Answer :
To write a linear function [tex]\( f(x) = mx + b \)[/tex] based on the given values, we need to determine the slope [tex]\( m \)[/tex] and the y-intercept [tex]\( b \)[/tex] for each function.
### 82. For [tex]\( f(0) = 7 \)[/tex] and [tex]\( f(3) = 1 \)[/tex]:
1. Identify two points: [tex]\( (0, 7) \)[/tex] and [tex]\( (3, 1) \)[/tex].
2. Calculate the slope [tex]\( m \)[/tex]:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - 7}{3 - 0} = \frac{-6}{3} = -2
\][/tex]
3. Find the y-intercept [tex]\( b \)[/tex]:
Since [tex]\( f(0) = 7 \)[/tex], we know directly that [tex]\( b = 7 \)[/tex].
4. Write the function:
[tex]\[
f(x) = -2x + 7
\][/tex]
### 83. For [tex]\( f(0) = 4 \)[/tex] and [tex]\( f(1) = -4 \)[/tex]:
1. Identify two points: [tex]\( (0, 4) \)[/tex] and [tex]\( (1, -4) \)[/tex].
2. Calculate the slope [tex]\( m \)[/tex]:
[tex]\[
m = \frac{-4 - 4}{1 - 0} = \frac{-8}{1} = -8
\][/tex]
3. Find the y-intercept [tex]\( b \)[/tex]:
Since [tex]\( f(0) = 4 \)[/tex], we know directly that [tex]\( b = 4 \)[/tex].
4. Write the function:
[tex]\[
f(x) = -8x + 4
\][/tex]
### 84. For [tex]\( f(4) = -3 \)[/tex] and [tex]\( f(0) = -2 \)[/tex]:
1. Identify two points: [tex]\( (4, -3) \)[/tex] and [tex]\( (0, -2) \)[/tex].
2. Calculate the slope [tex]\( m \)[/tex]:
[tex]\[
m = \frac{-3 - (-2)}{4 - 0} = \frac{-3 + 2}{4} = \frac{-1}{4} = -0.25
\][/tex]
3. Find the y-intercept [tex]\( b \)[/tex]:
Since [tex]\( f(0) = -2 \)[/tex], we know directly that [tex]\( b = -2 \)[/tex].
4. Write the function:
[tex]\[
f(x) = -0.25x - 2
\][/tex]
These steps lead us to the following linear functions:
- [tex]\( f(x) = -2x + 7 \)[/tex]
- [tex]\( f(x) = -8x + 4 \)[/tex]
- [tex]\( f(x) = -0.25x - 2 \)[/tex]
### 82. For [tex]\( f(0) = 7 \)[/tex] and [tex]\( f(3) = 1 \)[/tex]:
1. Identify two points: [tex]\( (0, 7) \)[/tex] and [tex]\( (3, 1) \)[/tex].
2. Calculate the slope [tex]\( m \)[/tex]:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - 7}{3 - 0} = \frac{-6}{3} = -2
\][/tex]
3. Find the y-intercept [tex]\( b \)[/tex]:
Since [tex]\( f(0) = 7 \)[/tex], we know directly that [tex]\( b = 7 \)[/tex].
4. Write the function:
[tex]\[
f(x) = -2x + 7
\][/tex]
### 83. For [tex]\( f(0) = 4 \)[/tex] and [tex]\( f(1) = -4 \)[/tex]:
1. Identify two points: [tex]\( (0, 4) \)[/tex] and [tex]\( (1, -4) \)[/tex].
2. Calculate the slope [tex]\( m \)[/tex]:
[tex]\[
m = \frac{-4 - 4}{1 - 0} = \frac{-8}{1} = -8
\][/tex]
3. Find the y-intercept [tex]\( b \)[/tex]:
Since [tex]\( f(0) = 4 \)[/tex], we know directly that [tex]\( b = 4 \)[/tex].
4. Write the function:
[tex]\[
f(x) = -8x + 4
\][/tex]
### 84. For [tex]\( f(4) = -3 \)[/tex] and [tex]\( f(0) = -2 \)[/tex]:
1. Identify two points: [tex]\( (4, -3) \)[/tex] and [tex]\( (0, -2) \)[/tex].
2. Calculate the slope [tex]\( m \)[/tex]:
[tex]\[
m = \frac{-3 - (-2)}{4 - 0} = \frac{-3 + 2}{4} = \frac{-1}{4} = -0.25
\][/tex]
3. Find the y-intercept [tex]\( b \)[/tex]:
Since [tex]\( f(0) = -2 \)[/tex], we know directly that [tex]\( b = -2 \)[/tex].
4. Write the function:
[tex]\[
f(x) = -0.25x - 2
\][/tex]
These steps lead us to the following linear functions:
- [tex]\( f(x) = -2x + 7 \)[/tex]
- [tex]\( f(x) = -8x + 4 \)[/tex]
- [tex]\( f(x) = -0.25x - 2 \)[/tex]
