Graph f(x)>-2x^2+8x-12

$f (x) > - 2 x^{2} + 8 x - 12$

Step 1

Subtract $f (x)$ from both sides of the inequality.

$0 > - 2 x^{2} + 8 x - 12 - f (x)$

Step 2

Find the slope and the y-intercept for the boundary line.

Rewrite in slope-intercept form.

The slope-intercept form is $y = m x + b$ , where $m$ is the slope and $b$ is the y-intercept.

$y = m x + b$

Rewrite so $x$ is on the left side of the inequality.

$- 2 x^{2} + 8 x - 12 - f (x) < 0$

Convert the inequality to an equation.

$- 2 x^{2} + 8 x - 12 - f (x) = 0$

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = - 2$ , $b = 8$ , and $c = - 12 - f (x)$ into the quadratic formula and solve for $x$ .

$\frac{- 8 \pm \sqrt{8^{2} - 4 \cdot (- 2 \cdot (- 12 - f (x)))}}{2 \cdot - 2}$

Simplify.

Simplify the numerator.

Raise $8$ to the power of $2$ .

$x = \frac{- 8 \pm \sqrt{64 - 4 \cdot - 2 \cdot (- 12 - f (x))}}{2 \cdot - 2}$

Multiply $- 4$ by $- 2$ .

$x = \frac{- 8 \pm \sqrt{64 + 8 \cdot (- 12 - f (x))}}{2 \cdot - 2}$

Apply the distributive property.

$x = \frac{- 8 \pm \sqrt{64 + 8 \cdot - 12 + 8 (- f (x))}}{2 \cdot - 2}$

Multiply $8$ by $- 12$ .

$x = \frac{- 8 \pm \sqrt{64 - 96 + 8 (- f (x))}}{2 \cdot - 2}$

Multiply $- 1$ by $8$ .

$x = \frac{- 8 \pm \sqrt{64 - 96 - 8 f x}}{2 \cdot - 2}$

Subtract $96$ from $64$ .

$x = \frac{- 8 \pm \sqrt{- 32 - 8 f x}}{2 \cdot - 2}$

Multiply $2$ by $- 2$ .

$x = \frac{- 8 \pm \sqrt{- 32 - 8 f x}}{- 4}$

Simplify $\frac{- 8 \pm \sqrt{- 32 - 8 f (x)}}{- 4}$ .

$x = \frac{8 \pm \sqrt{- 32 - 8 f x}}{4}$

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

Raise $8$ to the power of $2$ .

$x = \frac{- 8 \pm \sqrt{64 - 4 \cdot - 2 \cdot (- 12 - f (x))}}{2 \cdot - 2}$

Multiply $- 4$ by $- 2$ .

$x = \frac{- 8 \pm \sqrt{64 + 8 \cdot (- 12 - f (x))}}{2 \cdot - 2}$

Apply the distributive property.

$x = \frac{- 8 \pm \sqrt{64 + 8 \cdot - 12 + 8 (- f (x))}}{2 \cdot - 2}$

Multiply $8$ by $- 12$ .

$x = \frac{- 8 \pm \sqrt{64 - 96 + 8 (- f (x))}}{2 \cdot - 2}$

Multiply $- 1$ by $8$ .

$x = \frac{- 8 \pm \sqrt{64 - 96 - 8 f x}}{2 \cdot - 2}$

Subtract $96$ from $64$ .

$x = \frac{- 8 \pm \sqrt{- 32 - 8 f x}}{2 \cdot - 2}$

Multiply $2$ by $- 2$ .

$x = \frac{- 8 \pm \sqrt{- 32 - 8 f x}}{- 4}$

Simplify $\frac{- 8 \pm \sqrt{- 32 - 8 f (x)}}{- 4}$ .

$x = \frac{8 \pm \sqrt{- 32 - 8 f x}}{4}$

Change the $\pm$ to $+$ .

$x = \frac{8 + \sqrt{- 32 - 8 f x}}{4}$

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

Raise $8$ to the power of $2$ .

$x = \frac{- 8 \pm \sqrt{64 - 4 \cdot - 2 \cdot (- 12 - f (x))}}{2 \cdot - 2}$

Multiply $- 4$ by $- 2$ .

$x = \frac{- 8 \pm \sqrt{64 + 8 \cdot (- 12 - f (x))}}{2 \cdot - 2}$

Apply the distributive property.

$x = \frac{- 8 \pm \sqrt{64 + 8 \cdot - 12 + 8 (- f (x))}}{2 \cdot - 2}$

Multiply $8$ by $- 12$ .

$x = \frac{- 8 \pm \sqrt{64 - 96 + 8 (- f (x))}}{2 \cdot - 2}$

Multiply $- 1$ by $8$ .

$x = \frac{- 8 \pm \sqrt{64 - 96 - 8 f x}}{2 \cdot - 2}$

Subtract $96$ from $64$ .

$x = \frac{- 8 \pm \sqrt{- 32 - 8 f x}}{2 \cdot - 2}$

Multiply $2$ by $- 2$ .

$x = \frac{- 8 \pm \sqrt{- 32 - 8 f x}}{- 4}$

Simplify $\frac{- 8 \pm \sqrt{- 32 - 8 f (x)}}{- 4}$ .

$x = \frac{8 \pm \sqrt{- 32 - 8 f x}}{4}$

Change the $\pm$ to $-$ .

$x = \frac{8 - \sqrt{- 32 - 8 f x}}{4}$

Consolidate the solutions.

$x = \frac{8 + \sqrt{- 32 - 8 f x}}{4}, \frac{8 - \sqrt{- 32 - 8 f x}}{4}$

Arrange the polynomial to follow the $y = m x + b$ form for slope and y-intercept.

$- 2 x^{2} + 8 x = f (x) + 12$

The equation is not linear, so a constant slope does not exist.

Not Linear

Step 3

Graph a dashed line, then shade the area above the boundary line since $y$ is greater than $- 2 x^{2} + 8 x - 12 - f (x)$ .

$0 > - 2 x^{2} + 8 x - 12 - f (x)$

Step 4

Graph f(x)>-2x^2+8x-12

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