Calculus Examples

$y = 4 x^{2} + 3 x - 5$

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 $4 x^{2} + 3 x - 5$ .

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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 $c$ .

$a = 4$

$b = 3$

$c = - 5$

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

$d = \frac{3}{2 \cdot 4}$

Step 1.1.1.3.2

Multiply $2$ by $4$ .

$d = \frac{3}{8}$

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

$e = - 5 - \frac{3^{2}}{4 \cdot 4}$

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 $3$ to the power of $2$ .

$e = - 5 - \frac{9}{4 \cdot 4}$

Step 1.1.1.4.2.1.2

Multiply $4$ by $4$ .

$e = - 5 - \frac{9}{16}$

Step 1.1.1.4.2.2

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$e = - 5 \cdot \frac{16}{16} - \frac{9}{16}$

Step 1.1.1.4.2.3

Combine $- 5$ and $\frac{16}{16}$ .

$e = \frac{- 5 \cdot 16}{16} - \frac{9}{16}$

Step 1.1.1.4.2.4

Combine the numerators over the common denominator.

$e = \frac{- 5 \cdot 16 - 9}{16}$

Step 1.1.1.4.2.5

Simplify the numerator.

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Step 1.1.1.4.2.5.1

Multiply $- 5$ by $16$ .

$e = \frac{- 80 - 9}{16}$

Step 1.1.1.4.2.5.2

Subtract $9$ from $- 80$ .

$e = \frac{- 89}{16}$

Step 1.1.1.4.2.6

Move the negative in front of the fraction.

$e = - \frac{89}{16}$

Step 1.1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $4 {(x + \frac{3}{8})}^{2} - \frac{89}{16}$ .

$4 {(x + \frac{3}{8})}^{2} - \frac{89}{16}$

Step 1.1.2

Set $y$ equal to the new right side.

$y = 4 {(x + \frac{3}{8})}^{2} - \frac{89}{16}$

Step 1.2

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

$a = 4$

$h = - \frac{3}{8}$

$k = - \frac{89}{16}$

Step 1.3

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

Opens Up

Step 1.4

Find the vertex $(h, k)$ .

$(- \frac{3}{8}, - \frac{89}{16})$

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.

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

Step 1.5.3

Multiply $4$ by $4$ .

$\frac{1}{16}$

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.

$(- \frac{3}{8}, - \frac{11}{2})$

Step 1.7

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

$x = - \frac{3}{8}$

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.

$y = - \frac{45}{8}$

Step 1.9

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

Direction: Opens Up

Vertex: $(- \frac{3}{8}, - \frac{89}{16})$

Focus: $(- \frac{3}{8}, - \frac{11}{2})$

Axis of Symmetry: $x = - \frac{3}{8}$

Directrix: $y = - \frac{45}{8}$

Direction: Opens Up

Vertex: $(- \frac{3}{8}, - \frac{89}{16})$

Focus: $(- \frac{3}{8}, - \frac{11}{2})$

Axis of Symmetry: $x = - \frac{3}{8}$

Directrix: $y = - \frac{45}{8}$

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 $- 1$ in the expression.

$f (- 1) = 4 {(- 1)}^{2} + 3 (- 1) - 5$

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

$f (- 1) = 4 \cdot 1 + 3 (- 1) - 5$

Step 2.2.1.2

Multiply $4$ by $1$ .

$f (- 1) = 4 + 3 (- 1) - 5$

Step 2.2.1.3

Multiply $3$ by $- 1$ .

$f (- 1) = 4 - 3 - 5$

Step 2.2.2

Simplify by subtracting numbers.

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Step 2.2.2.1

Subtract $3$ from $4$ .

$f (- 1) = 1 - 5$

Step 2.2.2.2

Subtract $5$ from $1$ .

$f (- 1) = - 4$

Step 2.2.3

The final answer is $- 4$ .

$- 4$

Step 2.3

The $y$ value at $x = - 1$ is $- 4$ .

$y = - 4$

Step 2.4

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

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

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

$f (- 2) = 4 \cdot 4 + 3 (- 2) - 5$

Step 2.5.1.2

Multiply $4$ by $4$ .

$f (- 2) = 16 + 3 (- 2) - 5$

Step 2.5.1.3

Multiply $3$ by $- 2$ .

$f (- 2) = 16 - 6 - 5$

Step 2.5.2

Simplify by subtracting numbers.

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Step 2.5.2.1

Subtract $6$ from $16$ .

$f (- 2) = 10 - 5$

Step 2.5.2.2

Subtract $5$ from $10$ .

$f (- 2) = 5$

Step 2.5.3

The final answer is $5$ .

$5$

Step 2.6

The $y$ value at $x = - 2$ is $5$ .

$y = 5$

Step 2.7

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

$f (1) = 4 {(1)}^{2} + 3 (1) - 5$

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

One to any power is one.

$f (1) = 4 \cdot 1 + 3 (1) - 5$

Step 2.8.1.2

Multiply $4$ by $1$ .

$f (1) = 4 + 3 (1) - 5$

Step 2.8.1.3

Multiply $3$ by $1$ .

$f (1) = 4 + 3 - 5$

Step 2.8.2

Simplify by adding and subtracting.

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Step 2.8.2.1

Add $4$ and $3$ .

$f (1) = 7 - 5$

Step 2.8.2.2

Subtract $5$ from $7$ .

$f (1) = 2$

Step 2.8.3

The final answer is $2$ .

$2$

Step 2.9

The $y$ value at $x = 1$ is $2$ .

$y = 2$

Step 2.10

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

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

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

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

$f (2) = 4 \cdot 4 + 3 (2) - 5$

Step 2.11.1.2

Multiply $4$ by $4$ .

$f (2) = 16 + 3 (2) - 5$

Step 2.11.1.3

Multiply $3$ by $2$ .

$f (2) = 16 + 6 - 5$

Step 2.11.2

Simplify by adding and subtracting.

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Step 2.11.2.1

Add $16$ and $6$ .

$f (2) = 22 - 5$

Step 2.11.2.2

Subtract $5$ from $22$ .

$f (2) = 17$

Step 2.11.3

The final answer is $17$ .

$17$

Step 2.12

The $y$ value at $x = 2$ is $17$ .

$y = 17$

Step 2.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 2 & 5 \\ - 1 & - 4 \\ - \frac{3}{8} & - \frac{89}{16} \\ 1 & 2 \\ 2 & 17 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(- \frac{3}{8}, - \frac{89}{16})$

Focus: $(- \frac{3}{8}, - \frac{11}{2})$

Axis of Symmetry: $x = - \frac{3}{8}$

Directrix: $y = - \frac{45}{8}$

$\begin{array}{c} x & y \\ - 2 & 5 \\ - 1 & - 4 \\ - \frac{3}{8} & - \frac{89}{16} \\ 1 & 2 \\ 2 & 17 \end{array}$

Step 4