Precalculus Examples

$y = - \frac{1}{2} x^{2} + 3 x - \frac{5}{2}$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Combine $x^{2}$ and $\frac{1}{2}$ .

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

Step 1.2

Complete the square for $- \frac{x^{2}}{2} + 3 x - \frac{5}{2}$ .

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Step 1.2.1

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

$a = - \frac{1}{2}$

$b = 3$

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

Step 1.2.2

Consider the vertex form of a parabola.

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

Step 1.2.3

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

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Step 1.2.3.1

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

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

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Simplify the denominator.

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Step 1.2.3.2.1.1

Multiply $- 1$ by $2$ .

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

Step 1.2.3.2.1.2

Combine $- 2$ and $\frac{1}{2}$ .

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

Step 1.2.3.2.2

Divide $- 2$ by $2$ .

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

Step 1.2.3.2.3

Divide $3$ by $- 1$ .

$d = - 3$

Step 1.2.4

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

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Step 1.2.4.1

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

$e = - \frac{5}{2} - \frac{3^{2}}{4 (- \frac{1}{2})}$

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Simplify each term.

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Step 1.2.4.2.1.1

Raise $3$ to the power of $2$ .

$e = - \frac{5}{2} - \frac{9}{4 (- \frac{1}{2})}$

Step 1.2.4.2.1.2

Simplify the denominator.

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Step 1.2.4.2.1.2.1

Multiply $- 1$ by $4$ .

$e = - \frac{5}{2} - \frac{9}{- 4 (\frac{1}{2})}$

Step 1.2.4.2.1.2.2

Combine $- 4$ and $\frac{1}{2}$ .

$e = - \frac{5}{2} - \frac{9}{\frac{- 4}{2}}$

Step 1.2.4.2.1.3

Divide $- 4$ by $2$ .

$e = - \frac{5}{2} - \frac{9}{- 2}$

Step 1.2.4.2.1.4

Move the negative in front of the fraction.

$e = - \frac{5}{2} - - \frac{9}{2}$

Step 1.2.4.2.1.5

Multiply $- - \frac{9}{2}$ .

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Step 1.2.4.2.1.5.1

Multiply $- 1$ by $- 1$ .

$e = - \frac{5}{2} + 1 (\frac{9}{2})$

Step 1.2.4.2.1.5.2

Multiply $\frac{9}{2}$ by $1$ .

$e = - \frac{5}{2} + \frac{9}{2}$

Step 1.2.4.2.2

Combine the numerators over the common denominator.

$e = \frac{- 5 + 9}{2}$

Step 1.2.4.2.3

Add $- 5$ and $9$ .

$e = \frac{4}{2}$

Step 1.2.4.2.4

Divide $4$ by $2$ .

$e = 2$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{2} {(x - 3)}^{2} + 2$ .

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

Step 1.3

Set $y$ equal to the new right side.

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

Step 2

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

$a = - \frac{1}{2}$

$h = 3$

$k = 2$

Step 3

Since the value of $a$ is negative, the parabola opens down.

Opens Down

Step 4

Find the vertex $(h, k)$ .

$(3, 2)$

Step 5

Find $p$ , the distance from the vertex to the focus.

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Step 5.1

Find the distance from the vertex to a focus of the parabola by using the following formula.

$\frac{1}{4 a}$

Step 5.2

Substitute the value of $a$ into the formula.

$\frac{1}{4 \cdot (- \frac{1}{2})}$

Step 5.3

Simplify.

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Step 5.3.1

Cancel the common factor of $1$ and $- 1$ .

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Step 5.3.1.1

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

$\frac{- 1 (- 1)}{4 \cdot (- \frac{1}{2})}$

Step 5.3.1.2

Move the negative in front of the fraction.

$- \frac{1}{4 (\frac{1}{2})}$

Step 5.3.2

Combine $4$ and $\frac{1}{2}$ .

$- \frac{1}{\frac{4}{2}}$

Step 5.3.3

Divide $4$ by $2$ .

$- \frac{1}{2}$

Step 6

Find the focus.

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Step 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 6.2

Substitute the known values of $h$ , $p$ , and $k$ into the formula and simplify.

$(3, \frac{3}{2})$

Step 7

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

$x = 3$

Step 8