Calculus Examples

$f (x) = \frac{5}{2} x^{2}$

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

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

$y = \frac{5 x^{2}}{2}$

Step 1.1.2

Complete the square for $\frac{5 x^{2}}{2}$ .

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Step 1.1.2.1

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

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

$b = 0$

$c = 0$

Step 1.1.2.2

Consider the vertex form of a parabola.

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

Step 1.1.2.3

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

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Step 1.1.2.3.1

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

$d = \frac{0}{2 (\frac{5}{2})}$

Step 1.1.2.3.2

Simplify the right side.

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Step 1.1.2.3.2.1

Cancel the common factor of $0$ and $2$ .

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Step 1.1.2.3.2.1.1

Factor $2$ out of $0$ .

$d = \frac{2 (0)}{2 (\frac{5}{2})}$

Step 1.1.2.3.2.1.2

Cancel the common factors.

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Step 1.1.2.3.2.1.2.1

Cancel the common factor.

$d = \frac{2 \cdot 0}{2 (\frac{5}{2})}$

Step 1.1.2.3.2.1.2.2

Rewrite the expression.

$d = \frac{0}{\frac{5}{2}}$

Step 1.1.2.3.2.2

Multiply the numerator by the reciprocal of the denominator.

$d = 0 (\frac{2}{5})$

Step 1.1.2.3.2.3

Multiply $0$ by $\frac{2}{5}$ .

$d = 0$

Step 1.1.2.4

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

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Step 1.1.2.4.1

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

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

Step 1.1.2.4.2

Simplify the right side.

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Step 1.1.2.4.2.1

Simplify each term.

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Step 1.1.2.4.2.1.1

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

$e = 0 - \frac{0}{4 (\frac{5}{2})}$

Step 1.1.2.4.2.1.2

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

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

Step 1.1.2.4.2.1.3

Multiply $4$ by $5$ .

$e = 0 - \frac{0}{\frac{20}{2}}$

Step 1.1.2.4.2.1.4

Divide $20$ by $2$ .

$e = 0 - \frac{0}{10}$

Step 1.1.2.4.2.1.5

Divide $0$ by $10$ .

$e = 0 - 0$

Step 1.1.2.4.2.1.6

Multiply $- 1$ by $0$ .

$e = 0 + 0$

Step 1.1.2.4.2.2

Add $0$ and $0$ .

$e = 0$

Step 1.1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{5}{2} x^{2}$ .

$\frac{5}{2} x^{2}$

Step 1.1.3

Set $y$ equal to the new right side.

$y = \frac{5}{2} x^{2}$

Step 1.2

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

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

$h = 0$

$k = 0$

Step 1.3

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

Opens Up

Step 1.4

Find the vertex $(h, k)$ .

$(0, 0)$

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.

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

Step 1.5.3

Simplify.

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Step 1.5.3.1

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

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

Step 1.5.3.2

Simplify the expression.

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Step 1.5.3.2.1

Multiply $4$ by $5$ .

$\frac{1}{\frac{20}{2}}$

Step 1.5.3.2.2

Divide $20$ by $2$ .

$\frac{1}{10}$

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.

$(0, \frac{1}{10})$

Step 1.7

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

$x = 0$

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.

$y = - \frac{1}{10}$

Step 1.9

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

Direction: Opens Up

Vertex: $(0, 0)$

Focus: $(0, \frac{1}{10})$

Axis of Symmetry: $x = 0$

Directrix: $y = - \frac{1}{10}$

Direction: Opens Up

Vertex: $(0, 0)$

Focus: $(0, \frac{1}{10})$

Axis of Symmetry: $x = 0$

Directrix: $y = - \frac{1}{10}$

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.

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

Step 2.2

Simplify the result.

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Step 2.2.1

Cancel the common factor of ${(- 2)}^{2}$ and $2$ .

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Step 2.2.1.1

Rewrite $- 2$ as $- 1 (2)$ .

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

Step 2.2.1.2

Apply the product rule to $- 1 (2)$ .

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

Step 2.2.1.3

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

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

Step 2.2.1.4

Multiply $2^{2}$ by $1$ .

$f (- 2) = \frac{5 \cdot 2^{2}}{2}$

Step 2.2.1.5

Factor $2$ out of $5 \cdot 2^{2}$ .

$f (- 2) = \frac{2 (5 \cdot 2)}{2}$

Step 2.2.1.6

Cancel the common factors.

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Step 2.2.1.6.1

Factor $2$ out of $2$ .

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

Step 2.2.1.6.2

Cancel the common factor.

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

Step 2.2.1.6.3

Rewrite the expression.

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

Step 2.2.1.6.4

Divide $5 \cdot 2$ by $1$ .

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

Step 2.2.2

Multiply $5$ by $2$ .

$f (- 2) = 10$

Step 2.2.3

The final answer is $10$ .

$10$

Step 2.3

The $y$ value at $x = - 2$ is $10$ .

$y = 10$

Step 2.4

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

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

Step 2.5

Simplify the result.

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Step 2.5.1

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

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

Step 2.5.2

Multiply $5$ by $1$ .

$f (- 1) = \frac{5}{2}$

Step 2.5.3

The final answer is $\frac{5}{2}$ .

$\frac{5}{2}$

Step 2.6

The $y$ value at $x = - 1$ is $\frac{5}{2}$ .

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

Step 2.7

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

$f (2) = \frac{5 {(2)}^{2}}{2}$

Step 2.8

Simplify the result.

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Step 2.8.1

Cancel the common factor of ${(2)}^{2}$ and $2$ .

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Step 2.8.1.1

Factor $2$ out of $5 {(2)}^{2}$ .

$f (2) = \frac{2 (5 \cdot 2)}{2}$

Step 2.8.1.2

Cancel the common factors.

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Step 2.8.1.2.1

Factor $2$ out of $2$ .

$f (2) = \frac{2 (5 \cdot 2)}{2 (1)}$

Step 2.8.1.2.2

Cancel the common factor.

$f (2) = \frac{2 (5 \cdot 2)}{2 \cdot 1}$

Step 2.8.1.2.3

Rewrite the expression.

$f (2) = \frac{5 \cdot 2}{1}$

Step 2.8.1.2.4

Divide $5 \cdot 2$ by $1$ .

$f (2) = 5 \cdot 2$

Step 2.8.2

Multiply $5$ by $2$ .

$f (2) = 10$

Step 2.8.3

The final answer is $10$ .

$10$

Step 2.9

The $y$ value at $x = 2$ is $10$ .

$y = 10$

Step 2.10

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

$f (1) = \frac{5 {(1)}^{2}}{2}$

Step 2.11

Simplify the result.

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Step 2.11.1

One to any power is one.

$f (1) = \frac{5 \cdot 1}{2}$

Step 2.11.2

Multiply $5$ by $1$ .

$f (1) = \frac{5}{2}$

Step 2.11.3

The final answer is $\frac{5}{2}$ .

$\frac{5}{2}$

Step 2.12

The $y$ value at $x = 1$ is $\frac{5}{2}$ .

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

Step 2.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 2 & 10 \\ - 1 & \frac{5}{2} \\ 0 & 0 \\ 1 & \frac{5}{2} \\ 2 & 10 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(0, 0)$

Focus: $(0, \frac{1}{10})$

Axis of Symmetry: $x = 0$

Directrix: $y = - \frac{1}{10}$

$\begin{array}{c} x & y \\ - 2 & 10 \\ - 1 & \frac{5}{2} \\ 0 & 0 \\ 1 & \frac{5}{2} \\ 2 & 10 \end{array}$

Step 4