Find the Equation Using Two Points f(-2)=6 and f(0)=-4

$f (- 2) = 6$ and $f (0) = - 4$

$f (- 2) = 6$ , which means $(- 2, 6)$ is a point on the line. $f (0) = - 4$ , which means $(0, - 4)$ is a point on the line, too.

$(- 2, 6), (0, - 4)$

Find the slope of the line between $(- 2, 6)$ and $(0, - 4)$ 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.

$m = \frac{- 4 - (6)}{0 - (- 2)}$

Simplify.

Reduce the expression by cancelling the common factors.

Cancel the common factor of $- 4 - (6)$ and $0 - (- 2)$ .

Rewrite $- 4$ as $- 1 (4)$ .

$m = \frac{- 1 \cdot 4 - (6)}{0 - (- 2)}$

Factor $- 1$ out of $- 1 (4) - (6)$ .

$m = \frac{- 1 (4 + 6)}{0 - (- 2)}$

Reorder terms.

$m = \frac{- 1 (4 + 6)}{0 - 2 \cdot - 1}$

Factor $2$ out of $- 1 (4 + 6)$ .

$m = \frac{2 (- 1 (2 + 3))}{0 - 2 \cdot - 1}$

Cancel the common factors.

Factor $2$ out of $0$ .

$m = \frac{2 (- 1 (2 + 3))}{2 (0) - 2 \cdot - 1}$

Factor $2$ out of $- 2 \cdot - 1$ .

$m = \frac{2 (- 1 (2 + 3))}{2 (0) + 2 (1)}$

Factor $2$ out of $2 (0) + 2 (- - 1)$ .

$m = \frac{2 (- 1 (2 + 3))}{2 (0 + 1)}$

Cancel the common factor.

$m = \frac{2 (- 1 (2 + 3))}{2 (0 + 1)}$

Rewrite the expression.

$m = \frac{- 1 (2 + 3)}{0 + 1}$

Add $2$ and $3$ .

$m = \frac{- 1 \cdot 5}{0 + 1}$

Simplify the denominator.

Multiply $- 1$ by $- 1$ .

$m = \frac{- 1 \cdot 5}{0 + 1}$

Add $0$ and $1$ .

$m = \frac{- 1 \cdot 5}{1}$

Simplify the expression.

Multiply $- 1$ by $5$ .

$m = \frac{- 5}{1}$

Divide $- 5$ by $1$ .

$m = - 5$

Use the slope $- 5$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (6) = - 5 \cdot (x - (- 2))$

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

$y - 6 = - 5 \cdot (x + 2)$

Solve for $y$ .

Simplify $- 5 \cdot (x + 2)$ .

Rewrite.

$y - 6 = 0 + 0 - 5 \cdot (x + 2)$

Simplify by adding zeros.

$y - 6 = - 5 \cdot (x + 2)$

Apply the distributive property.

$y - 6 = - 5 x - 5 \cdot 2$

Multiply $- 5$ by $2$ .

$y - 6 = - 5 x - 10$

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

Add $6$ to both sides of the equation.

$y = - 5 x - 10 + 6$

Add $- 10$ and $6$ .

$y = - 5 x - 4$

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

$f (x) = - 5 x - 4$

Find the Equation Using Two Points f(-2)=6 and f(0)=-4

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