Question

Write an equation for the linear function f such that f(1)= 2 and f (3)=-2

Answer 1

To find the linear function \( f(x) \) that satisfies the conditions \( f(1) = 2 \) and \( f(3) = -2 \), we can start by using the point-slope form of a linear function.

A linear function can be expressed in the form:

\[
f(x) = mx + b
\]

where \( m \) is the slope and \( b \) is the y-intercept.

First, we calculate the slope \( m \) using the two points \( (1, 2) \) and \( (3, -2) \):

\[
m = \frac{f(3) - f(1)}{3 - 1} = \frac{-2 - 2}{3 - 1} = \frac{-4}{2} = -2
\]

Now that we have the slope \( m = -2 \), we can use one of the points to find the y-intercept \( b \). We'll use the point \( (1, 2) \):

\[
f(1) = m \cdot 1 + b \implies 2 = -2 \cdot 1 + b \implies 2 = -2 + b \implies b = 4
\]

Now, we can write the equation of the linear function:

\[
f(x) = -2x + 4
\]

Thus, the equation for the linear function is:

\[
f(x) = -2x + 4
\]