# How do you write a linear function f with the values f(2)=-1 and f(5)=4?

Jan 1, 2017

Use the two points to compute the slope, m, then use one of the points in the form $y = m \left(x\right) + b$ to find the value of b.

#### Explanation:

The equation for the slope, m, of a line is:

$m = \frac{{y}_{1} - {y}_{0}}{{x}_{1} - {x}_{0}} \text{ [1]}$

The equation $f \left(2\right) = - 1$ tells us that ${x}_{0} = 2 \mathmr{and} {y}_{0} = - 1$; substitute this into equation [1]:

$m = \frac{{y}_{1} - - 1}{{x}_{1} - 2} \text{ [2]}$

The equation $f \left(5\right) = 4$ tells us that ${x}_{1} = 5 \mathmr{and} {y}_{1} = 4$; substitute this into equation [2]:

$m = \frac{4 - - 1}{5 - 2} \text{ [3]}$

$m = \frac{5}{3}$

Substitute $\frac{5}{3}$ for m into the equation $y = m \left(x\right) + b$

$y = \frac{5}{3} x + b \text{ [4]}$

Substitute 2 for x and -1 for y and the solve for b:

$- 1 = \frac{5}{3} \left(2\right) + b$

$b = - \frac{13}{3}$

Substitute $- \frac{13}{3}$ for b in equation [4]:

$y = \frac{5}{3} x - \frac{13}{3} \text{ [5]}$

Check:

$- 1 = \frac{5}{3} \left(2\right) - \frac{13}{3}$
$4 = \frac{5}{3} \left(5\right) - \frac{13}{3}$

$- 1 = - 1$
$4 = 4$

This checks