Graph y=-3/4x^2+6x+6

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

Step 1

Simplify each term.

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

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

Move $3$ to the left of $x^{2}$ .

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

Step 2

Find the properties of the given parabola.

Rewrite the equation in vertex form.

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

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

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

$b = 6$

$c = 6$

Consider the vertex form of a parabola.

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

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

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

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

Simplify the right side.

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

Factor $2$ out of $6$ .

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

Cancel the common factors.

Cancel the common factor.

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

Rewrite the expression.

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

Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{4}{3}$ into the numerator.

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

Cancel the common factor.

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

Rewrite the expression.

$d = - 4$

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

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

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

Simplify the right side.

Simplify each term.

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

$e = 6 - \frac{36}{4 (- \frac{3}{4})}$

Simplify the denominator.

Multiply $- 1$ by $4$ .

$e = 6 - \frac{36}{- 4 (\frac{3}{4})}$

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

$e = 6 - \frac{36}{\frac{- 4 \cdot 3}{4}}$

Multiply $- 4$ by $3$ .

$e = 6 - \frac{36}{\frac{- 12}{4}}$

Divide $- 12$ by $4$ .

$e = 6 - \frac{36}{- 3}$

Divide $36$ by $- 3$ .

$e = 6 - - 12$

Multiply $- 1$ by $- 12$ .

$e = 6 + 12$

Add $6$ and $12$ .

$e = 18$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{3}{4} {(x - 4)}^{2} + 18$ .

$- \frac{3}{4} {(x - 4)}^{2} + 18$

Set $y$ equal to the new right side.

$y = - \frac{3}{4} \cdot {(x - 4)}^{2} + 18$

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

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

$h = 4$

$k = 18$

Since the value of $a$ is negative, the parabola opens down.

Opens Down

Find the vertex $(h, k)$ .

$(4, 18)$

Find $p$ , the distance from the vertex to the focus.

Find the distance from the vertex to a focus of the parabola by using the following formula.

$\frac{1}{4 a}$

Substitute the value of $a$ into the formula.

$\frac{1}{4 \cdot (- \frac{3}{4})}$

Simplify.

Cancel the common factor of $1$ and $- 1$ .

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- 1 (- 1)}{4 \cdot (- \frac{3}{4})}$

Move the negative in front of the fraction.

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

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

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

Simplify the expression.

Multiply $4$ by $3$ .

$- \frac{1}{\frac{12}{4}}$

Divide $12$ by $4$ .

$- \frac{1}{3}$

Find the focus.

The focus of a parabola can be found by adding $p$ to the y-coordinate $k$ if the parabola opens up or down.

$(h, k + p)$

Substitute the known values of $h$ , $p$ , and $k$ into the formula and simplify.

$(4, \frac{53}{3})$

Find the axis of symmetry by finding the line that passes through the vertex and the focus.

$x = 4$

Find the directrix.

The directrix of a parabola is the horizontal line found by subtracting $p$ from the y-coordinate $k$ of the vertex if the parabola opens up or down.

$y = k - p$

Substitute the known values of $p$ and $k$ into the formula and simplify.

$y = \frac{55}{3}$

Use the properties of the parabola to analyze and graph the parabola.

Direction: Opens Down

Vertex: $(4, 18)$

Focus: $(4, \frac{53}{3})$

Axis of Symmetry: $x = 4$

Directrix: $y = \frac{55}{3}$

Direction: Opens Down

Vertex: $(4, 18)$

Focus: $(4, \frac{53}{3})$

Axis of Symmetry: $x = 4$

Directrix: $y = \frac{55}{3}$

Step 3

Select a few $x$ values, and plug them into the equation to find the corresponding $y$ values. The $x$ values should be selected around the vertex.

Replace the variable $x$ with $2$ in the expression.

$f (2) = - \frac{3 {(2)}^{2}}{4} + 6 (2) + 6$

Simplify the result.

Simplify each term.

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

$f (2) = - \frac{3 \cdot 4}{4} + 6 (2) + 6$

Multiply $3$ by $4$ .

$f (2) = - \frac{12}{4} + 6 (2) + 6$

Divide $12$ by $4$ .

$f (2) = - 1 \cdot 3 + 6 (2) + 6$

Multiply $- 1$ by $3$ .

$f (2) = - 3 + 6 (2) + 6$

Multiply $6$ by $2$ .

$f (2) = - 3 + 12 + 6$

Simplify by adding numbers.

Add $- 3$ and $12$ .

$f (2) = 9 + 6$

Add $9$ and $6$ .

$f (2) = 15$

The final answer is $15$ .

$15$

The $y$ value at $x = 2$ is $15$ .

$y = 15$

Replace the variable $x$ with $3$ in the expression.

$f (3) = - \frac{3 {(3)}^{2}}{4} + 6 (3) + 6$

Simplify the result.

Simplify each term.

Multiply $3$ by ${(3)}^{2}$ by adding the exponents.

Multiply $3$ by ${(3)}^{2}$ .

Raise $3$ to the power of $1$ .

$f (3) = - \frac{3 {(3)}^{2}}{4} + 6 (3) + 6$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (3) = - \frac{3^{1 + 2}}{4} + 6 (3) + 6$

Add $1$ and $2$ .

$f (3) = - \frac{3^{3}}{4} + 6 (3) + 6$

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

$f (3) = - \frac{27}{4} + 6 (3) + 6$

Multiply $6$ by $3$ .

$f (3) = - \frac{27}{4} + 18 + 6$

Find the common denominator.

Write $18$ as a fraction with denominator $1$ .

$f (3) = - \frac{27}{4} + \frac{18}{1} + 6$

Multiply $\frac{18}{1}$ by $\frac{4}{4}$ .

$f (3) = - \frac{27}{4} + \frac{18}{1} \cdot \frac{4}{4} + 6$

Multiply $\frac{18}{1}$ by $\frac{4}{4}$ .

$f (3) = - \frac{27}{4} + \frac{18 \cdot 4}{4} + 6$

Write $6$ as a fraction with denominator $1$ .

$f (3) = - \frac{27}{4} + \frac{18 \cdot 4}{4} + \frac{6}{1}$

Multiply $\frac{6}{1}$ by $\frac{4}{4}$ .

$f (3) = - \frac{27}{4} + \frac{18 \cdot 4}{4} + \frac{6}{1} \cdot \frac{4}{4}$

Multiply $\frac{6}{1}$ by $\frac{4}{4}$ .

$f (3) = - \frac{27}{4} + \frac{18 \cdot 4}{4} + \frac{6 \cdot 4}{4}$

Combine the numerators over the common denominator.

$f (3) = \frac{- 27 + 18 \cdot 4 + 6 \cdot 4}{4}$

Simplify each term.

Multiply $18$ by $4$ .

$f (3) = \frac{- 27 + 72 + 6 \cdot 4}{4}$

Multiply $6$ by $4$ .

$f (3) = \frac{- 27 + 72 + 24}{4}$

Simplify by adding numbers.

Add $- 27$ and $72$ .

$f (3) = \frac{45 + 24}{4}$

Add $45$ and $24$ .

$f (3) = \frac{69}{4}$

The final answer is $\frac{69}{4}$ .

$\frac{69}{4}$

The $y$ value at $x = 3$ is $\frac{69}{4}$ .

$y = \frac{69}{4}$

Replace the variable $x$ with $6$ in the expression.

$f (6) = - \frac{3 {(6)}^{2}}{4} + 6 (6) + 6$

Simplify the result.

Find the common denominator.

Write $6 (6)$ as a fraction with denominator $1$ .

$f (6) = - \frac{3 {(6)}^{2}}{4} + \frac{6 (6)}{1} + 6$

Multiply $\frac{6 (6)}{1}$ by $\frac{4}{4}$ .

$f (6) = - \frac{3 {(6)}^{2}}{4} + \frac{6 (6)}{1} \cdot \frac{4}{4} + 6$

Multiply $\frac{6 (6)}{1}$ by $\frac{4}{4}$ .

$f (6) = - \frac{3 {(6)}^{2}}{4} + \frac{6 (6) \cdot 4}{4} + 6$

Write $6$ as a fraction with denominator $1$ .

$f (6) = - \frac{3 {(6)}^{2}}{4} + \frac{6 (6) \cdot 4}{4} + \frac{6}{1}$

Multiply $\frac{6}{1}$ by $\frac{4}{4}$ .

$f (6) = - \frac{3 {(6)}^{2}}{4} + \frac{6 (6) \cdot 4}{4} + \frac{6}{1} \cdot \frac{4}{4}$

Multiply $\frac{6}{1}$ by $\frac{4}{4}$ .

$f (6) = - \frac{3 {(6)}^{2}}{4} + \frac{6 (6) \cdot 4}{4} + \frac{6 \cdot 4}{4}$

Combine the numerators over the common denominator.

$f (6) = \frac{- 3 {(6)}^{2} + 6 (6) \cdot 4 + 6 \cdot 4}{4}$

Simplify each term.

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

$f (6) = \frac{- 3 \cdot 36 + 6 (6) \cdot 4 + 6 \cdot 4}{4}$

Multiply $- 3$ by $36$ .

$f (6) = \frac{- 108 + 6 (6) \cdot 4 + 6 \cdot 4}{4}$

Multiply $6 (6) \cdot 4$ .

Multiply $6$ by $6$ .

$f (6) = \frac{- 108 + 36 \cdot 4 + 6 \cdot 4}{4}$

Multiply $36$ by $4$ .

$f (6) = \frac{- 108 + 144 + 6 \cdot 4}{4}$

Multiply $6$ by $4$ .

$f (6) = \frac{- 108 + 144 + 24}{4}$

Simplify the expression.

Add $- 108$ and $144$ .

$f (6) = \frac{36 + 24}{4}$

Add $36$ and $24$ .

$f (6) = \frac{60}{4}$

Divide $60$ by $4$ .

$f (6) = 15$

The final answer is $15$ .

$15$

The $y$ value at $x = 6$ is $15$ .

$y = 15$

Replace the variable $x$ with $5$ in the expression.

$f (5) = - \frac{3 {(5)}^{2}}{4} + 6 (5) + 6$

Simplify the result.

Simplify each term.

Raise $5$ to the power of $2$ .

$f (5) = - \frac{3 \cdot 25}{4} + 6 (5) + 6$

Multiply $3$ by $25$ .

$f (5) = - \frac{75}{4} + 6 (5) + 6$

Multiply $6$ by $5$ .

$f (5) = - \frac{75}{4} + 30 + 6$

Find the common denominator.

Write $30$ as a fraction with denominator $1$ .

$f (5) = - \frac{75}{4} + \frac{30}{1} + 6$

Multiply $\frac{30}{1}$ by $\frac{4}{4}$ .

$f (5) = - \frac{75}{4} + \frac{30}{1} \cdot \frac{4}{4} + 6$

Multiply $\frac{30}{1}$ by $\frac{4}{4}$ .

$f (5) = - \frac{75}{4} + \frac{30 \cdot 4}{4} + 6$

Write $6$ as a fraction with denominator $1$ .

$f (5) = - \frac{75}{4} + \frac{30 \cdot 4}{4} + \frac{6}{1}$

Multiply $\frac{6}{1}$ by $\frac{4}{4}$ .

$f (5) = - \frac{75}{4} + \frac{30 \cdot 4}{4} + \frac{6}{1} \cdot \frac{4}{4}$

Multiply $\frac{6}{1}$ by $\frac{4}{4}$ .

$f (5) = - \frac{75}{4} + \frac{30 \cdot 4}{4} + \frac{6 \cdot 4}{4}$

Combine the numerators over the common denominator.

$f (5) = \frac{- 75 + 30 \cdot 4 + 6 \cdot 4}{4}$

Simplify each term.

Multiply $30$ by $4$ .

$f (5) = \frac{- 75 + 120 + 6 \cdot 4}{4}$

Multiply $6$ by $4$ .

$f (5) = \frac{- 75 + 120 + 24}{4}$

Simplify by adding numbers.

Add $- 75$ and $120$ .

$f (5) = \frac{45 + 24}{4}$

Add $45$ and $24$ .

$f (5) = \frac{69}{4}$

The final answer is $\frac{69}{4}$ .

$\frac{69}{4}$

The $y$ value at $x = 5$ is $\frac{69}{4}$ .

$y = \frac{69}{4}$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 2 & 15 \\ 3 & \frac{69}{4} \\ 4 & 18 \\ 5 & \frac{69}{4} \\ 6 & 15 \end{array}$

Step 4

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(4, 18)$

Focus: $(4, \frac{53}{3})$

Axis of Symmetry: $x = 4$

Directrix: $y = \frac{55}{3}$

$\begin{array}{c} x & y \\ 2 & 15 \\ 3 & \frac{69}{4} \\ 4 & 18 \\ 5 & \frac{69}{4} \\ 6 & 15 \end{array}$

Step 5

Graph y=-3/4x^2+6x+6

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