Find the Equation Using Two Points f(-4)=2 and f(6)=-3

$f (- 4) = 2$ and $f (6) = - 3$

Step 1

$f (- 4) = 2$ , which means $(- 4, 2)$ is a point on the line. $f (6) = - 3$ , which means $(6, - 3)$ is a point on the line, too.

$(- 4, 2), (6, - 3)$

Step 2

Find the slope of the line between $(- 4, 2)$ and $(6, - 3)$ 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.

$m = \frac{- 3 - (2)}{6 - (- 4)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $2$ .

$m = \frac{- 3 - 2}{6 - (- 4)}$

Subtract $2$ from $- 3$ .

$m = \frac{- 5}{6 - (- 4)}$

Simplify the denominator.

Multiply $- 1$ by $- 4$ .

$m = \frac{- 5}{6 + 4}$

Add $6$ and $4$ .

$m = \frac{- 5}{10}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $- 5$ and $10$ .

Factor $5$ out of $- 5$ .

$m = \frac{5 (- 1)}{10}$

Cancel the common factors.

Factor $5$ out of $10$ .

$m = \frac{5 \cdot - 1}{5 \cdot 2}$

Cancel the common factor.

$m = \frac{5 \cdot - 1}{5 \cdot 2}$

Rewrite the expression.

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

Move the negative in front of the fraction.

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

Step 3

Use the slope $- \frac{1}{2}$ and a given point $(- 4, 2)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (2) = - \frac{1}{2} \cdot (x - (- 4))$

Step 4

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

$y - 2 = - \frac{1}{2} \cdot (x + 4)$

Step 5

Solve for $y$ .

Simplify $- \frac{1}{2} \cdot (x + 4)$ .

Rewrite.

$y - 2 = 0 + 0 - \frac{1}{2} \cdot (x + 4)$

Simplify by adding zeros.

$y - 2 = - \frac{1}{2} \cdot (x + 4)$

Apply the distributive property.

$y - 2 = - \frac{1}{2} x - \frac{1}{2} \cdot 4$

Combine $x$ and $\frac{1}{2}$ .

$y - 2 = - \frac{x}{2} - \frac{1}{2} \cdot 4$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$y - 2 = - \frac{x}{2} + \frac{- 1}{2} \cdot 4$

Factor $2$ out of $4$ .

$y - 2 = - \frac{x}{2} + \frac{- 1}{2} \cdot (2 (2))$

Cancel the common factor.

$y - 2 = - \frac{x}{2} + \frac{- 1}{2} \cdot (2 \cdot 2)$

Rewrite the expression.

$y - 2 = - \frac{x}{2} - 1 \cdot 2$

Multiply $- 1$ by $2$ .

$y - 2 = - \frac{x}{2} - 2$

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

Add $2$ to both sides of the equation.

$y = - \frac{x}{2} - 2 + 2$

Combine the opposite terms in $- \frac{x}{2} - 2 + 2$ .

Add $- 2$ and $2$ .

$y = - \frac{x}{2} + 0$

Add $- \frac{x}{2}$ and $0$ .

$y = - \frac{x}{2}$

Step 6

The final answer is the equation in slope-intercept form.

$y = - \frac{1}{2} x$

Step 7

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

$f (x) = - \frac{1}{2} x$

Step 8

Find the Equation Using Two Points f(-4)=2 and f(6)=-3

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