Trigonometry Examples

$y = 2 {(x + 3)}^{2} + 10$

Step 1

Find the x-intercepts.

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Step 1.1

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

$0 = 2 {(x + 3)}^{2} + 10$

Step 1.2

Solve the equation.

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Step 1.2.1

Remove parentheses.

$0 = 2 {(x + 3)}^{2} + 10$

Step 1.2.2

Simplify $2 {(x + 3)}^{2} + 10$ .

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Step 1.2.2.1

Simplify each term.

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Step 1.2.2.1.1

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

$0 = 2 ((x + 3) (x + 3)) + 10$

Step 1.2.2.1.2

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

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Step 1.2.2.1.2.1

Apply the distributive property.

$0 = 2 (x (x + 3) + 3 (x + 3)) + 10$

Step 1.2.2.1.2.2

Apply the distributive property.

$0 = 2 (x \cdot x + x \cdot 3 + 3 (x + 3)) + 10$

Step 1.2.2.1.2.3

Apply the distributive property.

$0 = 2 (x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3) + 10$

Step 1.2.2.1.3

Simplify and combine like terms.

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Step 1.2.2.1.3.1

Simplify each term.

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Step 1.2.2.1.3.1.1

Multiply $x$ by $x$ .

$0 = 2 (x^{2} + x \cdot 3 + 3 x + 3 \cdot 3) + 10$

Step 1.2.2.1.3.1.2

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

$0 = 2 (x^{2} + 3 \cdot x + 3 x + 3 \cdot 3) + 10$

Step 1.2.2.1.3.1.3

Multiply $3$ by $3$ .

$0 = 2 (x^{2} + 3 x + 3 x + 9) + 10$

Step 1.2.2.1.3.2

Add $3 x$ and $3 x$ .

$0 = 2 (x^{2} + 6 x + 9) + 10$

Step 1.2.2.1.4

Apply the distributive property.

$0 = 2 x^{2} + 2 (6 x) + 2 \cdot 9 + 10$

Step 1.2.2.1.5

Simplify.

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Step 1.2.2.1.5.1

Multiply $6$ by $2$ .

$0 = 2 x^{2} + 12 x + 2 \cdot 9 + 10$

Step 1.2.2.1.5.2

Multiply $2$ by $9$ .

$0 = 2 x^{2} + 12 x + 18 + 10$

Step 1.2.2.2

Add $18$ and $10$ .

$0 = 2 x^{2} + 12 x + 28$

Step 1.2.3

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

No solution

Step 1.3

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.

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Step 2.1

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

$y = 2 {((0) + 3)}^{2} + 10$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 2 {(0 + 3)}^{2} + 10$

Step 2.2.2

Remove parentheses.

$y = 2 {((0) + 3)}^{2} + 10$

Step 2.2.3

Simplify $2 {((0) + 3)}^{2} + 10$ .

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Add $0$ and $3$ .

$y = 2 \cdot 3^{2} + 10$

Step 2.2.3.1.2

Raise $3$ to the power of $2$ .

$y = 2 \cdot 9 + 10$

Step 2.2.3.1.3

Multiply $2$ by $9$ .

$y = 18 + 10$

Step 2.2.3.2

Add $18$ and $10$ .

$y = 28$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 28)$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, 28)$

Step 4