Find the Equation Using Two Points f(2)=6 , f(-1)=-3

$f (2) = 6$ , $f (- 1) = - 3$

Step 1

$f (2) = 6$ , which means $(2, 6)$ is a point on the line. $f (- 1) = - 3$ , which means $(- 1, - 3)$ is a point on the line, too.

$(2, 6), (- 1, - 3)$

Step 2

Find the slope of the line between $(2, 6)$ and $(- 1, - 3)$ using $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ , which is the change of $y$ over the change of $x$ .

Slope is equal to the change in $y$ over the change in $x$ , or rise over run.

$m = \frac{change in y}{change in x}$

The change in $x$ is equal to the difference in x-coordinates (also called run), and the change in $y$ is equal to the difference in y-coordinates (also called rise).

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

Substitute in the values of $x$ and $y$ into the equation to find the slope.

$m = \frac{- 3 - (6)}{- 1 - (2)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $6$ .

$m = \frac{- 3 - 6}{- 1 - (2)}$

Subtract $6$ from $- 3$ .

$m = \frac{- 9}{- 1 - (2)}$

Simplify the denominator.

Multiply $- 1$ by $2$ .

$m = \frac{- 9}{- 1 - 2}$

Subtract $2$ from $- 1$ .

$m = \frac{- 9}{- 3}$

Divide $- 9$ by $- 3$ .

$m = 3$

Step 3

Use the slope $3$ and a given point $(2, 6)$ to substitute for $x_{1}$ and $y_{1}$ in the point-slope form $y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (6) = 3 \cdot (x - (2))$

Step 4

Simplify the equation and keep it in point-slope form.

$y - 6 = 3 \cdot (x - 2)$

Step 5

Solve for $y$ .

Simplify $3 \cdot (x - 2)$ .

Rewrite.

$y - 6 = 0 + 0 + 3 \cdot (x - 2)$

Simplify by adding zeros.

$y - 6 = 3 \cdot (x - 2)$

Apply the distributive property.

$y - 6 = 3 x + 3 \cdot - 2$

Multiply $3$ by $- 2$ .

$y - 6 = 3 x - 6$

Move all terms not containing $y$ to the right side of the equation.

Add $6$ to both sides of the equation.

$y = 3 x - 6 + 6$

Combine the opposite terms in $3 x - 6 + 6$ .

Add $- 6$ and $6$ .

$y = 3 x + 0$

Add $3 x$ and $0$ .

$y = 3 x$

Step 6

Replace $y$ by $f (x)$ .

$f (x) = 3 x$

Step 7

Find the Equation Using Two Points f(2)=6 , f(-1)=-3

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