Trigonometry Examples

$y = - \frac{1}{4} x^{2} + 4 x - 19$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Combine $x^{2}$ and $\frac{1}{4}$ .

$y = - \frac{x^{2}}{4} + 4 x - 19$

Step 1.2

Complete the square for $- \frac{x^{2}}{4} + 4 x - 19$ .

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Step 1.2.1

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = - \frac{1}{4}$

$b = 4$

$c = - 19$

Step 1.2.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 1.2.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 1.2.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{4}{2 (- \frac{1}{4})}$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Cancel the common factor of $4$ and $2$ .

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Step 1.2.3.2.1.1

Factor $2$ out of $4$ .

$d = \frac{2 \cdot 2}{2 (- \frac{1}{4})}$

Step 1.2.3.2.1.2

Cancel the common factors.

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Step 1.2.3.2.1.2.1

Cancel the common factor.

$d = \frac{2 \cdot 2}{2 (- \frac{1}{4})}$

Step 1.2.3.2.1.2.2

Rewrite the expression.

$d = \frac{2}{- \frac{1}{4}}$

Step 1.2.3.2.2

Multiply the numerator by the reciprocal of the denominator.

$d = 2 (- 1 \cdot 4)$

Step 1.2.3.2.3

Multiply $2 (- 1 \cdot 4)$ .

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Step 1.2.3.2.3.1

Multiply $- 1$ by $4$ .

$d = 2 \cdot - 4$

Step 1.2.3.2.3.2

Multiply $2$ by $- 4$ .

$d = - 8$

Step 1.2.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 1.2.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = - 19 - \frac{4^{2}}{4 (- \frac{1}{4})}$

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Simplify each term.

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Step 1.2.4.2.1.1

Cancel the common factor of $4^{2}$ and $4$ .

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Step 1.2.4.2.1.1.1

Factor $4$ out of $4^{2}$ .

$e = - 19 - \frac{4 \cdot 4}{4 (- \frac{1}{4})}$

Step 1.2.4.2.1.1.2

Cancel the common factors.

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Step 1.2.4.2.1.1.2.1

Cancel the common factor.

$e = - 19 - \frac{4 \cdot 4}{4 (- \frac{1}{4})}$

Step 1.2.4.2.1.1.2.2

Rewrite the expression.

$e = - 19 - \frac{4}{- \frac{1}{4}}$

Step 1.2.4.2.1.2

Multiply the numerator by the reciprocal of the denominator.

$e = - 19 - (4 (- 1 \cdot 4))$

Step 1.2.4.2.1.3

Multiply $- (4 (- 1 \cdot 4))$ .

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Step 1.2.4.2.1.3.1

Multiply $- 1$ by $4$ .

$e = - 19 - (4 \cdot - 4)$

Step 1.2.4.2.1.3.2

Multiply $4$ by $- 4$ .

$e = - 19 - - 16$

Step 1.2.4.2.1.3.3

Multiply $- 1$ by $- 16$ .

$e = - 19 + 16$

Step 1.2.4.2.2

Add $- 19$ and $16$ .

$e = - 3$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{4} {(x - 8)}^{2} - 3$ .

$- \frac{1}{4} {(x - 8)}^{2} - 3$

Step 1.3

Set $y$ equal to the new right side.

$y = - \frac{1}{4} \cdot {(x - 8)}^{2} - 3$

Step 2

Use the vertex form, $y = a {(x - h)}^{2} + k$ , to determine the values of $a$ , $h$ , and $k$ .

$a = - \frac{1}{4}$

$h = 8$

$k = - 3$

Step 3

Find the vertex $(h, k)$ .

$(8, - 3)$

Step 4