Precalculus Examples

$f (x) = 4 x^{2} + 20 x - 24$

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

$4 x^{2} + 20 x - 24$ .

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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

$c$ .

$a = 4$

$b = 20$

$c = - 24$

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

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

$d = \frac{20}{2 \cdot 4}$

Step 1.1.1.3.2

Simplify the right side.

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Step 1.1.1.3.2.1

Cancel the common factor of

$20$ and

$2$ .

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Step 1.1.1.3.2.1.1

Factor

$2$ out of

$20$ .

$d = \frac{2 \cdot 10}{2 \cdot 4}$

Step 1.1.1.3.2.1.2

Cancel the common factors.

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Step 1.1.1.3.2.1.2.1

Factor

$2$ out of

$2 \cdot 4$ .

$d = \frac{2 \cdot 10}{2 (4)}$

Step 1.1.1.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot 10}{2 \cdot 4}$

Step 1.1.1.3.2.1.2.3

Rewrite the expression.

$d = \frac{10}{4}$

Step 1.1.1.3.2.2

Cancel the common factor of

$10$ and

$4$ .

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Step 1.1.1.3.2.2.1

Factor

$2$ out of

$10$ .

$d = \frac{2 (5)}{4}$

Step 1.1.1.3.2.2.2

Cancel the common factors.

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Step 1.1.1.3.2.2.2.1

Factor

$2$ out of

$4$ .

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

Step 1.1.1.3.2.2.2.2

Cancel the common factor.

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

Step 1.1.1.3.2.2.2.3

Rewrite the expression.

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

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

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

$e = - 24 - \frac{20^{2}}{4 \cdot 4}$

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

$20$ to the power of

$2$ .

$e = - 24 - \frac{400}{4 \cdot 4}$

Step 1.1.1.4.2.1.2

Multiply

$4$ by

$4$ .

$e = - 24 - \frac{400}{16}$

Step 1.1.1.4.2.1.3

Divide

$400$ by

$16$ .

$e = - 24 - 1 \cdot 25$

Step 1.1.1.4.2.1.4

Multiply

$- 1$ by

$25$ .

$e = - 24 - 25$

Step 1.1.1.4.2.2

Subtract

$25$ from

$- 24$ .

$e = - 49$

Step 1.1.1.5

Substitute the values of

$a$ ,

$d$ , and

$e$ into the vertex form

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

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

Step 1.1.2

Set

$y$ equal to the new right side.

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

Step 1.2

Use the vertex form,

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

$a$ ,

$h$ , and

$k$ .

$a = 4$

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

$k = - 49$

Step 1.3

Since the value of

$a$ is positive, the parabola opens up.

Opens Up

Step 1.4

Find the vertex

$(h, k)$ .

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

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.

$(- \frac{5}{2}, - \frac{783}{16})$

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.

$y = - \frac{785}{16}$

Step 1.9

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

Direction: Opens Up

Vertex:

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

Focus:

$(- \frac{5}{2}, - \frac{783}{16})$

Axis of Symmetry:

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

Directrix:

$y = - \frac{785}{16}$

Direction: Opens Up

Vertex:

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

Focus:

$(- \frac{5}{2}, - \frac{783}{16})$

Axis of Symmetry:

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

Directrix:

$y = - \frac{785}{16}$

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.

$f (- 3) = 4 {(- 3)}^{2} + 20 (- 3) - 24$

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

$2$ .

$f (- 3) = 4 \cdot 9 + 20 (- 3) - 24$

Step 2.2.1.2

Multiply

$4$ by

$9$ .

$f (- 3) = 36 + 20 (- 3) - 24$

Step 2.2.1.3

Multiply

$20$ by

$- 3$ .

$f (- 3) = 36 - 60 - 24$

Step 2.2.2

Simplify by subtracting numbers.

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Step 2.2.2.1

Subtract

$60$ from

$36$ .

$f (- 3) = - 24 - 24$

Step 2.2.2.2

Subtract

$24$ from

$- 24$ .

$f (- 3) = - 48$

Step 2.2.3

The final answer is

$- 48$ .

$- 48$

Step 2.3

The

$y$ value at

$x = - 3$ is

$- 48$ .

$y = - 48$

Step 2.4

Replace the variable

$x$ with

$- 4$ in the expression.

$f (- 4) = 4 {(- 4)}^{2} + 20 (- 4) - 24$

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

$2$ .

$f (- 4) = 4 \cdot 16 + 20 (- 4) - 24$

Step 2.5.1.2

Multiply

$4$ by

$16$ .

$f (- 4) = 64 + 20 (- 4) - 24$

Step 2.5.1.3

Multiply

$20$ by

$- 4$ .

$f (- 4) = 64 - 80 - 24$

Step 2.5.2

Simplify by subtracting numbers.

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Step 2.5.2.1

Subtract

$80$ from

$64$ .

$f (- 4) = - 16 - 24$

Step 2.5.2.2

Subtract

$24$ from

$- 16$ .

$f (- 4) = - 40$

Step 2.5.3

The final answer is

$- 40$ .

$- 40$

Step 2.6

The

$y$ value at

$x = - 4$ is

$- 40$ .

$y = - 40$

Step 2.7

Replace the variable

$x$ with

$- 1$ in the expression.

$f (- 1) = 4 {(- 1)}^{2} + 20 (- 1) - 24$

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

Raise

$- 1$ to the power of

$2$ .

$f (- 1) = 4 \cdot 1 + 20 (- 1) - 24$

Step 2.8.1.2

Multiply

$4$ by

$1$ .

$f (- 1) = 4 + 20 (- 1) - 24$

Step 2.8.1.3

Multiply

$20$ by

$- 1$ .

$f (- 1) = 4 - 20 - 24$

Step 2.8.2

Simplify by subtracting numbers.

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Step 2.8.2.1

Subtract

$20$ from

$4$ .

$f (- 1) = - 16 - 24$

Step 2.8.2.2

Subtract

$24$ from

$- 16$ .

$f (- 1) = - 40$

Step 2.8.3

The final answer is

$- 40$ .

$- 40$

Step 2.9

The

$y$ value at

$x = - 1$ is

$- 40$ .

$y = - 40$

Step 2.10

Replace the variable

$x$ with

$0$ in the expression.

$f (0) = 4 {(0)}^{2} + 20 (0) - 24$

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

$0$ .

$f (0) = 4 \cdot 0 + 20 (0) - 24$

Step 2.11.1.2

Multiply

$4$ by

$0$ .

$f (0) = 0 + 20 (0) - 24$

Step 2.11.1.3

Multiply

$20$ by

$0$ .

$f (0) = 0 + 0 - 24$

Step 2.11.2

Simplify by adding and subtracting.

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Step 2.11.2.1

Add

$0$ and

$0$ .

$f (0) = 0 - 24$

Step 2.11.2.2

Subtract

$24$ from

$0$ .

$f (0) = - 24$

Step 2.11.3

The final answer is

$- 24$ .

$- 24$

Step 2.12

The

$y$ value at

$x = 0$ is

$- 24$ .

$y = - 24$

Step 2.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 4 & - 40 \\ - 3 & - 48 \\ - \frac{5}{2} & - 49 \\ - 1 & - 40 \\ 0 & - 24 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex:

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

Focus:

$(- \frac{5}{2}, - \frac{783}{16})$

Axis of Symmetry:

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

Directrix:

$y = - \frac{785}{16}$

$\begin{array}{c} x & y \\ - 4 & - 40 \\ - 3 & - 48 \\ - \frac{5}{2} & - 49 \\ - 1 & - 40 \\ 0 & - 24 \end{array}$

Step 4