Precalculus Examples

$x^{2} - 2 y = 8 x - 10$

Step 1

Rewrite the equation in vertex form.

Tap for more steps...

Step 1.1

Isolate $y$ to the left side of the equation.

Tap for more steps...

Step 1.1.1

Subtract $x^{2}$ from both sides of the equation.

$- 2 y = 8 x - 10 - x^{2}$

Step 1.1.2

Divide each term in $- 2 y = 8 x - 10 - x^{2}$ by $- 2$ and simplify.

Tap for more steps...

Step 1.1.2.1

Divide each term in $- 2 y = 8 x - 10 - x^{2}$ by $- 2$ .

$\frac{- 2 y}{- 2} = \frac{8 x}{- 2} + \frac{- 10}{- 2} + \frac{- x^{2}}{- 2}$

Step 1.1.2.2

Simplify the left side.

Tap for more steps...

Step 1.1.2.2.1

Cancel the common factor of $- 2$ .

Tap for more steps...

Step 1.1.2.2.1.1

Cancel the common factor.

$\frac{- 2 y}{- 2} = \frac{8 x}{- 2} + \frac{- 10}{- 2} + \frac{- x^{2}}{- 2}$

Step 1.1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{8 x}{- 2} + \frac{- 10}{- 2} + \frac{- x^{2}}{- 2}$

Step 1.1.2.3

Simplify the right side.

Tap for more steps...

Step 1.1.2.3.1

Simplify each term.

Tap for more steps...

Step 1.1.2.3.1.1

Cancel the common factor of $8$ and $- 2$ .

Tap for more steps...

Step 1.1.2.3.1.1.1

Factor $2$ out of $8 x$ .

$y = \frac{2 (4 x)}{- 2} + \frac{- 10}{- 2} + \frac{- x^{2}}{- 2}$

Step 1.1.2.3.1.1.2

Move the negative one from the denominator of $\frac{4 x}{- 1}$ .

$y = - 1 \cdot (4 x) + \frac{- 10}{- 2} + \frac{- x^{2}}{- 2}$

Step 1.1.2.3.1.2

Rewrite $- 1 \cdot (4 x)$ as $- (4 x)$ .

$y = - (4 x) + \frac{- 10}{- 2} + \frac{- x^{2}}{- 2}$

Step 1.1.2.3.1.3

Multiply $4$ by $- 1$ .

$y = - 4 x + \frac{- 10}{- 2} + \frac{- x^{2}}{- 2}$

Step 1.1.2.3.1.4

Divide $- 10$ by $- 2$ .

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

Step 1.1.2.3.1.5

Dividing two negative values results in a positive value.

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

Step 1.2

Complete the square for $- 4 x + 5 + \frac{x^{2}}{2}$ .

Tap for more steps...

Step 1.2.1

Simplify the expression.

Tap for more steps...

Step 1.2.1.1

Move $5$ .

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

Step 1.2.1.2

Reorder $- 4 x$ and $\frac{x^{2}}{2}$ .

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

Step 1.2.2

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

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

$b = - 4$

$c = 5$

Step 1.2.3

Consider the vertex form of a parabola.

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

Step 1.2.4

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

Tap for more steps...

Step 1.2.4.1

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

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

Step 1.2.4.2

Simplify the right side.

Tap for more steps...

Step 1.2.4.2.1

Cancel the common factor of $- 4$ and $2$ .

Tap for more steps...

Step 1.2.4.2.1.1

Factor $2$ out of $- 4$ .

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

Step 1.2.4.2.1.2

Cancel the common factors.

Tap for more steps...

Step 1.2.4.2.1.2.1

Cancel the common factor.

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

Step 1.2.4.2.1.2.2

Rewrite the expression.

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

Step 1.2.4.2.2

Multiply the numerator by the reciprocal of the denominator.

$d = - 2 \cdot 2$

Step 1.2.4.2.3

Multiply $- 2$ by $2$ .

$d = - 4$

Step 1.2.5

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

Tap for more steps...

Step 1.2.5.1

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

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

Step 1.2.5.2

Simplify the right side.

Tap for more steps...

Step 1.2.5.2.1

Simplify each term.

Tap for more steps...

Step 1.2.5.2.1.1

Cancel the common factor of ${(- 4)}^{2}$ and $4$ .

Tap for more steps...

Step 1.2.5.2.1.1.1

Rewrite $- 4$ as $- 1 (4)$ .

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

Step 1.2.5.2.1.1.2

Apply the product rule to $- 1 (4)$ .

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

Step 1.2.5.2.1.1.3

Raise $- 1$ to the power of $2$ .

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

Step 1.2.5.2.1.1.4

Multiply $4^{2}$ by $1$ .

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

Step 1.2.5.2.1.1.5

Factor $4$ out of $4^{2}$ .

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

Step 1.2.5.2.1.1.6

Cancel the common factors.

Tap for more steps...

Step 1.2.5.2.1.1.6.1

Cancel the common factor.

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

Step 1.2.5.2.1.1.6.2

Rewrite the expression.

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

Step 1.2.5.2.1.2

Multiply the numerator by the reciprocal of the denominator.

$e = 5 - (4 \cdot 2)$

Step 1.2.5.2.1.3

Multiply $- (4 \cdot 2)$ .

Tap for more steps...

Step 1.2.5.2.1.3.1

Multiply $4$ by $2$ .

$e = 5 - 1 \cdot 8$

Step 1.2.5.2.1.3.2

Multiply $- 1$ by $8$ .

$e = 5 - 8$

Step 1.2.5.2.2

Subtract $8$ from $5$ .

$e = - 3$

Step 1.2.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{2} {(x - 4)}^{2} - 3$ .

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

Step 1.3

Set $y$ equal to the new right side.

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

Step 2

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

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

$h = 4$

$k = - 3$

Step 3

Since the value of $a$ is positive, the parabola opens up.

Opens Up

Step 4

Find the vertex $(h, k)$ .

$(4, - 3)$

Step 5

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

Tap for more steps...

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.

Tap for more steps...

Step 5.3.1

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

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

Step 5.3.2

Divide $4$ by $2$ .

$\frac{1}{2}$

Step 6

Find the focus.

Tap for more steps...

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.

$(4, - \frac{5}{2})$

Step 7

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

$x = 4$

Step 8

Find the directrix.

Tap for more steps...

Step 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 8.2

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

$y = - \frac{7}{2}$

Step 9

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

Direction: Opens Up

Vertex: $(4, - 3)$

Focus: $(4, - \frac{5}{2})$

Axis of Symmetry: $x = 4$

Directrix: $y = - \frac{7}{2}$

Step 10