Find the x and y Intercepts -3(x-4)^2+3

$- 3 {(x - 4)}^{2} + 3$

Step 1

Write $- 3 {(x - 4)}^{2} + 3$ as an equation.

$y = - 3 {(x - 4)}^{2} + 3$

Step 2

Find the x-intercepts.

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

$0 = - 3 {(x - 4)}^{2} + 3$

Solve the equation.

Remove parentheses.

$0 = - 3 {(x - 4)}^{2} + 3$

Simplify $- 3 {(x - 4)}^{2} + 3$ .

Simplify each term.

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

$0 = - 3 ((x - 4) (x - 4)) + 3$

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

Apply the distributive property.

$0 = - 3 (x (x - 4) - 4 (x - 4)) + 3$

Apply the distributive property.

$0 = - 3 (x \cdot x + x \cdot - 4 - 4 (x - 4)) + 3$

Apply the distributive property.

$0 = - 3 (x \cdot x + x \cdot - 4 - 4 x - 4 \cdot - 4) + 3$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$0 = - 3 (x^{2} + x \cdot - 4 - 4 x - 4 \cdot - 4) + 3$

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

$0 = - 3 (x^{2} - 4 \cdot x - 4 x - 4 \cdot - 4) + 3$

Multiply $- 4$ by $- 4$ .

$0 = - 3 (x^{2} - 4 x - 4 x + 16) + 3$

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

$0 = - 3 (x^{2} - 8 x + 16) + 3$

Apply the distributive property.

$0 = - 3 x^{2} - 3 (- 8 x) - 3 \cdot 16 + 3$

Simplify.

Multiply $- 8$ by $- 3$ .

$0 = - 3 x^{2} + 24 x - 3 \cdot 16 + 3$

Multiply $- 3$ by $16$ .

$0 = - 3 x^{2} + 24 x - 48 + 3$

Add $- 48$ and $3$ .

$0 = - 3 x^{2} + 24 x - 45$

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

$x = 3, 5$

x-intercept(s) in point form.

x-intercept(s): $(3, 0), (5, 0)$

Step 3

Find the y-intercepts.

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

$y = - 3 {((0) - 4)}^{2} + 3$

Solve the equation.

Remove parentheses.

$y = - 3 {(0 - 4)}^{2} + 3$

Remove parentheses.

$y = - 3 {((0) - 4)}^{2} + 3$

Simplify $- 3 {((0) - 4)}^{2} + 3$ .

Simplify each term.

Subtract $4$ from $0$ .

$y = - 3 {(- 4)}^{2} + 3$

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

$y = - 3 \cdot 16 + 3$

Multiply $- 3$ by $16$ .

$y = - 48 + 3$

Add $- 48$ and $3$ .

$y = - 45$

y-intercept(s) in point form.

y-intercept(s): $(0, - 45)$

Step 4

List the intersections.

x-intercept(s): $(3, 0), (5, 0)$

y-intercept(s): $(0, - 45)$

Step 5

Find the x and y Intercepts -3(x-4)^2+3

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