Question

1. create a linear function using the given function evaluations. f(0) = -7 and f(5) = 3

Explanation:

Step1: Recall linear function form

A linear function is in the form \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y - intercept.

Step2: Find the y - intercept (\( b \))

We know that \( f(0)=-7 \). Substitute \( x = 0 \) and \( f(0)=-7 \) into \( f(x)=mx + b \):
\( f(0)=m(0)+b=-7 \), so \( b=-7 \).

Step3: Find the slope (\( m \))

We know that \( f(5) = 3 \), \( b=-7 \) and use the linear function formula \( f(x)=mx + b \). Substitute \( x = 5 \), \( f(5)=3 \) and \( b = - 7 \) into the formula:
\( 3=m(5)+(-7) \)
First, add 7 to both sides of the equation: \( 3 + 7=5m-7 + 7 \), which simplifies to \( 10 = 5m \).
Then, divide both sides by 5: \( m=\frac{10}{5}=2 \).

Step4: Write the linear function

Now that we have \( m = 2 \) and \( b=-7 \), substitute these values into the linear function form \( f(x)=mx + b \). We get \( f(x)=2x-7 \).

Answer:

\( f(x)=2x - 7 \)