Precalculus Examples

$f (x) = x^{2} + 3 x + \frac{1}{4}$

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 $x^{2} + 3 x + \frac{1}{4}$ .

Tap for more steps...

Step 1.1.1.1

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

$a = 1$

$b = 3$

$c = \frac{1}{4}$

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

Tap for more steps...

Step 1.1.1.3.1

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

$d = \frac{3}{2 \cdot 1}$

Step 1.1.1.3.2

Multiply $2$ by $1$ .

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

Step 1.1.1.4

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

Tap for more steps...

Step 1.1.1.4.1

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

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

Step 1.1.1.4.2

Simplify the right side.

Tap for more steps...

Step 1.1.1.4.2.1

Simplify each term.

Tap for more steps...

Step 1.1.1.4.2.1.1

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

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

Step 1.1.1.4.2.1.2

Multiply $4$ by $1$ .

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

Step 1.1.1.4.2.2

Combine the numerators over the common denominator.

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

Step 1.1.1.4.2.3

Subtract $9$ from $1$ .

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

Step 1.1.1.4.2.4

Divide $- 8$ by $4$ .

$e = - 2$

Step 1.1.1.5

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

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

Step 1.1.2

Set $y$ equal to the new right side.

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

Step 1.2

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

$a = 1$

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

$k = - 2$

Step 1.3

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

Opens Up

Step 1.4

Find the vertex $(h, k)$ .

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

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.

$(- \frac{3}{2}, - \frac{7}{4})$

Step 1.7

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

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

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.

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

Step 1.9

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

Direction: Opens Up

Vertex: $(- \frac{3}{2}, - 2)$

Focus: $(- \frac{3}{2}, - \frac{7}{4})$

Axis of Symmetry: $x = - \frac{3}{2}$

Directrix: $y = - \frac{9}{4}$

Direction: Opens Up

Vertex: $(- \frac{3}{2}, - 2)$

Focus: $(- \frac{3}{2}, - \frac{7}{4})$

Axis of Symmetry: $x = - \frac{3}{2}$

Directrix: $y = - \frac{9}{4}$

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.

Tap for more steps...

Step 2.1

Replace the variable $x$ with $- 3$ in the expression.

$f (- 3) = {(- 3)}^{2} + 3 (- 3) + \frac{1}{4}$

Step 2.2

Simplify the result.

Tap for more steps...

Step 2.2.1

Simplify each term.

Tap for more steps...

Step 2.2.1.1

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

$f (- 3) = 9 + 3 (- 3) + \frac{1}{4}$

Step 2.2.1.2

Multiply $3$ by $- 3$ .

$f (- 3) = 9 - 9 + \frac{1}{4}$

Step 2.2.2

Find the common denominator.

Tap for more steps...

Step 2.2.2.1

Write $9$ as a fraction with denominator $1$ .

$f (- 3) = \frac{9}{1} - 9 + \frac{1}{4}$

Step 2.2.2.2

Multiply $\frac{9}{1}$ by $\frac{4}{4}$ .

$f (- 3) = \frac{9}{1} \cdot \frac{4}{4} - 9 + \frac{1}{4}$

Step 2.2.2.3

Multiply $\frac{9}{1}$ by $\frac{4}{4}$ .

$f (- 3) = \frac{9 \cdot 4}{4} - 9 + \frac{1}{4}$

Step 2.2.2.4

Write $- 9$ as a fraction with denominator $1$ .

$f (- 3) = \frac{9 \cdot 4}{4} + \frac{- 9}{1} + \frac{1}{4}$

Step 2.2.2.5

Multiply $\frac{- 9}{1}$ by $\frac{4}{4}$ .

$f (- 3) = \frac{9 \cdot 4}{4} + \frac{- 9}{1} \cdot \frac{4}{4} + \frac{1}{4}$

Step 2.2.2.6

Multiply $\frac{- 9}{1}$ by $\frac{4}{4}$ .

$f (- 3) = \frac{9 \cdot 4}{4} + \frac{- 9 \cdot 4}{4} + \frac{1}{4}$

Step 2.2.3

Combine the numerators over the common denominator.

$f (- 3) = \frac{9 \cdot 4 - 9 \cdot 4 + 1}{4}$

Step 2.2.4

Simplify each term.

Tap for more steps...

Step 2.2.4.1

Multiply $9$ by $4$ .

$f (- 3) = \frac{36 - 9 \cdot 4 + 1}{4}$

Step 2.2.4.2

Multiply $- 9$ by $4$ .

$f (- 3) = \frac{36 - 36 + 1}{4}$

Step 2.2.5

Simplify by adding and subtracting.

Tap for more steps...

Step 2.2.5.1

Subtract $36$ from $36$ .

$f (- 3) = \frac{0 + 1}{4}$

Step 2.2.5.2

Add $0$ and $1$ .

$f (- 3) = \frac{1}{4}$

Step 2.2.6

The final answer is $\frac{1}{4}$ .

$\frac{1}{4}$

Step 2.3

The $y$ value at $x = - 3$ is $\frac{1}{4}$ .

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

Step 2.4

Replace the variable $x$ with $- 4$ in the expression.

$f (- 4) = {(- 4)}^{2} + 3 (- 4) + \frac{1}{4}$

Step 2.5

Simplify the result.

Tap for more steps...

Step 2.5.1

Simplify each term.

Tap for more steps...

Step 2.5.1.1

Raise $- 4$ to the power of $2$ .

$f (- 4) = 16 + 3 (- 4) + \frac{1}{4}$

Step 2.5.1.2

Multiply $3$ by $- 4$ .

$f (- 4) = 16 - 12 + \frac{1}{4}$

Step 2.5.2

Find the common denominator.

Tap for more steps...

Step 2.5.2.1

Write $16$ as a fraction with denominator $1$ .

$f (- 4) = \frac{16}{1} - 12 + \frac{1}{4}$

Step 2.5.2.2

Multiply $\frac{16}{1}$ by $\frac{4}{4}$ .

$f (- 4) = \frac{16}{1} \cdot \frac{4}{4} - 12 + \frac{1}{4}$

Step 2.5.2.3

Multiply $\frac{16}{1}$ by $\frac{4}{4}$ .

$f (- 4) = \frac{16 \cdot 4}{4} - 12 + \frac{1}{4}$

Step 2.5.2.4

Write $- 12$ as a fraction with denominator $1$ .

$f (- 4) = \frac{16 \cdot 4}{4} + \frac{- 12}{1} + \frac{1}{4}$

Step 2.5.2.5

Multiply $\frac{- 12}{1}$ by $\frac{4}{4}$ .

$f (- 4) = \frac{16 \cdot 4}{4} + \frac{- 12}{1} \cdot \frac{4}{4} + \frac{1}{4}$

Step 2.5.2.6

Multiply $\frac{- 12}{1}$ by $\frac{4}{4}$ .

$f (- 4) = \frac{16 \cdot 4}{4} + \frac{- 12 \cdot 4}{4} + \frac{1}{4}$

Step 2.5.3

Combine the numerators over the common denominator.

$f (- 4) = \frac{16 \cdot 4 - 12 \cdot 4 + 1}{4}$

Step 2.5.4

Simplify each term.

Tap for more steps...

Step 2.5.4.1

Multiply $16$ by $4$ .

$f (- 4) = \frac{64 - 12 \cdot 4 + 1}{4}$

Step 2.5.4.2

Multiply $- 12$ by $4$ .

$f (- 4) = \frac{64 - 48 + 1}{4}$

Step 2.5.5

Simplify by adding and subtracting.

Tap for more steps...

Step 2.5.5.1

Subtract $48$ from $64$ .

$f (- 4) = \frac{16 + 1}{4}$

Step 2.5.5.2

Add $16$ and $1$ .

$f (- 4) = \frac{17}{4}$

Step 2.5.6

The final answer is $\frac{17}{4}$ .

$\frac{17}{4}$

Step 2.6

The $y$ value at $x = - 4$ is $\frac{17}{4}$ .

$y = \frac{17}{4}$

Step 2.7

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

$f (- 1) = {(- 1)}^{2} + 3 (- 1) + \frac{1}{4}$

Step 2.8

Simplify the result.

Tap for more steps...

Step 2.8.1

Simplify each term.

Tap for more steps...

Step 2.8.1.1

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

$f (- 1) = 1 + 3 (- 1) + \frac{1}{4}$

Step 2.8.1.2

Multiply $3$ by $- 1$ .

$f (- 1) = 1 - 3 + \frac{1}{4}$

Step 2.8.2

Find the common denominator.

Tap for more steps...

Step 2.8.2.1

Write $1$ as a fraction with denominator $1$ .

$f (- 1) = \frac{1}{1} - 3 + \frac{1}{4}$

Step 2.8.2.2

Multiply $\frac{1}{1}$ by $\frac{4}{4}$ .

$f (- 1) = \frac{1}{1} \cdot \frac{4}{4} - 3 + \frac{1}{4}$

Step 2.8.2.3

Multiply $\frac{1}{1}$ by $\frac{4}{4}$ .

$f (- 1) = \frac{4}{4} - 3 + \frac{1}{4}$

Step 2.8.2.4

Write $- 3$ as a fraction with denominator $1$ .

$f (- 1) = \frac{4}{4} + \frac{- 3}{1} + \frac{1}{4}$

Step 2.8.2.5

Multiply $\frac{- 3}{1}$ by $\frac{4}{4}$ .

$f (- 1) = \frac{4}{4} + \frac{- 3}{1} \cdot \frac{4}{4} + \frac{1}{4}$

Step 2.8.2.6

Multiply $\frac{- 3}{1}$ by $\frac{4}{4}$ .

$f (- 1) = \frac{4}{4} + \frac{- 3 \cdot 4}{4} + \frac{1}{4}$

Step 2.8.3

Combine the numerators over the common denominator.

$f (- 1) = \frac{4 - 3 \cdot 4 + 1}{4}$

Step 2.8.4

Simplify the expression.

Tap for more steps...

Step 2.8.4.1

Multiply $- 3$ by $4$ .

$f (- 1) = \frac{4 - 12 + 1}{4}$

Step 2.8.4.2

Subtract $12$ from $4$ .

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

Step 2.8.4.3

Add $- 8$ and $1$ .

$f (- 1) = \frac{- 7}{4}$

Step 2.8.4.4

Move the negative in front of the fraction.

$f (- 1) = - \frac{7}{4}$

Step 2.8.5

The final answer is $- \frac{7}{4}$ .

$- \frac{7}{4}$

Step 2.9

The $y$ value at $x = - 1$ is $- \frac{7}{4}$ .

$y = - \frac{7}{4}$

Step 2.10

Replace the variable $x$ with $0$ in the expression.

$f (0) = {(0)}^{2} + 3 (0) + \frac{1}{4}$

Step 2.11

Simplify the result.

Tap for more steps...

Step 2.11.1

Simplify each term.

Tap for more steps...

Step 2.11.1.1

Raising $0$ to any positive power yields $0$ .

$f (0) = 0 + 3 (0) + \frac{1}{4}$

Step 2.11.1.2

Multiply $3$ by $0$ .

$f (0) = 0 + 0 + \frac{1}{4}$

Step 2.11.2

Simplify by adding numbers.

Tap for more steps...

Step 2.11.2.1

Add $0$ and $0$ .

$f (0) = 0 + \frac{1}{4}$

Step 2.11.2.2

Add $0$ and $\frac{1}{4}$ .

$f (0) = \frac{1}{4}$

Step 2.11.3

The final answer is $\frac{1}{4}$ .

$\frac{1}{4}$

Step 2.12

The $y$ value at $x = 0$ is $\frac{1}{4}$ .

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

Step 2.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 4 & \frac{17}{4} \\ - 3 & \frac{1}{4} \\ - \frac{3}{2} & - 2 \\ - 1 & - \frac{7}{4} \\ 0 & \frac{1}{4} \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(- \frac{3}{2}, - 2)$

Focus: $(- \frac{3}{2}, - \frac{7}{4})$

Axis of Symmetry: $x = - \frac{3}{2}$

Directrix: $y = - \frac{9}{4}$

$\begin{array}{c} x & y \\ - 4 & \frac{17}{4} \\ - 3 & \frac{1}{4} \\ - \frac{3}{2} & - 2 \\ - 1 & - \frac{7}{4} \\ 0 & \frac{1}{4} \end{array}$

Step 4