by a linear equation. You can use function notation to show that the output value of the function f, written f(x), corresponds to the input value x. Important: Any linear function can be written in slope-intercept form f(x) = mx +b. f(x) just replaces the y.
Example 1: Identifying Linear Functions Determine whether the function f(x) = –2x3 is linear. If so, give the slope and y-intercept of the function’s graph. The function is not linear because x has an exponent other than 1. The function cannot be written in the form f(x) = mx + b.
Check It Out: Example 2
Determine whether the function f(x) = –2x + 4 + x is linear. If so, give the slope and y-intercept of the function’s graph. f(x) = –2x + 4 + x f(x) = –x + 4 The function is linear because it can be written in the form f(x) = mx + b. The slope is –1 and the y-intercept is 4.
Additional Example 3: Writing the Equation for a Linear Function
Write a rule for the linear function. Step 1 Identify the y-intercept b from the graph. b=2 Step 2 Locate another point on the graph, such as (1, 4). Step 3 Substitute the x- and yvalues of the point into the equation, f(x) = mx + b, and solve for m.
Additional Example 3 Continued
f(x) = mx + b 4 = m(1) + 2
(x, y) = (1, 4)
4=m+2 –2 –2 2=m The rule is f(x) = 2x + 2.
Check It Out: Example 4
Write a rule for the linear function. y
4
2
x -4
-2
2
-4
4
Step 1 Identify the y-intercept b from the graph. b=1 Step 2 Locate another point on the graph, such as (5, 2). Step 3 Substitute the x- and yvalues of the point into the equation, f(x) = mx + b, and solve for m.
Check It Out: Example 4 Continued f(x) = mx + b 2 = m(5) + 1
(x, y) = (5, 2)
2 = 5m + 1 –1 –1 1 = 5m
m= 1 5 1 The rule is f(x) = x + 1. 5
Additional Example 5: Writing the Equation for a Linear Function
Write a rule for the linear function. x
y
Step 1 Locate two points.
–3 –8
(1, 4) and (3, 10)
–1 –2
Step 2 Find the slope m.
1
4
3
10
y2 – y1 m= x2 – x1
10 – 4 = = 3–1
6=3 2
Step 3 Substitute the x- and y-values of the point into the equation, f(x) = mx + b, and solve for b.
Additional Example 5 Continued f(x) = mx + b
4 = 3(1) + b
(x, y) = (1, 4)
4= 3+b –3 –3 1= b
The rule is f(x) = 3x + 1.
Check It Out: Example 6
Write a rule for the linear function. x
y
Step 1 Locate two points.
0
5
(0, 5) and (1, 6)
1
6
Step 2 Find the slope m.
2
7
–1
4
y2 – y1 6–5 1 m= x –x = = 1 =1 1 – 0 2 1 Step 3 Substitute the x- and y-values of the point into the equation, f(x) = mx + b, and solve for b.
Check It Out: Example 6 Continued f(x) = mx + b 5 = 1(0) + b (x, y) = (0, 5) 5= b The rule is f(x) = x + 5.
ASSIGNMENT PGS. 702 – 703 (1 – 12) ALL Extra Credit: (22 – 29) All Show all work to receive full credit. Calculators may be used. Use grid paper to make neat graphs. Use a ruler to draw straight lines.
