Precalculus Examples

$p (x) = (x - 1) (x + 2)$

Step 1

Find the properties of the given parabola.

Tap for more steps...

Step 1.1

Rewrite the equation in vertex form.

Tap for more steps...

Step 1.1.1

Complete the square for $(x - 1) (x + 2)$ .

Tap for more steps...

Step 1.1.1.1

Simplify the expression.

Tap for more steps...

Step 1.1.1.1.1

Expand $(x - 1) (x + 2)$ using the FOIL Method.

Tap for more steps...

Step 1.1.1.1.1.1

Apply the distributive property.

$x (x + 2) - 1 (x + 2)$

Step 1.1.1.1.1.2

Apply the distributive property.

$x \cdot x + x \cdot 2 - 1 (x + 2)$

Step 1.1.1.1.1.3

Apply the distributive property.

$x \cdot x + x \cdot 2 - 1 x - 1 \cdot 2$

Step 1.1.1.1.2

Simplify and combine like terms.

Tap for more steps...

Step 1.1.1.1.2.1

Simplify each term.

Tap for more steps...

Step 1.1.1.1.2.1.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot 2 - 1 x - 1 \cdot 2$

Step 1.1.1.1.2.1.2

Move $2$ to the left of $x$ .

$x^{2} + 2 \cdot x - 1 x - 1 \cdot 2$

Step 1.1.1.1.2.1.3

Rewrite $- 1 x$ as $- x$ .

$x^{2} + 2 x - x - 1 \cdot 2$

Step 1.1.1.1.2.1.4

Multiply $- 1$ by $2$ .

$x^{2} + 2 x - x - 2$

Step 1.1.1.1.2.2

Subtract $x$ from $2 x$ .

$x^{2} + x - 2$

Step 1.1.1.2

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

$a = 1$

$b = 1$

$c = - 2$

Step 1.1.1.3

Consider the vertex form of a parabola.

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

Step 1.1.1.4

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

Tap for more steps...

Step 1.1.1.4.1

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

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

Step 1.1.1.4.2

Cancel the common factor of $1$ .

Tap for more steps...

Step 1.1.1.4.2.1

Cancel the common factor.

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

Step 1.1.1.4.2.2

Rewrite the expression.

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

Step 1.1.1.5

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

Tap for more steps...

Step 1.1.1.5.1

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

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

Step 1.1.1.5.2

Simplify the right side.

Tap for more steps...

Step 1.1.1.5.2.1

Simplify each term.

Tap for more steps...

Step 1.1.1.5.2.1.1

One to any power is one.

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

Step 1.1.1.5.2.1.2

Multiply $4$ by $1$ .

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

Step 1.1.1.5.2.2

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 1.1.1.5.2.3

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

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

Step 1.1.1.5.2.4

Combine the numerators over the common denominator.

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

Step 1.1.1.5.2.5

Simplify the numerator.

Tap for more steps...

Step 1.1.1.5.2.5.1

Multiply $- 2$ by $4$ .

$e = \frac{- 8 - 1}{4}$

Step 1.1.1.5.2.5.2

Subtract $1$ from $- 8$ .

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

Step 1.1.1.5.2.6

Move the negative in front of the fraction.

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

Step 1.1.1.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x + \frac{1}{2})}^{2} - \frac{9}{4}$ .

${(x + \frac{1}{2})}^{2} - \frac{9}{4}$

Step 1.1.2

Set $y$ equal to the new right side.

$y = {(x + \frac{1}{2})}^{2} - \frac{9}{4}$

Step 1.2

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

$a = 1$

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

$k = - \frac{9}{4}$

Step 1.3

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

Opens Up

Step 1.4

Find the vertex $(h, k)$ .

$(- \frac{1}{2}, - \frac{9}{4})$

Step 1.5

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

Tap for more steps...

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.

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

Step 1.5.3

Cancel the common factor of $1$ .

Tap for more steps...

Step 1.5.3.1

Cancel the common factor.

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

Step 1.5.3.2

Rewrite the expression.

$\frac{1}{4}$

Step 1.6

Find the focus.

Tap for more steps...

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.

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

Step 1.7

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

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

Step 1.8

Find the directrix.

Tap for more steps...

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.

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

Step 1.9

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

Direction: Opens Up

Vertex: $(- \frac{1}{2}, - \frac{9}{4})$

Focus: $(- \frac{1}{2}, - 2)$

Axis of Symmetry: $x = - \frac{1}{2}$

Directrix: $y = - \frac{5}{2}$

Direction: Opens Up

Vertex: $(- \frac{1}{2}, - \frac{9}{4})$

Focus: $(- \frac{1}{2}, - 2)$

Axis of Symmetry: $x = - \frac{1}{2}$

Directrix: $y = - \frac{5}{2}$

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.

Tap for more steps...

Step 2.1

Replace the variable $x$ with $- 1$ in the expression.

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

Step 2.2

Simplify the result.

Tap for more steps...

Step 2.2.1

Subtract $1$ from $- 1$ .

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

Step 2.2.2

Add $- 1$ and $2$ .

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

Step 2.2.3

Multiply $- 2$ by $1$ .

$f (- 1) = - 2$

Step 2.2.4

The final answer is $- 2$ .

$- 2$

Step 2.3

The $y$ value at $x = - 1$ is $- 2$ .

$y = - 2$

Step 2.4

Replace the variable $x$ with $- 2$ in the expression.

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

Step 2.5

Simplify the result.

Tap for more steps...

Step 2.5.1

Subtract $1$ from $- 2$ .

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

Step 2.5.2

Add $- 2$ and $2$ .

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

Step 2.5.3

Multiply $- 3$ by $0$ .

$f (- 2) = 0$

Step 2.5.4

The final answer is $0$ .

$0$

Step 2.6

The $y$ value at $x = - 2$ is $0$ .

$y = 0$

Step 2.7

Replace the variable $x$ with $1$ in the expression.

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

Step 2.8

Simplify the result.

Tap for more steps...

Step 2.8.1

Subtract $1$ from $1$ .

$f (1) = 0 ((1) + 2)$

Step 2.8.2

Add $1$ and $2$ .

$f (1) = 0 \cdot 3$

Step 2.8.3

Multiply $0$ by $3$ .

$f (1) = 0$

Step 2.8.4

The final answer is $0$ .

$0$

Step 2.9

The $y$ value at $x = 1$ is $0$ .

$y = 0$

Step 2.10

Replace the variable $x$ with $2$ in the expression.

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

Step 2.11

Simplify the result.

Tap for more steps...

Step 2.11.1

Subtract $1$ from $2$ .

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

Step 2.11.2

Multiply $(2) + 2$ by $1$ .

$f (2) = (2) + 2$

Step 2.11.3

Add $2$ and $2$ .

$f (2) = 4$

Step 2.11.4

The final answer is $4$ .

$4$

Step 2.12

The $y$ value at $x = 2$ is $4$ .

$y = 4$

Step 2.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 2 & 0 \\ - 1 & - 2 \\ - \frac{1}{2} & - \frac{9}{4} \\ 1 & 0 \\ 2 & 4 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(- \frac{1}{2}, - \frac{9}{4})$

Focus: $(- \frac{1}{2}, - 2)$

Axis of Symmetry: $x = - \frac{1}{2}$

Directrix: $y = - \frac{5}{2}$

$\begin{array}{c} x & y \\ - 2 & 0 \\ - 1 & - 2 \\ - \frac{1}{2} & - \frac{9}{4} \\ 1 & 0 \\ 2 & 4 \end{array}$

Step 4