High School

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Write a linear function [tex]f[/tex] with the given values.

82. [tex]f(0) = 7, \, f(3) = 1[/tex]

83. [tex]f(0) = 4, \, f(1) = -4[/tex]

84. [tex]f(4) = -3, \, f(0) = -2[/tex]

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]

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