Find the x and y Intercepts f(x)=9-(x-4)^2

$f (x) = 9 - {(x - 4)}^{2}$

Step 1

Find the x-intercepts.

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = 9 - {(x - 4)}^{2}$

Solve the equation.

Remove parentheses.

$0 = 9 - {(x - 4)}^{2}$

Simplify $9 - {(x - 4)}^{2}$ .

Simplify each term.

Rewrite ${(x - 4)}^{2}$ as $(x - 4) (x - 4)$ .

$0 = 9 - ((x - 4) (x - 4))$

Expand $(x - 4) (x - 4)$ using the FOIL Method.

Apply the distributive property.

$0 = 9 - (x (x - 4) - 4 (x - 4))$

Apply the distributive property.

$0 = 9 - (x \cdot x + x \cdot - 4 - 4 (x - 4))$

Apply the distributive property.

$0 = 9 - (x \cdot x + x \cdot - 4 - 4 x - 4 \cdot - 4)$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$0 = 9 - (x^{2} + x \cdot - 4 - 4 x - 4 \cdot - 4)$

Move $- 4$ to the left of $x$ .

$0 = 9 - (x^{2} - 4 \cdot x - 4 x - 4 \cdot - 4)$

Multiply $- 4$ by $- 4$ .

$0 = 9 - (x^{2} - 4 x - 4 x + 16)$

Subtract $4 x$ from $- 4 x$ .

$0 = 9 - (x^{2} - 8 x + 16)$

Apply the distributive property.

$0 = 9 - x^{2} - (- 8 x) - 1 \cdot 16$

Simplify.

Multiply $- 8$ by $- 1$ .

$0 = 9 - x^{2} + 8 x - 1 \cdot 16$

Multiply $- 1$ by $16$ .

$0 = 9 - x^{2} + 8 x - 16$

Subtract $16$ from $9$ .

$0 = - x^{2} + 8 x - 7$

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x = 1, 7$

x-intercept(s) in point form.

x-intercept(s): $(1, 0), (7, 0)$

Step 2

Find the y-intercepts.

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = 9 - {((0) - 4)}^{2}$

Solve the equation.

Remove parentheses.

$y = 9 - {(0 - 4)}^{2}$

Remove parentheses.

$y = 9 - {((0) - 4)}^{2}$

Simplify $9 - {((0) - 4)}^{2}$ .

Simplify each term.

Subtract $4$ from $0$ .

$y = 9 - {(- 4)}^{2}$

Raise $- 4$ to the power of $2$ .

$y = 9 - 1 \cdot 16$

Multiply $- 1$ by $16$ .

$y = 9 - 16$

Subtract $16$ from $9$ .

$y = - 7$

y-intercept(s) in point form.

y-intercept(s): $(0, - 7)$

Step 3

List the intersections.

x-intercept(s): $(1, 0), (7, 0)$

y-intercept(s): $(0, - 7)$

Step 4

Find the x and y Intercepts f(x)=9-(x-4)^2

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