Trigonometry Examples

$y = \frac{3}{4} x^{2} - 6 x + 15$

Step 1

Rewrite the equation in vertex form.

Tap for more steps...

Step 1.1

Combine $\frac{3}{4}$ and $x^{2}$ .

$y = \frac{3 x^{2}}{4} - 6 x + 15$

Step 1.2

Complete the square for $\frac{3 x^{2}}{4} - 6 x + 15$ .

Tap for more steps...

Step 1.2.1

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

$a = \frac{3}{4}$

$b = - 6$

$c = 15$

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 $d = \frac{b}{2 a}$ .

Tap for more steps...

Step 1.2.3.1

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

$d = \frac{- 6}{2 (\frac{3}{4})}$

Step 1.2.3.2

Simplify the right side.

Tap for more steps...

Step 1.2.3.2.1

Cancel the common factor of $- 6$ and $2$ .

Tap for more steps...

Step 1.2.3.2.1.1

Factor $2$ out of $- 6$ .

$d = \frac{2 \cdot - 3}{2 (\frac{3}{4})}$

Step 1.2.3.2.1.2

Cancel the common factors.

Tap for more steps...

Step 1.2.3.2.1.2.1

Cancel the common factor.

$d = \frac{2 \cdot - 3}{2 (\frac{3}{4})}$

Step 1.2.3.2.1.2.2

Rewrite the expression.

$d = \frac{- 3}{\frac{3}{4}}$

Step 1.2.3.2.2

Multiply the numerator by the reciprocal of the denominator.

$d = - 3 (\frac{4}{3})$

Step 1.2.3.2.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 1.2.3.2.3.1

Factor $3$ out of $- 3$ .

$d = 3 (- 1) \frac{4}{3}$

Step 1.2.3.2.3.2

Cancel the common factor.

$d = 3 \cdot - 1 \frac{4}{3}$

Step 1.2.3.2.3.3

Rewrite the expression.

$d = - 1 \cdot 4$

Step 1.2.3.2.4

Multiply $- 1$ by $4$ .

$d = - 4$

Step 1.2.4

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

Tap for more steps...

Step 1.2.4.1

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

$e = 15 - \frac{{(- 6)}^{2}}{4 (\frac{3}{4})}$

Step 1.2.4.2

Simplify the right side.

Tap for more steps...

Step 1.2.4.2.1

Simplify each term.

Tap for more steps...

Step 1.2.4.2.1.1

Raise $- 6$ to the power of $2$ .

$e = 15 - \frac{36}{4 (\frac{3}{4})}$

Step 1.2.4.2.1.2

Combine $4$ and $\frac{3}{4}$ .

$e = 15 - \frac{36}{\frac{4 \cdot 3}{4}}$

Step 1.2.4.2.1.3

Multiply $4$ by $3$ .

$e = 15 - \frac{36}{\frac{12}{4}}$

Step 1.2.4.2.1.4

Divide $12$ by $4$ .

$e = 15 - \frac{36}{3}$

Step 1.2.4.2.1.5

Divide $36$ by $3$ .

$e = 15 - 1 \cdot 12$

Step 1.2.4.2.1.6

Multiply $- 1$ by $12$ .

$e = 15 - 12$

Step 1.2.4.2.2

Subtract $12$ from $15$ .

$e = 3$

Step 1.2.5

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

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

Step 1.3

Set $y$ equal to the new right side.

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

Step 2

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

$a = \frac{3}{4}$

$h = 4$

$k = 3$

Step 3

Find the vertex $(h, k)$ .

$(4, 3)$

Step 4