Calculus Examples

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

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 $2 x^{2} + 5 x$ .

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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 $c$ .

$a = 2$

$b = 5$

$c = 0$

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

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

Step 1.1.1.3.2

Multiply $2$ by $2$ .

$d = \frac{5}{4}$

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 $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{5^{2}}{4 \cdot 2}$

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 $2$ .

$e = 0 - \frac{25}{4 \cdot 2}$

Step 1.1.1.4.2.1.2

Multiply $4$ by $2$ .

$e = 0 - \frac{25}{8}$

Step 1.1.1.4.2.2

Subtract $\frac{25}{8}$ from $0$ .

$e = - \frac{25}{8}$

Step 1.1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $2 {(x + \frac{5}{4})}^{2} - \frac{25}{8}$ .

$2 {(x + \frac{5}{4})}^{2} - \frac{25}{8}$

Step 1.1.2

Set $y$ equal to the new right side.

$y = 2 {(x + \frac{5}{4})}^{2} - \frac{25}{8}$

Step 1.2

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

$a = 2$

$h = - \frac{5}{4}$

$k = - \frac{25}{8}$

Step 1.3

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

Opens Up

Step 1.4

Find the vertex $(h, k)$ .

$(- \frac{5}{4}, - \frac{25}{8})$

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.

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

Step 1.5.3

Multiply $4$ by $2$ .

$\frac{1}{8}$

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.

$(- \frac{5}{4}, - 3)$

Step 1.7

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

$x = - \frac{5}{4}$

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.

$y = - \frac{13}{4}$

Step 1.9

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

Direction: Opens Up

Vertex: $(- \frac{5}{4}, - \frac{25}{8})$

Focus: $(- \frac{5}{4}, - 3)$

Axis of Symmetry: $x = - \frac{5}{4}$

Directrix: $y = - \frac{13}{4}$

Direction: Opens Up

Vertex: $(- \frac{5}{4}, - \frac{25}{8})$

Focus: $(- \frac{5}{4}, - 3)$

Axis of Symmetry: $x = - \frac{5}{4}$

Directrix: $y = - \frac{13}{4}$

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 $- 2$ in the expression.

$f (- 2) = 2 {(- 2)}^{2} + 5 (- 2)$

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

$f (- 2) = 2 \cdot 4 + 5 (- 2)$

Step 2.2.1.2

Multiply $2$ by $4$ .

$f (- 2) = 8 + 5 (- 2)$

Step 2.2.1.3

Multiply $5$ by $- 2$ .

$f (- 2) = 8 - 10$

Step 2.2.2

Subtract $10$ from $8$ .

$f (- 2) = - 2$

Step 2.2.3

The final answer is $- 2$ .

$- 2$

Step 2.3

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

$y = - 2$

Step 2.4

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

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

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

$f (- 3) = 2 \cdot 9 + 5 (- 3)$

Step 2.5.1.2

Multiply $2$ by $9$ .

$f (- 3) = 18 + 5 (- 3)$

Step 2.5.1.3

Multiply $5$ by $- 3$ .

$f (- 3) = 18 - 15$

Step 2.5.2

Subtract $15$ from $18$ .

$f (- 3) = 3$

Step 2.5.3

The final answer is $3$ .

$3$

Step 2.6

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

$y = 3$

Step 2.7

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

$f (0) = 2 {(0)}^{2} + 5 (0)$

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

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

$f (0) = 2 \cdot 0 + 5 (0)$

Step 2.8.1.2

Multiply $2$ by $0$ .

$f (0) = 0 + 5 (0)$

Step 2.8.1.3

Multiply $5$ by $0$ .

$f (0) = 0 + 0$

Step 2.8.2

Add $0$ and $0$ .

$f (0) = 0$

Step 2.8.3

The final answer is $0$ .

$0$

Step 2.9

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

$y = 0$

Step 2.10

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

$f (1) = 2 {(1)}^{2} + 5 (1)$

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

One to any power is one.

$f (1) = 2 \cdot 1 + 5 (1)$

Step 2.11.1.2

Multiply $2$ by $1$ .

$f (1) = 2 + 5 (1)$

Step 2.11.1.3

Multiply $5$ by $1$ .

$f (1) = 2 + 5$

Step 2.11.2

Add $2$ and $5$ .

$f (1) = 7$

Step 2.11.3

The final answer is $7$ .

$7$

Step 2.12

The $y$ value at $x = 1$ is $7$ .

$y = 7$

Step 2.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 3 & 3 \\ - 2 & - 2 \\ - \frac{5}{4} & - \frac{25}{8} \\ 0 & 0 \\ 1 & 7 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(- \frac{5}{4}, - \frac{25}{8})$

Focus: $(- \frac{5}{4}, - 3)$

Axis of Symmetry: $x = - \frac{5}{4}$

Directrix: $y = - \frac{13}{4}$

$\begin{array}{c} x & y \\ - 3 & 3 \\ - 2 & - 2 \\ - \frac{5}{4} & - \frac{25}{8} \\ 0 & 0 \\ 1 & 7 \end{array}$

Step 4