Graph y>=-2(x-3)^2+2

$y \geq - 2 {(x - 3)}^{2} + 2$

Step 1

Simplify $- 2 {(x - 3)}^{2} + 2$ .

Simplify each term.

Rewrite ${(x - 3)}^{2}$ as $(x - 3) (x - 3)$ .

$y \geq - 2 ((x - 3) (x - 3)) + 2$

Expand $(x - 3) (x - 3)$ using the FOIL Method.

Apply the distributive property.

$y \geq - 2 (x (x - 3) - 3 (x - 3)) + 2$

Apply the distributive property.

$y \geq - 2 (x \cdot x + x \cdot - 3 - 3 (x - 3)) + 2$

Apply the distributive property.

$y \geq - 2 (x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3) + 2$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$y \geq - 2 (x^{2} + x \cdot - 3 - 3 x - 3 \cdot - 3) + 2$

Move $- 3$ to the left of $x$ .

$y \geq - 2 (x^{2} - 3 \cdot x - 3 x - 3 \cdot - 3) + 2$

Multiply $- 3$ by $- 3$ .

$y \geq - 2 (x^{2} - 3 x - 3 x + 9) + 2$

Subtract $3 x$ from $- 3 x$ .

$y \geq - 2 (x^{2} - 6 x + 9) + 2$

Apply the distributive property.

$y \geq - 2 x^{2} - 2 (- 6 x) - 2 \cdot 9 + 2$

Simplify.

Multiply $- 6$ by $- 2$ .

$y \geq - 2 x^{2} + 12 x - 2 \cdot 9 + 2$

Multiply $- 2$ by $9$ .

$y \geq - 2 x^{2} + 12 x - 18 + 2$

Add $- 18$ and $2$ .

$y \geq - 2 x^{2} + 12 x - 16$

Step 2

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

Not Linear

Step 3

Graph a solid line, then shade the area above the boundary line since $y$ is greater than $- 2 x^{2} + 12 x - 16$ .

$y \geq - 2 x^{2} + 12 x - 16$

Step 4