Find the x and y Intercepts h(x)=0.5(x+4)^2+1

$h (x) = 0.5 {(x + 4)}^{2} + 1$

Step 1

Find the x-intercepts.

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

$0 = 0.5 {(x + 4)}^{2} + 1$

Solve the equation.

Remove parentheses.

$0 = 0.5 {(x + 4)}^{2} + 1$

Simplify $0.5 {(x + 4)}^{2} + 1$ .

Simplify each term.

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

$0 = 0.5 ((x + 4) (x + 4)) + 1$

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

Apply the distributive property.

$0 = 0.5 (x (x + 4) + 4 (x + 4)) + 1$

Apply the distributive property.

$0 = 0.5 (x \cdot x + x \cdot 4 + 4 (x + 4)) + 1$

Apply the distributive property.

$0 = 0.5 (x \cdot x + x \cdot 4 + 4 x + 4 \cdot 4) + 1$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$0 = 0.5 (x^{2} + x \cdot 4 + 4 x + 4 \cdot 4) + 1$

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

$0 = 0.5 (x^{2} + 4 \cdot x + 4 x + 4 \cdot 4) + 1$

Multiply $4$ by $4$ .

$0 = 0.5 (x^{2} + 4 x + 4 x + 16) + 1$

Add $4 x$ and $4 x$ .

$0 = 0.5 (x^{2} + 8 x + 16) + 1$

Apply the distributive property.

$0 = 0.5 x^{2} + 0.5 (8 x) + 0.5 \cdot 16 + 1$

Simplify.

Multiply $8$ by $0.5$ .

$0 = 0.5 x^{2} + 4 x + 0.5 \cdot 16 + 1$

Multiply $0.5$ by $16$ .

$0 = 0.5 x^{2} + 4 x + 8 + 1$

Add $8$ and $1$ .

$0 = 0.5 x^{2} + 4 x + 9$

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

No solution

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

x-intercept(s): $None$

Step 2

Find the y-intercepts.

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

$y = 0.5 {((0) + 4)}^{2} + 1$

Solve the equation.

Remove parentheses.

$y = 0.5 {(0 + 4)}^{2} + 1$

Remove parentheses.

$y = 0.5 {((0) + 4)}^{2} + 1$

Simplify $0.5 {((0) + 4)}^{2} + 1$ .

Simplify each term.

Add $0$ and $4$ .

$y = 0.5 \cdot 4^{2} + 1$

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

$y = 0.5 \cdot 16 + 1$

Multiply $0.5$ by $16$ .

$y = 8 + 1$

Add $8$ and $1$ .

$y = 9$

y-intercept(s) in point form.

y-intercept(s): $(0, 9)$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, 9)$

Step 4

Find the x and y Intercepts h(x)=0.5(x+4)^2+1

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