Trigonometry Examples

y = - 2 x^{2} - 2 x + 4

$y = - 2 x^{2} - 2 x + 4$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in

0

$0$ for

y

$y$ and solve for

x

$x$ .

0 = - 2 x^{2} - 2 x + 4

$0 = - 2 x^{2} - 2 x + 4$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as

- 2 x^{2} - 2 x + 4 = 0

$- 2 x^{2} - 2 x + 4 = 0$ .

- 2 x^{2} - 2 x + 4 = 0

$- 2 x^{2} - 2 x + 4 = 0$

Step 1.2.2

Factor the left side of the equation.

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Step 1.2.2.1

Factor

- 2

$- 2$ out of

- 2 x^{2} - 2 x + 4

$- 2 x^{2} - 2 x + 4$ .

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Step 1.2.2.1.1

Factor

- 2

$- 2$ out of

- 2 x^{2}

$- 2 x^{2}$ .

- 2 x^{2} - 2 x + 4 = 0

$- 2 x^{2} - 2 x + 4 = 0$

Step 1.2.2.1.2

Factor

- 2

$- 2$ out of

- 2 x

$- 2 x$ .

- 2 x^{2} - 2 x + 4 = 0

$- 2 x^{2} - 2 x + 4 = 0$

Step 1.2.2.1.3

Factor

- 2

$- 2$ out of

4

$4$ .

- 2 x^{2} - 2 x - 2 \cdot - 2 = 0

$- 2 x^{2} - 2 x - 2 \cdot - 2 = 0$

Step 1.2.2.1.4

Factor

- 2

$- 2$ out of

- 2 (x^{2}) - 2 (x)

$- 2 (x^{2}) - 2 (x)$ .

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

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

Step 1.2.2.1.5

Factor

- 2

$- 2$ out of

- 2 (x^{2} + x) - 2 (- 2)

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

- 2 (x^{2} + x - 2) = 0

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

- 2 (x^{2} + x - 2) = 0

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

Step 1.2.2.2

Factor.

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Step 1.2.2.2.1

Factor

x^{2} + x - 2

$x^{2} + x - 2$ using the AC method.

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Step 1.2.2.2.1.1

Consider the form

x^{2} + b x + c

$x^{2} + b x + c$ . Find a pair of integers whose product is

c

$c$ and whose sum is

b

$b$ . In this case, whose product is

- 2

$- 2$ and whose sum is

1

$1$ .

- 1, 2

$- 1, 2$

Step 1.2.2.2.1.2

Write the factored form using these integers.

- 2 ((x - 1) (x + 2)) = 0

$- 2 ((x - 1) (x + 2)) = 0$

- 2 ((x - 1) (x + 2)) = 0

$- 2 ((x - 1) (x + 2)) = 0$

Step 1.2.2.2.2

Remove unnecessary parentheses.

- 2 (x - 1) (x + 2) = 0

$- 2 (x - 1) (x + 2) = 0$

- 2 (x - 1) (x + 2) = 0

$- 2 (x - 1) (x + 2) = 0$

- 2 (x - 1) (x + 2) = 0

$- 2 (x - 1) (x + 2) = 0$

Step 1.2.3

If any individual factor on the left side of the equation is equal to

0

$0$ , the entire expression will be equal to

0

$0$ .

x - 1 = 0

$x - 1 = 0$

x + 2 = 0

$x + 2 = 0$

Step 1.2.4

Set

x - 1

$x - 1$ equal to

0

$0$ and solve for

x

$x$ .

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Step 1.2.4.1

Set

x - 1

$x - 1$ equal to

0

$0$ .

x - 1 = 0

$x - 1 = 0$

Step 1.2.4.2

Add

1

$1$ to both sides of the equation.

x = 1

$x = 1$

x = 1

$x = 1$

Step 1.2.5

Set

x + 2

$x + 2$ equal to

0

$0$ and solve for

x

$x$ .

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Step 1.2.5.1

Set

x + 2

$x + 2$ equal to

0

$0$ .

x + 2 = 0

$x + 2 = 0$

Step 1.2.5.2

Subtract

2

$2$ from both sides of the equation.

x = - 2

$x = - 2$

x = - 2

$x = - 2$

Step 1.2.6

The final solution is all the values that make

- 2 (x - 1) (x + 2) = 0

$- 2 (x - 1) (x + 2) = 0$ true.

x = 1, - 2

$x = 1, - 2$

x = 1, - 2

$x = 1, - 2$

Step 1.3

x-intercept(s) in point form.

x-intercept(s):

(1, 0), (- 2, 0)

$(1, 0), (- 2, 0)$

x-intercept(s):

(1, 0), (- 2, 0)

$(1, 0), (- 2, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in

0

$0$ for

x

$x$ and solve for

y

$y$ .

y = - 2 {(0)}^{2} - 2 \cdot 0 + 4

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

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

y = - 2 \cdot 0^{2} - 2 \cdot 0 + 4

$y = - 2 \cdot 0^{2} - 2 \cdot 0 + 4$

Step 2.2.2

Remove parentheses.

y = - 2 {(0)}^{2} - 2 \cdot 0 + 4

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

Step 2.2.3

Simplify

- 2 {(0)}^{2} - 2 \cdot 0 + 4

$- 2 {(0)}^{2} - 2 \cdot 0 + 4$ .

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Raising

0

$0$ to any positive power yields

0

$0$ .

y = - 2 \cdot 0 - 2 \cdot 0 + 4

$y = - 2 \cdot 0 - 2 \cdot 0 + 4$

Step 2.2.3.1.2

Multiply

- 2

$- 2$ by

0

$0$ .

y = 0 - 2 \cdot 0 + 4

$y = 0 - 2 \cdot 0 + 4$

Step 2.2.3.1.3

Multiply

- 2

$- 2$ by

0

$0$ .

y = 0 + 0 + 4

$y = 0 + 0 + 4$

y = 0 + 0 + 4

$y = 0 + 0 + 4$

Step 2.2.3.2

Simplify by adding numbers.

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Step 2.2.3.2.1

Add

0

$0$ and

0

$0$ .

y = 0 + 4

$y = 0 + 4$

Step 2.2.3.2.2

Add

0

$0$ and

4

$4$ .

y = 4

$y = 4$

y = 4

$y = 4$

y = 4

$y = 4$

y = 4

$y = 4$

Step 2.3

y-intercept(s) in point form.

y-intercept(s):

(0, 4)

$(0, 4)$

y-intercept(s):

(0, 4)

$(0, 4)$

Step 3

List the intersections.

x-intercept(s):

(1, 0), (- 2, 0)

$(1, 0), (- 2, 0)$

y-intercept(s):

(0, 4)

$(0, 4)$

Step 4