Trigonometry Examples

$4 x^{2} - 24 x + 4 x^{2} - 20 y = 3$

Step 1

Solve for $y$ .

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Step 1.1

Add $4 x^{2}$ and $4 x^{2}$ .

$8 x^{2} - 24 x - 20 y = 3$

Step 1.2

Move all terms not containing $y$ to the right side of the equation.

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Step 1.2.1

Subtract $8 x^{2}$ from both sides of the equation.

$- 24 x - 20 y = 3 - 8 x^{2}$

Step 1.2.2

Add $24 x$ to both sides of the equation.

$- 20 y = 3 - 8 x^{2} + 24 x$

Step 1.3

Divide each term in $- 20 y = 3 - 8 x^{2} + 24 x$ by $- 20$ and simplify.

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Step 1.3.1

Divide each term in $- 20 y = 3 - 8 x^{2} + 24 x$ by $- 20$ .

$\frac{- 20 y}{- 20} = \frac{3}{- 20} + \frac{- 8 x^{2}}{- 20} + \frac{24 x}{- 20}$

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Cancel the common factor of $- 20$ .

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Step 1.3.2.1.1

Cancel the common factor.

$\frac{- 20 y}{- 20} = \frac{3}{- 20} + \frac{- 8 x^{2}}{- 20} + \frac{24 x}{- 20}$

Step 1.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{3}{- 20} + \frac{- 8 x^{2}}{- 20} + \frac{24 x}{- 20}$

Step 1.3.3

Simplify the right side.

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Step 1.3.3.1

Simplify each term.

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Step 1.3.3.1.1

Move the negative in front of the fraction.

$y = - \frac{3}{20} + \frac{- 8 x^{2}}{- 20} + \frac{24 x}{- 20}$

Step 1.3.3.1.2

Cancel the common factor of $- 8$ and $- 20$ .

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Step 1.3.3.1.2.1

Factor $- 4$ out of $- 8 x^{2}$ .

$y = - \frac{3}{20} + \frac{- 4 (2 x^{2})}{- 20} + \frac{24 x}{- 20}$

Step 1.3.3.1.2.2

Cancel the common factors.

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Step 1.3.3.1.2.2.1

Factor $- 4$ out of $- 20$ .

$y = - \frac{3}{20} + \frac{- 4 (2 x^{2})}{- 4 (5)} + \frac{24 x}{- 20}$

Step 1.3.3.1.2.2.2

Cancel the common factor.

$y = - \frac{3}{20} + \frac{- 4 (2 x^{2})}{- 4 \cdot 5} + \frac{24 x}{- 20}$

Step 1.3.3.1.2.2.3

Rewrite the expression.

$y = - \frac{3}{20} + \frac{2 x^{2}}{5} + \frac{24 x}{- 20}$

Step 1.3.3.1.3

Cancel the common factor of $24$ and $- 20$ .

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Step 1.3.3.1.3.1

Factor $4$ out of $24 x$ .

$y = - \frac{3}{20} + \frac{2 x^{2}}{5} + \frac{4 (6 x)}{- 20}$

Step 1.3.3.1.3.2

Cancel the common factors.

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Step 1.3.3.1.3.2.1

Factor $4$ out of $- 20$ .

$y = - \frac{3}{20} + \frac{2 x^{2}}{5} + \frac{4 (6 x)}{4 (- 5)}$

Step 1.3.3.1.3.2.2

Cancel the common factor.

$y = - \frac{3}{20} + \frac{2 x^{2}}{5} + \frac{4 (6 x)}{4 \cdot - 5}$

Step 1.3.3.1.3.2.3

Rewrite the expression.

$y = - \frac{3}{20} + \frac{2 x^{2}}{5} + \frac{6 x}{- 5}$

Step 1.3.3.1.4

Move the negative in front of the fraction.

$y = - \frac{3}{20} + \frac{2 x^{2}}{5} - \frac{6 x}{5}$

Step 2

Find the properties of the given parabola.

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Step 2.1

Rewrite the equation in vertex form.

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Step 2.1.1

Move $- \frac{3}{20}$ .

$y = \frac{2 x^{2}}{5} - \frac{6 x}{5} - \frac{3}{20}$

Step 2.1.2

Complete the square for $\frac{2 x^{2}}{5} - \frac{6 x}{5} - \frac{3}{20}$ .

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Step 2.1.2.1

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

$a = \frac{2}{5}$

$b = - \frac{6}{5}$

$c = - \frac{3}{20}$

Step 2.1.2.2

Consider the vertex form of a parabola.

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

Step 2.1.2.3

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

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Step 2.1.2.3.1

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

$d = \frac{- \frac{6}{5}}{2 (\frac{2}{5})}$

Step 2.1.2.3.2

Simplify the right side.

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Step 2.1.2.3.2.1

Multiply the numerator by the reciprocal of the denominator.

$d = - \frac{6}{5} \cdot \frac{1}{2 (\frac{2}{5})}$

Step 2.1.2.3.2.2

Cancel the common factor of $2$ .

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Step 2.1.2.3.2.2.1

Move the leading negative in $- \frac{6}{5}$ into the numerator.

$d = \frac{- 6}{5} \cdot \frac{1}{2 (\frac{2}{5})}$

Step 2.1.2.3.2.2.2

Factor $2$ out of $- 6$ .

$d = \frac{2 (- 3)}{5} \cdot \frac{1}{2 (\frac{2}{5})}$

Step 2.1.2.3.2.2.3

Cancel the common factor.

$d = \frac{2 \cdot - 3}{5} \cdot \frac{1}{2 (\frac{2}{5})}$

Step 2.1.2.3.2.2.4

Rewrite the expression.

$d = \frac{- 3}{5} \cdot \frac{1}{\frac{2}{5}}$

Step 2.1.2.3.2.3

Multiply $\frac{- 3}{5}$ by $\frac{1}{\frac{2}{5}}$ .

$d = \frac{- 3}{5 (\frac{2}{5})}$

Step 2.1.2.3.2.4

Combine $5$ and $\frac{2}{5}$ .

$d = \frac{- 3}{\frac{5 \cdot 2}{5}}$

Step 2.1.2.3.2.5

Multiply $5$ by $2$ .

$d = \frac{- 3}{\frac{10}{5}}$

Step 2.1.2.3.2.6

Cancel the common factor of $10$ and $5$ .

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Step 2.1.2.3.2.6.1

Factor $5$ out of $10$ .

$d = \frac{- 3}{\frac{5 \cdot 2}{5}}$

Step 2.1.2.3.2.6.2

Cancel the common factors.

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Step 2.1.2.3.2.6.2.1

Factor $5$ out of $5$ .

$d = \frac{- 3}{\frac{5 \cdot 2}{5 (1)}}$

Step 2.1.2.3.2.6.2.2

Cancel the common factor.

$d = \frac{- 3}{\frac{5 \cdot 2}{5 \cdot 1}}$

Step 2.1.2.3.2.6.2.3

Rewrite the expression.

$d = \frac{- 3}{\frac{2}{1}}$

Step 2.1.2.3.2.6.2.4

Divide $2$ by $1$ .

$d = \frac{- 3}{2}$

Step 2.1.2.3.2.7

Move the negative in front of the fraction.

$d = - \frac{3}{2}$

Step 2.1.2.4

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

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Step 2.1.2.4.1

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

$e = - \frac{3}{20} - \frac{{(- \frac{6}{5})}^{2}}{4 (\frac{2}{5})}$

Step 2.1.2.4.2

Simplify the right side.

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Step 2.1.2.4.2.1

Simplify each term.

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Step 2.1.2.4.2.1.1

Simplify the numerator.

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Step 2.1.2.4.2.1.1.1

Apply the product rule to $- \frac{6}{5}$ .

$e = - \frac{3}{20} - \frac{{(- 1)}^{2} {(\frac{6}{5})}^{2}}{4 (\frac{2}{5})}$

Step 2.1.2.4.2.1.1.2

Raise $- 1$ to the power of $2$ .

$e = - \frac{3}{20} - \frac{1 {(\frac{6}{5})}^{2}}{4 (\frac{2}{5})}$

Step 2.1.2.4.2.1.1.3

Apply the product rule to $\frac{6}{5}$ .

$e = - \frac{3}{20} - \frac{1 \frac{6^{2}}{5^{2}}}{4 (\frac{2}{5})}$

Step 2.1.2.4.2.1.1.4

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

$e = - \frac{3}{20} - \frac{1 \frac{36}{5^{2}}}{4 (\frac{2}{5})}$

Step 2.1.2.4.2.1.1.5

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

$e = - \frac{3}{20} - \frac{1 (\frac{36}{25})}{4 (\frac{2}{5})}$

Step 2.1.2.4.2.1.1.6

Multiply $\frac{36}{25}$ by $1$ .

$e = - \frac{3}{20} - \frac{\frac{36}{25}}{4 (\frac{2}{5})}$

Step 2.1.2.4.2.1.2

Combine $4$ and $\frac{2}{5}$ .

$e = - \frac{3}{20} - \frac{\frac{36}{25}}{\frac{4 \cdot 2}{5}}$

Step 2.1.2.4.2.1.3

Multiply $4$ by $2$ .

$e = - \frac{3}{20} - \frac{\frac{36}{25}}{\frac{8}{5}}$

Step 2.1.2.4.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = - \frac{3}{20} - (\frac{36}{25} \cdot \frac{5}{8})$

Step 2.1.2.4.2.1.5

Cancel the common factor of $4$ .

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Step 2.1.2.4.2.1.5.1

Factor $4$ out of $36$ .

$e = - \frac{3}{20} - (\frac{4 (9)}{25} \cdot \frac{5}{8})$

Step 2.1.2.4.2.1.5.2

Factor $4$ out of $8$ .

$e = - \frac{3}{20} - (\frac{4 \cdot 9}{25} \cdot \frac{5}{4 \cdot 2})$

Step 2.1.2.4.2.1.5.3

Cancel the common factor.

$e = - \frac{3}{20} - (\frac{4 \cdot 9}{25} \cdot \frac{5}{4 \cdot 2})$

Step 2.1.2.4.2.1.5.4

Rewrite the expression.

$e = - \frac{3}{20} - (\frac{9}{25} \cdot \frac{5}{2})$

Step 2.1.2.4.2.1.6

Cancel the common factor of $5$ .

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Step 2.1.2.4.2.1.6.1

Factor $5$ out of $25$ .

$e = - \frac{3}{20} - (\frac{9}{5 (5)} \cdot \frac{5}{2})$

Step 2.1.2.4.2.1.6.2

Cancel the common factor.

$e = - \frac{3}{20} - (\frac{9}{5 \cdot 5} \cdot \frac{5}{2})$

Step 2.1.2.4.2.1.6.3

Rewrite the expression.

$e = - \frac{3}{20} - (\frac{9}{5} \cdot \frac{1}{2})$

Step 2.1.2.4.2.1.7

Multiply $\frac{9}{5}$ by $\frac{1}{2}$ .

$e = - \frac{3}{20} - \frac{9}{5 \cdot 2}$

Step 2.1.2.4.2.1.8

Multiply $5$ by $2$ .

$e = - \frac{3}{20} - \frac{9}{10}$

Step 2.1.2.4.2.2

To write $- \frac{9}{10}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$e = - \frac{3}{20} - \frac{9}{10} \cdot \frac{2}{2}$

Step 2.1.2.4.2.3

Write each expression with a common denominator of $20$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.1.2.4.2.3.1

Multiply $\frac{9}{10}$ by $\frac{2}{2}$ .

$e = - \frac{3}{20} - \frac{9 \cdot 2}{10 \cdot 2}$

Step 2.1.2.4.2.3.2

Multiply $10$ by $2$ .

$e = - \frac{3}{20} - \frac{9 \cdot 2}{20}$

Step 2.1.2.4.2.4

Combine the numerators over the common denominator.

$e = \frac{- 3 - 9 \cdot 2}{20}$

Step 2.1.2.4.2.5

Simplify the numerator.

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Step 2.1.2.4.2.5.1

Multiply $- 9$ by $2$ .

$e = \frac{- 3 - 18}{20}$

Step 2.1.2.4.2.5.2

Subtract $18$ from $- 3$ .

$e = \frac{- 21}{20}$

Step 2.1.2.4.2.6

Move the negative in front of the fraction.

$e = - \frac{21}{20}$

Step 2.1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{2}{5} {(x - \frac{3}{2})}^{2} - \frac{21}{20}$ .

$\frac{2}{5} {(x - \frac{3}{2})}^{2} - \frac{21}{20}$

Step 2.1.3

Set $y$ equal to the new right side.

$y = \frac{2}{5} \cdot {(x - \frac{3}{2})}^{2} - \frac{21}{20}$

Step 2.2

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

$a = \frac{2}{5}$

$h = \frac{3}{2}$

$k = - \frac{21}{20}$

Step 2.3

Since the value of $a$ is positive, the parabola opens up.

Opens Up

Step 2.4

Find the vertex $(h, k)$ .

$(\frac{3}{2}, - \frac{21}{20})$

Step 2.5

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

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Step 2.5.1

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

$\frac{1}{4 a}$

Step 2.5.2

Substitute the value of $a$ into the formula.

$\frac{1}{4 \cdot \frac{2}{5}}$

Step 2.5.3

Simplify.

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Step 2.5.3.1

Combine $4$ and $\frac{2}{5}$ .

$\frac{1}{\frac{4 \cdot 2}{5}}$

Step 2.5.3.2

Multiply $4$ by $2$ .

$\frac{1}{\frac{8}{5}}$

Step 2.5.3.3

Multiply the numerator by the reciprocal of the denominator.

$1 (\frac{5}{8})$

Step 2.5.3.4

Multiply $\frac{5}{8}$ by $1$ .

$\frac{5}{8}$

Step 2.6

Find the focus.

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Step 2.6.1

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

$(h, k + p)$

Step 2.6.2

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

$(\frac{3}{2}, - \frac{17}{40})$

Step 2.7

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

$x = \frac{3}{2}$

Step 2.8

Find the directrix.

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Step 2.8.1

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.

$y = k - p$

Step 2.8.2

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

$y = - \frac{67}{40}$

Step 2.9

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

Direction: Opens Up

Vertex: $(\frac{3}{2}, - \frac{21}{20})$

Focus: $(\frac{3}{2}, - \frac{17}{40})$

Axis of Symmetry: $x = \frac{3}{2}$

Directrix: $y = - \frac{67}{40}$

Direction: Opens Up

Vertex: $(\frac{3}{2}, - \frac{21}{20})$

Focus: $(\frac{3}{2}, - \frac{17}{40})$

Axis of Symmetry: $x = \frac{3}{2}$

Directrix: $y = - \frac{67}{40}$

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.

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Step 3.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = \frac{2 {(1)}^{2}}{5} - \frac{6 (1)}{5} - \frac{3}{20}$

Step 3.2

Simplify the result.

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Step 3.2.1

Combine the numerators over the common denominator.

$f (1) = \frac{2 {(1)}^{2} - 6 \cdot 1}{5} + \frac{- 3}{20}$

Step 3.2.2

Simplify each term.

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Step 3.2.2.1

One to any power is one.

$f (1) = \frac{2 \cdot 1 - 6 \cdot 1}{5} + \frac{- 3}{20}$

Step 3.2.2.2

Multiply $2$ by $1$ .

$f (1) = \frac{2 - 6 \cdot 1}{5} + \frac{- 3}{20}$

Step 3.2.2.3

Multiply $- 6$ by $1$ .

$f (1) = \frac{2 - 6}{5} + \frac{- 3}{20}$

Step 3.2.3

Subtract $6$ from $2$ .

$f (1) = \frac{- 4}{5} + \frac{- 3}{20}$

Step 3.2.4

Simplify each term.

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Step 3.2.4.1

Move the negative in front of the fraction.

$f (1) = - \frac{4}{5} + \frac{- 3}{20}$

Step 3.2.4.2

Move the negative in front of the fraction.

$f (1) = - \frac{4}{5} - \frac{3}{20}$

Step 3.2.5

To write $- \frac{4}{5}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$f (1) = - \frac{4}{5} \cdot \frac{4}{4} - \frac{3}{20}$

Step 3.2.6

Write each expression with a common denominator of $20$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.2.6.1

Multiply $\frac{4}{5}$ by $\frac{4}{4}$ .

$f (1) = - \frac{4 \cdot 4}{5 \cdot 4} - \frac{3}{20}$

Step 3.2.6.2

Multiply $5$ by $4$ .

$f (1) = - \frac{4 \cdot 4}{20} - \frac{3}{20}$

Step 3.2.7

Combine the numerators over the common denominator.

$f (1) = \frac{- 4 \cdot 4 - 3}{20}$

Step 3.2.8

Simplify the numerator.

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Step 3.2.8.1

Multiply $- 4$ by $4$ .

$f (1) = \frac{- 16 - 3}{20}$

Step 3.2.8.2

Subtract $3$ from $- 16$ .

$f (1) = \frac{- 19}{20}$

Step 3.2.9

Move the negative in front of the fraction.

$f (1) = - \frac{19}{20}$

Step 3.2.10

The final answer is $- \frac{19}{20}$ .

$- \frac{19}{20}$

Step 3.3

The $y$ value at $x = 1$ is $- \frac{19}{20}$ .

$y = - \frac{19}{20}$

Step 3.4

Replace the variable $x$ with $0$ in the expression.

$f (0) = \frac{2 {(0)}^{2}}{5} - \frac{6 (0)}{5} - \frac{3}{20}$

Step 3.5

Simplify the result.

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Step 3.5.1

Combine the numerators over the common denominator.

$f (0) = \frac{2 {(0)}^{2} - 6 \cdot 0}{5} + \frac{- 3}{20}$

Step 3.5.2

Simplify each term.

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Step 3.5.2.1

Raising $0$ to any positive power yields $0$ .

$f (0) = \frac{2 \cdot 0 - 6 \cdot 0}{5} + \frac{- 3}{20}$

Step 3.5.2.2

Multiply $2$ by $0$ .

$f (0) = \frac{0 - 6 \cdot 0}{5} + \frac{- 3}{20}$

Step 3.5.2.3

Multiply $- 6$ by $0$ .

$f (0) = \frac{0 + 0}{5} + \frac{- 3}{20}$

Step 3.5.3

Add $0$ and $0$ .

$f (0) = \frac{0}{5} + \frac{- 3}{20}$

Step 3.5.4

Simplify each term.

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Step 3.5.4.1

Divide $0$ by $5$ .

$f (0) = 0 + \frac{- 3}{20}$

Step 3.5.4.2

Move the negative in front of the fraction.

$f (0) = 0 - \frac{3}{20}$

Step 3.5.5

Subtract $\frac{3}{20}$ from $0$ .

$f (0) = - \frac{3}{20}$

Step 3.5.6

The final answer is $- \frac{3}{20}$ .

$- \frac{3}{20}$

Step 3.6

The $y$ value at $x = 0$ is $- \frac{3}{20}$ .

$y = - \frac{3}{20}$

Step 3.7

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

$f (3) = \frac{2 {(3)}^{2}}{5} - \frac{6 (3)}{5} - \frac{3}{20}$

Step 3.8

Simplify the result.

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Step 3.8.1

Combine the numerators over the common denominator.

$f (3) = \frac{2 {(3)}^{2} - 6 \cdot 3}{5} + \frac{- 3}{20}$

Step 3.8.2

Simplify each term.

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Step 3.8.2.1

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

$f (3) = \frac{2 \cdot 9 - 6 \cdot 3}{5} + \frac{- 3}{20}$

Step 3.8.2.2

Multiply $2$ by $9$ .

$f (3) = \frac{18 - 6 \cdot 3}{5} + \frac{- 3}{20}$

Step 3.8.2.3

Multiply $- 6$ by $3$ .

$f (3) = \frac{18 - 18}{5} + \frac{- 3}{20}$

Step 3.8.3

Subtract $18$ from $18$ .

$f (3) = \frac{0}{5} + \frac{- 3}{20}$

Step 3.8.4

Simplify each term.

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Step 3.8.4.1

Divide $0$ by $5$ .

$f (3) = 0 + \frac{- 3}{20}$

Step 3.8.4.2

Move the negative in front of the fraction.

$f (3) = 0 - \frac{3}{20}$

Step 3.8.5

Subtract $\frac{3}{20}$ from $0$ .

$f (3) = - \frac{3}{20}$

Step 3.8.6

The final answer is $- \frac{3}{20}$ .

$- \frac{3}{20}$

Step 3.9

The $y$ value at $x = 3$ is $- \frac{3}{20}$ .

$y = - \frac{3}{20}$

Step 3.10

Replace the variable $x$ with $4$ in the expression.

$f (4) = \frac{2 {(4)}^{2}}{5} - \frac{6 (4)}{5} - \frac{3}{20}$

Step 3.11

Simplify the result.

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Step 3.11.1

Combine the numerators over the common denominator.

$f (4) = \frac{2 {(4)}^{2} - 6 \cdot 4}{5} + \frac{- 3}{20}$

Step 3.11.2

Simplify each term.

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Step 3.11.2.1

Raise $4$ to the power of $2$ .

$f (4) = \frac{2 \cdot 16 - 6 \cdot 4}{5} + \frac{- 3}{20}$

Step 3.11.2.2

Multiply $2$ by $16$ .

$f (4) = \frac{32 - 6 \cdot 4}{5} + \frac{- 3}{20}$

Step 3.11.2.3

Multiply $- 6$ by $4$ .

$f (4) = \frac{32 - 24}{5} + \frac{- 3}{20}$

Step 3.11.3

Simplify the expression.

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Step 3.11.3.1

Subtract $24$ from $32$ .

$f (4) = \frac{8}{5} + \frac{- 3}{20}$

Step 3.11.3.2

Move the negative in front of the fraction.

$f (4) = \frac{8}{5} - \frac{3}{20}$

Step 3.11.4

To write $\frac{8}{5}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$f (4) = \frac{8}{5} \cdot \frac{4}{4} - \frac{3}{20}$

Step 3.11.5

Write each expression with a common denominator of $20$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.11.5.1

Multiply $\frac{8}{5}$ by $\frac{4}{4}$ .

$f (4) = \frac{8 \cdot 4}{5 \cdot 4} - \frac{3}{20}$

Step 3.11.5.2

Multiply $5$ by $4$ .

$f (4) = \frac{8 \cdot 4}{20} - \frac{3}{20}$

Step 3.11.6

Combine the numerators over the common denominator.

$f (4) = \frac{8 \cdot 4 - 3}{20}$

Step 3.11.7

Simplify the numerator.

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Step 3.11.7.1

Multiply $8$ by $4$ .

$f (4) = \frac{32 - 3}{20}$

Step 3.11.7.2

Subtract $3$ from $32$ .

$f (4) = \frac{29}{20}$

Step 3.11.8

The final answer is $\frac{29}{20}$ .

$\frac{29}{20}$

Step 3.12

The $y$ value at $x = 4$ is $\frac{29}{20}$ .

$y = \frac{29}{20}$

Step 3.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 0 & - \frac{3}{20} \\ 1 & - \frac{19}{20} \\ \frac{3}{2} & - \frac{21}{20} \\ 3 & - \frac{3}{20} \\ 4 & \frac{29}{20} \end{array}$

Step 4

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(\frac{3}{2}, - \frac{21}{20})$

Focus: $(\frac{3}{2}, - \frac{17}{40})$

Axis of Symmetry: $x = \frac{3}{2}$

Directrix: $y = - \frac{67}{40}$

$\begin{array}{c} x & y \\ 0 & - \frac{3}{20} \\ 1 & - \frac{19}{20} \\ \frac{3}{2} & - \frac{21}{20} \\ 3 & - \frac{3}{20} \\ 4 & \frac{29}{20} \end{array}$

Step 5