Calculus Examples

$y = - x^{2} - 5 x + 6$

Step 1

Find the properties of the given parabola.

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Step 1.1

Rewrite the equation in vertex form.

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Step 1.1.1

Complete the square for $- x^{2} - 5 x + 6$ .

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Step 1.1.1.1

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

$a = - 1$

$b = - 5$

$c = 6$

Step 1.1.1.2

Consider the vertex form of a parabola.

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

Step 1.1.1.3

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

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Step 1.1.1.3.1

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

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

Step 1.1.1.3.2

Simplify the right side.

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Step 1.1.1.3.2.1

Multiply $2$ by $- 1$ .

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

Step 1.1.1.3.2.2

Dividing two negative values results in a positive value.

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

Step 1.1.1.4

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

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Step 1.1.1.4.1

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

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

Step 1.1.1.4.2

Simplify the right side.

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Step 1.1.1.4.2.1

Simplify each term.

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Step 1.1.1.4.2.1.1

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

$e = 6 - \frac{25}{4 \cdot - 1}$

Step 1.1.1.4.2.1.2

Multiply $4$ by $- 1$ .

$e = 6 - \frac{25}{- 4}$

Step 1.1.1.4.2.1.3

Move the negative in front of the fraction.

$e = 6 - - \frac{25}{4}$

Step 1.1.1.4.2.1.4

Multiply $- - \frac{25}{4}$ .

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Step 1.1.1.4.2.1.4.1

Multiply $- 1$ by $- 1$ .

$e = 6 + 1 (\frac{25}{4})$

Step 1.1.1.4.2.1.4.2

Multiply $\frac{25}{4}$ by $1$ .

$e = 6 + \frac{25}{4}$

Step 1.1.1.4.2.2

To write $6$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$e = 6 \cdot \frac{4}{4} + \frac{25}{4}$

Step 1.1.1.4.2.3

Combine $6$ and $\frac{4}{4}$ .

$e = \frac{6 \cdot 4}{4} + \frac{25}{4}$

Step 1.1.1.4.2.4

Combine the numerators over the common denominator.

$e = \frac{6 \cdot 4 + 25}{4}$

Step 1.1.1.4.2.5

Simplify the numerator.

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Step 1.1.1.4.2.5.1

Multiply $6$ by $4$ .

$e = \frac{24 + 25}{4}$

Step 1.1.1.4.2.5.2

Add $24$ and $25$ .

$e = \frac{49}{4}$

Step 1.1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- {(x + \frac{5}{2})}^{2} + \frac{49}{4}$ .

$- {(x + \frac{5}{2})}^{2} + \frac{49}{4}$

Step 1.1.2

Set $y$ equal to the new right side.

$y = - {(x + \frac{5}{2})}^{2} + \frac{49}{4}$

Step 1.2

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

$a = - 1$

$h = - \frac{5}{2}$

$k = \frac{49}{4}$

Step 1.3

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

Opens Down

Step 1.4

Find the vertex $(h, k)$ .

$(- \frac{5}{2}, \frac{49}{4})$

Step 1.5

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

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Step 1.5.1

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

$\frac{1}{4 a}$

Step 1.5.2

Substitute the value of $a$ into the formula.

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

Step 1.5.3

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

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Step 1.5.3.1

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

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

Step 1.5.3.2

Move the negative in front of the fraction.

$- \frac{1}{4}$

Step 1.6

Find the focus.

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Step 1.6.1

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 1.6.2

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

$(- \frac{5}{2}, 12)$

Step 1.7

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

$x = - \frac{5}{2}$

Step 1.8

Find the directrix.

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Step 1.8.1

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 1.8.2

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

$y = \frac{25}{2}$

Step 1.9

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

Direction: Opens Down

Vertex: $(- \frac{5}{2}, \frac{49}{4})$

Focus: $(- \frac{5}{2}, 12)$

Axis of Symmetry: $x = - \frac{5}{2}$

Directrix: $y = \frac{25}{2}$

Direction: Opens Down

Vertex: $(- \frac{5}{2}, \frac{49}{4})$

Focus: $(- \frac{5}{2}, 12)$

Axis of Symmetry: $x = - \frac{5}{2}$

Directrix: $y = \frac{25}{2}$

Step 2

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 2.1

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

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

Step 2.2

Simplify the result.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

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

$f (- 3) = - 1 \cdot 9 - 5 \cdot - 3 + 6$

Step 2.2.1.2

Multiply $- 1$ by $9$ .

$f (- 3) = - 9 - 5 \cdot - 3 + 6$

Step 2.2.1.3

Multiply $- 5$ by $- 3$ .

$f (- 3) = - 9 + 15 + 6$

Step 2.2.2

Simplify by adding numbers.

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Step 2.2.2.1

Add $- 9$ and $15$ .

$f (- 3) = 6 + 6$

Step 2.2.2.2

Add $6$ and $6$ .

$f (- 3) = 12$

Step 2.2.3

The final answer is $12$ .

$12$

Step 2.3

The $y$ value at $x = - 3$ is $12$ .

$y = 12$

Step 2.4

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

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

Step 2.5

Simplify the result.

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Step 2.5.1

Simplify each term.

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Step 2.5.1.1

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

$f (- 4) = - 1 \cdot 16 - 5 \cdot - 4 + 6$

Step 2.5.1.2

Multiply $- 1$ by $16$ .

$f (- 4) = - 16 - 5 \cdot - 4 + 6$

Step 2.5.1.3

Multiply $- 5$ by $- 4$ .

$f (- 4) = - 16 + 20 + 6$

Step 2.5.2

Simplify by adding numbers.

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Step 2.5.2.1

Add $- 16$ and $20$ .

$f (- 4) = 4 + 6$

Step 2.5.2.2

Add $4$ and $6$ .

$f (- 4) = 10$

Step 2.5.3

The final answer is $10$ .

$10$

Step 2.6

The $y$ value at $x = - 4$ is $10$ .

$y = 10$

Step 2.7

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

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

Step 2.8

Simplify the result.

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Step 2.8.1

Simplify each term.

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Step 2.8.1.1

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 2.8.1.1.1

Multiply $- 1$ by ${(- 1)}^{2}$ .

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Step 2.8.1.1.1.1

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

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

Step 2.8.1.1.1.2

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

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

Step 2.8.1.1.2

Add $1$ and $2$ .

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

Step 2.8.1.2

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

$f (- 1) = - 1 - 5 \cdot - 1 + 6$

Step 2.8.1.3

Multiply $- 5$ by $- 1$ .

$f (- 1) = - 1 + 5 + 6$

Step 2.8.2

Simplify by adding numbers.

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Step 2.8.2.1

Add $- 1$ and $5$ .

$f (- 1) = 4 + 6$

Step 2.8.2.2

Add $4$ and $6$ .

$f (- 1) = 10$

Step 2.8.3

The final answer is $10$ .

$10$

Step 2.9

The $y$ value at $x = - 1$ is $10$ .

$y = 10$

Step 2.10

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

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

Step 2.11

Simplify the result.

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Step 2.11.1

Simplify each term.

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Step 2.11.1.1

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

$f (0) = - 0 - 5 \cdot 0 + 6$

Step 2.11.1.2

Multiply $- 1$ by $0$ .

$f (0) = 0 - 5 \cdot 0 + 6$

Step 2.11.1.3

Multiply $- 5$ by $0$ .

$f (0) = 0 + 0 + 6$

Step 2.11.2

Simplify by adding numbers.

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Step 2.11.2.1

Add $0$ and $0$ .

$f (0) = 0 + 6$

Step 2.11.2.2

Add $0$ and $6$ .

$f (0) = 6$

Step 2.11.3

The final answer is $6$ .

$6$

Step 2.12

The $y$ value at $x = 0$ is $6$ .

$y = 6$

Step 2.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 4 & 10 \\ - 3 & 12 \\ - \frac{5}{2} & \frac{49}{4} \\ - 1 & 10 \\ 0 & 6 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(- \frac{5}{2}, \frac{49}{4})$

Focus: $(- \frac{5}{2}, 12)$

Axis of Symmetry: $x = - \frac{5}{2}$

Directrix: $y = \frac{25}{2}$

$\begin{array}{c} x & y \\ - 4 & 10 \\ - 3 & 12 \\ - \frac{5}{2} & \frac{49}{4} \\ - 1 & 10 \\ 0 & 6 \end{array}$

Step 4