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Example 7 from Algebra and Trigonometry, 4.1 Linear Functions

Writing an Equation for a Linear Function Given Two Points

If $f$ is a linear function, with $f (3) = −2,$ and $f (8) = 1,$ find an equation for the function in slope-intercept form.

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
We can write the given points using coordinates.

$\begin{matrix} f (3) & = & - 2 \to (3, −2) \\ f (8) & = & 1 \to (8, 1) \end{matrix}$
We can then use the points to calculate the slope.

$\begin{matrix} m & = & \frac{y_{2} - y_{1}}{x_{2} - x_{1}} \\ = & \frac{1 - (−2)}{8 - 3} \\ = & \frac{3}{5} \end{matrix}$
Substitute the slope and the coordinates of one of the points into the point-slope form.

$\begin{matrix} y - y_{1} & = & m (x - x_{1}) \\ y - (−2) & = & \frac{3}{5} (x - 3) \end{matrix}$
We can use algebra to rewrite the equation in the slope-intercept form.

$\begin{matrix} y + 2 & = & \frac{3}{5} (x - 3) \\ y + 2 & = & \frac{3}{5} x - \frac{9}{5} \\ y & = & \frac{3}{5} x - \frac{19}{5} \end{matrix}$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

If $f (x)$ is a linear function, with $f (2) = –11,$ and $f (4) = −25,$ write an equation for the function in slope-intercept form.

How did it go?

PreviousWriting an Equation for a Linear Cost Function Next Using a Linear Function to Determine the Number of Songs in a Music Collection

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 4.1, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.