Algebra Examples

f (x) = 2 x^{2} - 3 x + 1

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

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

$2 x^{2} - 3 x + 1$ .

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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

$c$ .

$a = 2$

$b = - 3$

$c = 1$

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

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

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

Step 1.1.1.3.2

Simplify the right side.

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Step 1.1.1.3.2.1

Multiply

$2$ by

$2$ .

$d = \frac{- 3}{4}$

Step 1.1.1.3.2.2

Move the negative in front of the fraction.

$d = - \frac{3}{4}$

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

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

$e = 1 - \frac{{(- 3)}^{2}}{4 \cdot 2}$

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

$- 3$ to the power of

$2$ .

$e = 1 - \frac{9}{4 \cdot 2}$

Step 1.1.1.4.2.1.2

Multiply

$4$ by

$2$ .

$e = 1 - \frac{9}{8}$

Step 1.1.1.4.2.2

Write

$1$ as a fraction with a common denominator.

$e = \frac{8}{8} - \frac{9}{8}$

Step 1.1.1.4.2.3

Combine the numerators over the common denominator.

$e = \frac{8 - 9}{8}$

Step 1.1.1.4.2.4

Subtract

$9$ from

$8$ .

$e = \frac{- 1}{8}$

Step 1.1.1.4.2.5

Move the negative in front of the fraction.

$e = - \frac{1}{8}$

Step 1.1.1.5

Substitute the values of

$a$ ,

$d$ , and

$e$ into the vertex form

$2 {(x - \frac{3}{4})}^{2} - \frac{1}{8}$ .

$2 {(x - \frac{3}{4})}^{2} - \frac{1}{8}$

Step 1.1.2

Set

$y$ equal to the new right side.

$y = 2 {(x - \frac{3}{4})}^{2} - \frac{1}{8}$

Step 1.2

Use the vertex form,

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

$a$ ,

$h$ , and

$k$ .

$a = 2$

$h = \frac{3}{4}$

$k = - \frac{1}{8}$

Step 1.3

Since the value of

$a$ is positive, the parabola opens up.

Opens Up

Step 1.4

Find the vertex

$(h, k)$ .

$(\frac{3}{4}, - \frac{1}{8})$

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.

$(\frac{3}{4}, 0)$

Step 1.7

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

$x = \frac{3}{4}$

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.

$y = - \frac{1}{4}$

Step 1.9

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

Direction: Opens Up

Vertex:

$(\frac{3}{4}, - \frac{1}{8})$

Focus:

$(\frac{3}{4}, 0)$

Axis of Symmetry:

$x = \frac{3}{4}$

Directrix:

$y = - \frac{1}{4}$

Direction: Opens Up

Vertex:

$(\frac{3}{4}, - \frac{1}{8})$

Focus:

$(\frac{3}{4}, 0)$

Axis of Symmetry:

$x = \frac{3}{4}$

Directrix:

$y = - \frac{1}{4}$

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

$0$ in the expression.

$f (0) = 2 {(0)}^{2} - 3 \cdot 0 + 1$

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

Raising

$0$ to any positive power yields

$0$ .

$f (0) = 2 \cdot 0 - 3 \cdot 0 + 1$

Step 2.2.1.2

Multiply

$2$ by

$0$ .

$f (0) = 0 - 3 \cdot 0 + 1$

Step 2.2.1.3

Multiply

$- 3$ by

$0$ .

$f (0) = 0 + 0 + 1$

Step 2.2.2

Simplify by adding numbers.

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Step 2.2.2.1

Add

$0$ and

$0$ .

$f (0) = 0 + 1$

Step 2.2.2.2

Add

$0$ and

$1$ .

$f (0) = 1$

Step 2.2.3

The final answer is

$1$ .

$1$

Step 2.3

The

$y$ value at

$x = 0$ is

$1$ .

$y = 1$

Step 2.4

Replace the variable

$x$ with

$- 1$ in the expression.

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

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

$- 1$ to the power of

$2$ .

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

Step 2.5.1.2

Multiply

$2$ by

$1$ .

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

Step 2.5.1.3

Multiply

$- 3$ by

$- 1$ .

$f (- 1) = 2 + 3 + 1$

Step 2.5.2

Simplify by adding numbers.

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Step 2.5.2.1

Add

$2$ and

$3$ .

$f (- 1) = 5 + 1$

Step 2.5.2.2

Add

$5$ and

$1$ .

$f (- 1) = 6$

Step 2.5.3

The final answer is

$6$ .

$6$

Step 2.6

The

$y$ value at

$x = - 1$ is

$6$ .

$y = 6$

Step 2.7

Replace the variable

$x$ with

$2$ in the expression.

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

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

Multiply

$2$ by

${(2)}^{2}$ by adding the exponents.

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Step 2.8.1.1.1

Multiply

$2$ by

${(2)}^{2}$ .

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Step 2.8.1.1.1.1

Raise

$2$ to the power of

$1$ .

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

Step 2.8.1.1.1.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.8.1.1.2

Add

$1$ and

$2$ .

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

Step 2.8.1.2

Raise

$2$ to the power of

$3$ .

$f (2) = 8 - 3 \cdot 2 + 1$

Step 2.8.1.3

Multiply

$- 3$ by

$2$ .

$f (2) = 8 - 6 + 1$

Step 2.8.2

Simplify by adding and subtracting.

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Step 2.8.2.1

Subtract

$6$ from

$8$ .

$f (2) = 2 + 1$

Step 2.8.2.2

Add

$2$ and

$1$ .

$f (2) = 3$

Step 2.8.3

The final answer is

$3$ .

$3$

Step 2.9

The

$y$ value at

$x = 2$ is

$3$ .

$y = 3$

Step 2.10

Replace the variable

$x$ with

$3$ in the expression.

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

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

Raise

$3$ to the power of

$2$ .

$f (3) = 2 \cdot 9 - 3 \cdot 3 + 1$

Step 2.11.1.2

Multiply

$2$ by

$9$ .

$f (3) = 18 - 3 \cdot 3 + 1$

Step 2.11.1.3

Multiply

$- 3$ by

$3$ .

$f (3) = 18 - 9 + 1$

Step 2.11.2

Simplify by adding and subtracting.

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Step 2.11.2.1

Subtract

$9$ from

$18$ .

$f (3) = 9 + 1$

Step 2.11.2.2

Add

$9$ and

$1$ .

$f (3) = 10$

Step 2.11.3

The final answer is

$10$ .

$10$

Step 2.12

The

$y$ value at

$x = 3$ is

$10$ .

$y = 10$

Step 2.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 1 & 6 \\ 0 & 1 \\ \frac{3}{4} & - \frac{1}{8} \\ 2 & 3 \\ 3 & 10 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex:

$(\frac{3}{4}, - \frac{1}{8})$

Focus:

$(\frac{3}{4}, 0)$

Axis of Symmetry:

$x = \frac{3}{4}$

Directrix:

$y = - \frac{1}{4}$

$\begin{array}{c} x & y \\ - 1 & 6 \\ 0 & 1 \\ \frac{3}{4} & - \frac{1}{8} \\ 2 & 3 \\ 3 & 10 \end{array}$

Step 4