Question

write a linear function f with the given values.

1. f(10) = 5, f(2) = -3
2. f(6) = 8, f(9) = 3

Explanation:

Problem 4: \( f(10) = 5 \), \( f(2) = -3 \)

Step 1: Find the slope \( m \)

The slope formula is \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Let \( (x_1,y_1)=(2, - 3) \) and \( (x_2,y_2)=(10,5) \). Then \( m=\frac{5-(-3)}{10 - 2}=\frac{5 + 3}{8}=\frac{8}{8}=1 \).

Step 2: Use point - slope form to find the equation

The point - slope form of a line is \( y - y_1=m(x - x_1) \). Using the point \( (2,-3) \) and \( m = 1 \), we have \( y-(-3)=1\times(x - 2) \), which simplifies to \( y + 3=x - 2 \). Then \( y=x-2 - 3=x - 5 \). So the linear function is \( f(x)=x - 5 \).

Problem 5: \( f(6)=8 \), \( f(9)=3 \)

Step 1: Find the slope \( m \)

Using the slope formula \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Let \( (x_1,y_1)=(6,8) \) and \( (x_2,y_2)=(9,3) \). Then \( m=\frac{3 - 8}{9 - 6}=\frac{-5}{3}=-\frac{5}{3} \).

Step 2: Use point - slope form to find the equation

Using the point - slope form \( y - y_1=m(x - x_1) \) with the point \( (6,8) \) and \( m=-\frac{5}{3} \), we get \( y - 8=-\frac{5}{3}(x - 6) \).
Expand the right - hand side: \( y - 8=-\frac{5}{3}x+10 \).
Add 8 to both sides: \( y=-\frac{5}{3}x + 10+8=-\frac{5}{3}x+18 \). So the linear function is \( f(x)=-\frac{5}{3}x + 18 \).

Answer:

s:
Problem 4: \( f(x)=x - 5 \)
Problem 5: \( f(x)=-\frac{5}{3}x + 18 \)