Precalculus Examples

f (x) = 3 x^{2} - 2 x

$f (x) = 3 x^{2} - 2 x$

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

3 x^{2} - 2 x

$3 x^{2} - 2 x$ .

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Step 1.1.1.1

Use the form

a x^{2} + b x + c

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

a

$a$ ,

b

$b$ , and

c

$c$ .

a = 3

$a = 3$

b = - 2

$b = - 2$

c = 0

$c = 0$

Step 1.1.1.2

Consider the vertex form of a parabola.

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

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

Step 1.1.1.3

Find the value of

d

$d$ using the formula

d = \frac{b}{2 a}

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

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Step 1.1.1.3.1

Substitute the values of

a

$a$ and

b

$b$ into the formula

d = \frac{b}{2 a}

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

d = \frac{- 2}{2 \cdot 3}

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

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

- 2

$- 2$ and

2

$2$ .

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Step 1.1.1.3.2.1.1

Factor

2

$2$ out of

- 2

$- 2$ .

d = \frac{2 \cdot - 1}{2 \cdot 3}

$d = \frac{2 \cdot - 1}{2 \cdot 3}$

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

$2$ out of

2 \cdot 3

$2 \cdot 3$ .

d = \frac{2 \cdot - 1}{2 (3)}

$d = \frac{2 \cdot - 1}{2 (3)}$

Step 1.1.1.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 1}{2 \cdot 3}$

Step 1.1.1.3.2.1.2.3

Rewrite the expression.

$d = \frac{- 1}{3}$

Step 1.1.1.3.2.2

Move the negative in front of the fraction.

$d = - \frac{1}{3}$

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

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

$e = 0 - \frac{{(- 2)}^{2}}{4 \cdot 3}$

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

$- 2$ to the power of

$2$ .

$e = 0 - \frac{4}{4 \cdot 3}$

Step 1.1.1.4.2.1.2

Multiply

$4$ by

$3$ .

$e = 0 - \frac{4}{12}$

Step 1.1.1.4.2.1.3

Cancel the common factor of

$4$ and

$12$ .

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Step 1.1.1.4.2.1.3.1

Factor

$4$ out of

$4$ .

$e = 0 - \frac{4 (1)}{12}$

Step 1.1.1.4.2.1.3.2

Cancel the common factors.

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Step 1.1.1.4.2.1.3.2.1

Factor

$4$ out of

$12$ .

$e = 0 - \frac{4 \cdot 1}{4 \cdot 3}$

Step 1.1.1.4.2.1.3.2.2

Cancel the common factor.

$e = 0 - \frac{4 \cdot 1}{4 \cdot 3}$

Step 1.1.1.4.2.1.3.2.3

Rewrite the expression.

$e = 0 - \frac{1}{3}$

Step 1.1.1.4.2.2

Subtract

$\frac{1}{3}$ from

$0$ .

$e = - \frac{1}{3}$

Step 1.1.1.5

Substitute the values of

$a$ ,

$d$ , and

$e$ into the vertex form

$3 {(x - \frac{1}{3})}^{2} - \frac{1}{3}$ .

$3 {(x - \frac{1}{3})}^{2} - \frac{1}{3}$

Step 1.1.2

Set

$y$ equal to the new right side.

$y = 3 {(x - \frac{1}{3})}^{2} - \frac{1}{3}$

Step 1.2

Use the vertex form,

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

$a$ ,

$h$ , and

$k$ .

$a = 3$

$h = \frac{1}{3}$

$k = - \frac{1}{3}$

Step 1.3

Since the value of

$a$ is positive, the parabola opens up.

Opens Up

Step 1.4

Find the vertex

$(h, k)$ .

$(\frac{1}{3}, - \frac{1}{3})$

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.

$\frac{1}{4 \cdot 3}$

Step 1.5.3

Multiply

$4$ by

$3$ .

$\frac{1}{12}$

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.

$(\frac{1}{3}, - \frac{1}{4})$

Step 1.7

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

$x = \frac{1}{3}$

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.

$y = - \frac{5}{12}$

Step 1.9

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

Direction: Opens Up

Vertex:

$(\frac{1}{3}, - \frac{1}{3})$

Focus:

$(\frac{1}{3}, - \frac{1}{4})$

Axis of Symmetry:

$x = \frac{1}{3}$

Directrix:

$y = - \frac{5}{12}$

Direction: Opens Up

Vertex:

$(\frac{1}{3}, - \frac{1}{3})$

Focus:

$(\frac{1}{3}, - \frac{1}{4})$

Axis of Symmetry:

$x = \frac{1}{3}$

Directrix:

$y = - \frac{5}{12}$

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.

$f (- 1) = 3 {(- 1)}^{2} - 2 \cdot - 1$

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

$2$ .

$f (- 1) = 3 \cdot 1 - 2 \cdot - 1$

Step 2.2.1.2

Multiply

$3$ by

$1$ .

$f (- 1) = 3 - 2 \cdot - 1$

Step 2.2.1.3

Multiply

$- 2$ by

$- 1$ .

$f (- 1) = 3 + 2$

Step 2.2.2

Add

$3$ and

$2$ .

$f (- 1) = 5$

Step 2.2.3

The final answer is

$5$ .

$5$

Step 2.3

The

$y$ value at

$x = - 1$ is

$5$ .

$y = 5$

Step 2.4

Replace the variable

$x$ with

$- 2$ in the expression.

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

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

$2$ .

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

Step 2.5.1.2

Multiply

$3$ by

$4$ .

$f (- 2) = 12 - 2 \cdot - 2$

Step 2.5.1.3

Multiply

$- 2$ by

$- 2$ .

$f (- 2) = 12 + 4$

Step 2.5.2

Add

$12$ and

$4$ .

$f (- 2) = 16$

Step 2.5.3

The final answer is

$16$ .

$16$

Step 2.6

The

$y$ value at

$x = - 2$ is

$16$ .

$y = 16$

Step 2.7

Replace the variable

$x$ with

$1$ in the expression.

$f (1) = 3 {(1)}^{2} - 2 \cdot 1$

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.

$f (1) = 3 \cdot 1 - 2 \cdot 1$

Step 2.8.1.2

Multiply

$3$ by

$1$ .

$f (1) = 3 - 2 \cdot 1$

Step 2.8.1.3

Multiply

$- 2$ by

$1$ .

$f (1) = 3 - 2$

Step 2.8.2

Subtract

$2$ from

$3$ .

$f (1) = 1$

Step 2.8.3

The final answer is

$1$ .

$1$

Step 2.9

The

$y$ value at

$x = 1$ is

$1$ .

$y = 1$

Step 2.10

Replace the variable

$x$ with

$2$ in the expression.

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

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

$2$ .

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

Step 2.11.1.2

Multiply

$3$ by

$4$ .

$f (2) = 12 - 2 \cdot 2$

Step 2.11.1.3

Multiply

$- 2$ by

$2$ .

$f (2) = 12 - 4$

Step 2.11.2

Subtract

$4$ from

$12$ .

$f (2) = 8$

Step 2.11.3

The final answer is

$8$ .

$8$

Step 2.12

The

$y$ value at

$x = 2$ is

$8$ .

$y = 8$

Step 2.13

Graph the parabola using its properties and the selected points.

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

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex:

$(\frac{1}{3}, - \frac{1}{3})$

Focus:

$(\frac{1}{3}, - \frac{1}{4})$

Axis of Symmetry:

$x = \frac{1}{3}$

Directrix:

$y = - \frac{5}{12}$

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

Step 4