Algebra Examples

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

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Complete the square for $- 2 x^{2} + 3 x - 4$ .

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Step 1.1.1

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

$a = - 2$

$b = 3$

$c = - 4$

Step 1.1.2

Consider the vertex form of a parabola.

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

Step 1.1.3

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

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Step 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 1.1.3.2

Simplify the right side.

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Step 1.1.3.2.1

Multiply $2$ by $- 2$ .

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

Step 1.1.3.2.2

Move the negative in front of the fraction.

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

Step 1.1.4

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

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Step 1.1.4.1

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

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

Step 1.1.4.2

Simplify the right side.

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Step 1.1.4.2.1

Simplify each term.

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Step 1.1.4.2.1.1

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

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

Step 1.1.4.2.1.2

Multiply $4$ by $- 2$ .

$e = - 4 - \frac{9}{- 8}$

Step 1.1.4.2.1.3

Move the negative in front of the fraction.

$e = - 4 - - \frac{9}{8}$

Step 1.1.4.2.1.4

Multiply $- - \frac{9}{8}$ .

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Step 1.1.4.2.1.4.1

Multiply $- 1$ by $- 1$ .

$e = - 4 + 1 (\frac{9}{8})$

Step 1.1.4.2.1.4.2

Multiply $\frac{9}{8}$ by $1$ .

$e = - 4 + \frac{9}{8}$

Step 1.1.4.2.2

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$e = - 4 \cdot \frac{8}{8} + \frac{9}{8}$

Step 1.1.4.2.3

Combine $- 4$ and $\frac{8}{8}$ .

$e = \frac{- 4 \cdot 8}{8} + \frac{9}{8}$

Step 1.1.4.2.4

Combine the numerators over the common denominator.

$e = \frac{- 4 \cdot 8 + 9}{8}$

Step 1.1.4.2.5

Simplify the numerator.

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Step 1.1.4.2.5.1

Multiply $- 4$ by $8$ .

$e = \frac{- 32 + 9}{8}$

Step 1.1.4.2.5.2

Add $- 32$ and $9$ .

$e = \frac{- 23}{8}$

Step 1.1.4.2.6

Move the negative in front of the fraction.

$e = - \frac{23}{8}$

Step 1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 2 {(x - \frac{3}{4})}^{2} - \frac{23}{8}$ .

$- 2 {(x - \frac{3}{4})}^{2} - \frac{23}{8}$

Step 1.2

Set $y$ equal to the new right side.

$y = - 2 {(x - \frac{3}{4})}^{2} - \frac{23}{8}$

Step 2

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

$a = - 2$

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

$k = - \frac{23}{8}$

Step 3

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

Opens Down

Step 4

Find the vertex $(h, k)$ .

$(\frac{3}{4}, - \frac{23}{8})$

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.

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

Step 5.3

Simplify.

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Step 5.3.1

Multiply $4$ by $- 2$ .

$\frac{1}{- 8}$

Step 5.3.2

Move the negative in front of the fraction.

$- \frac{1}{8}$

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.

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

Step 7

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

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

Step 8