Precalculus Examples

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

Step 1

Solve for $y$ .

Tap for more steps...

Step 1.1

Rewrite the equation as $4 (y - 1) = {(x + \frac{1}{2})}^{2}$ .

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

Step 1.2

Divide each term in $4 (y - 1) = {(x + \frac{1}{2})}^{2}$ by $4$ and simplify.

Tap for more steps...

Step 1.2.1

Divide each term in $4 (y - 1) = {(x + \frac{1}{2})}^{2}$ by $4$ .

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

Step 1.2.2

Simplify the left side.

Tap for more steps...

Step 1.2.2.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 1.2.2.1.1

Cancel the common factor.

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

Step 1.2.2.1.2

Divide $y - 1$ by $1$ .

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

Step 1.2.3

Simplify the right side.

Tap for more steps...

Step 1.2.3.1

Simplify the numerator.

Tap for more steps...

Step 1.2.3.1.1

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

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

Step 1.2.3.1.2

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

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

Step 1.2.3.1.3

Combine the numerators over the common denominator.

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

Step 1.2.3.1.4

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

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

Step 1.2.3.1.5

Apply the product rule to $\frac{2 x + 1}{2}$ .

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

Step 1.2.3.1.6

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

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

Step 1.2.3.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.2.3.3

Combine.

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

Step 1.2.3.4

Multiply.

Tap for more steps...

Step 1.2.3.4.1

Multiply ${(2 x + 1)}^{2}$ by $1$ .

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

Step 1.2.3.4.2

Multiply $4$ by $4$ .

$y - 1 = \frac{{(2 x + 1)}^{2}}{16}$

Step 1.3

Add $1$ to both sides of the equation.

$y = \frac{{(2 x + 1)}^{2}}{16} + 1$

Step 2

Find the properties of the given parabola.

Tap for more steps...

Step 2.1

Rewrite the equation in vertex form.

Tap for more steps...

Step 2.1.1

Reorder terms.

$y = \frac{1}{16} \cdot {(2 x + 1)}^{2} + 1$

Step 2.1.2

Complete the square for $\frac{1}{16} \cdot {(2 x + 1)}^{2} + 1$ .

Tap for more steps...

Step 2.1.2.1

Simplify the expression.

Tap for more steps...

Step 2.1.2.1.1

Simplify each term.

Tap for more steps...

Step 2.1.2.1.1.1

Rewrite ${(2 x + 1)}^{2}$ as $(2 x + 1) (2 x + 1)$ .

$\frac{1}{16} \cdot ((2 x + 1) (2 x + 1)) + 1$

Step 2.1.2.1.1.2

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

Tap for more steps...

Step 2.1.2.1.1.2.1

Apply the distributive property.

$\frac{1}{16} \cdot (2 x (2 x + 1) + 1 (2 x + 1)) + 1$

Step 2.1.2.1.1.2.2

Apply the distributive property.

$\frac{1}{16} \cdot (2 x (2 x) + 2 x \cdot 1 + 1 (2 x + 1)) + 1$

Step 2.1.2.1.1.2.3

Apply the distributive property.

$\frac{1}{16} \cdot (2 x (2 x) + 2 x \cdot 1 + 1 (2 x) + 1 \cdot 1) + 1$

Step 2.1.2.1.1.3

Simplify and combine like terms.

Tap for more steps...

Step 2.1.2.1.1.3.1

Simplify each term.

Tap for more steps...

Step 2.1.2.1.1.3.1.1

Rewrite using the commutative property of multiplication.

$\frac{1}{16} \cdot (2 \cdot 2 x \cdot x + 2 x \cdot 1 + 1 (2 x) + 1 \cdot 1) + 1$

Step 2.1.2.1.1.3.1.2

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 2.1.2.1.1.3.1.2.1

Move $x$ .

$\frac{1}{16} \cdot (2 \cdot 2 (x \cdot x) + 2 x \cdot 1 + 1 (2 x) + 1 \cdot 1) + 1$

Step 2.1.2.1.1.3.1.2.2

Multiply $x$ by $x$ .

$\frac{1}{16} \cdot (2 \cdot 2 x^{2} + 2 x \cdot 1 + 1 (2 x) + 1 \cdot 1) + 1$

Step 2.1.2.1.1.3.1.3

Multiply $2$ by $2$ .

$\frac{1}{16} \cdot (4 x^{2} + 2 x \cdot 1 + 1 (2 x) + 1 \cdot 1) + 1$

Step 2.1.2.1.1.3.1.4

Multiply $2$ by $1$ .

$\frac{1}{16} \cdot (4 x^{2} + 2 x + 1 (2 x) + 1 \cdot 1) + 1$

Step 2.1.2.1.1.3.1.5

Multiply $2 x$ by $1$ .

$\frac{1}{16} \cdot (4 x^{2} + 2 x + 2 x + 1 \cdot 1) + 1$

Step 2.1.2.1.1.3.1.6

Multiply $1$ by $1$ .

$\frac{1}{16} \cdot (4 x^{2} + 2 x + 2 x + 1) + 1$

Step 2.1.2.1.1.3.2

Add $2 x$ and $2 x$ .

$\frac{1}{16} \cdot (4 x^{2} + 4 x + 1) + 1$

Step 2.1.2.1.1.4

Apply the distributive property.

$\frac{1}{16} (4 x^{2}) + \frac{1}{16} (4 x) + \frac{1}{16} \cdot 1 + 1$

Step 2.1.2.1.1.5

Simplify.

Tap for more steps...

Step 2.1.2.1.1.5.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 2.1.2.1.1.5.1.1

Factor $4$ out of $16$ .

$\frac{1}{4 (4)} (4 x^{2}) + \frac{1}{16} (4 x) + \frac{1}{16} \cdot 1 + 1$

Step 2.1.2.1.1.5.1.2

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

$\frac{1}{4 (4)} (4 (x^{2})) + \frac{1}{16} (4 x) + \frac{1}{16} \cdot 1 + 1$

Step 2.1.2.1.1.5.1.3

Cancel the common factor.

$\frac{1}{4 \cdot 4} (4 x^{2}) + \frac{1}{16} (4 x) + \frac{1}{16} \cdot 1 + 1$

Step 2.1.2.1.1.5.1.4

Rewrite the expression.

$\frac{1}{4} x^{2} + \frac{1}{16} (4 x) + \frac{1}{16} \cdot 1 + 1$

Step 2.1.2.1.1.5.2

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

$\frac{x^{2}}{4} + \frac{1}{16} (4 x) + \frac{1}{16} \cdot 1 + 1$

Step 2.1.2.1.1.5.3

Cancel the common factor of $4$ .

Tap for more steps...

Step 2.1.2.1.1.5.3.1

Factor $4$ out of $16$ .

$\frac{x^{2}}{4} + \frac{1}{4 (4)} (4 x) + \frac{1}{16} \cdot 1 + 1$

Step 2.1.2.1.1.5.3.2

Factor $4$ out of $4 x$ .

$\frac{x^{2}}{4} + \frac{1}{4 (4)} (4 (x)) + \frac{1}{16} \cdot 1 + 1$

Step 2.1.2.1.1.5.3.3

Cancel the common factor.

$\frac{x^{2}}{4} + \frac{1}{4 \cdot 4} (4 x) + \frac{1}{16} \cdot 1 + 1$

Step 2.1.2.1.1.5.3.4

Rewrite the expression.

$\frac{x^{2}}{4} + \frac{1}{4} x + \frac{1}{16} \cdot 1 + 1$

Step 2.1.2.1.1.5.4

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

$\frac{x^{2}}{4} + \frac{x}{4} + \frac{1}{16} \cdot 1 + 1$

Step 2.1.2.1.1.5.5

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

$\frac{x^{2}}{4} + \frac{x}{4} + \frac{1}{16} + 1$

Step 2.1.2.1.2

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

$\frac{x^{2}}{4} + \frac{x}{4} + \frac{1}{16} + \frac{16}{16}$

Step 2.1.2.1.3

Combine the numerators over the common denominator.

$\frac{x^{2}}{4} + \frac{x}{4} + \frac{1 + 16}{16}$

Step 2.1.2.1.4

Add $1$ and $16$ .

$\frac{x^{2}}{4} + \frac{x}{4} + \frac{17}{16}$

Step 2.1.2.2

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

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

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

$c = \frac{17}{16}$

Step 2.1.2.3

Consider the vertex form of a parabola.

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

Step 2.1.2.4

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

Tap for more steps...

Step 2.1.2.4.1

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

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

Step 2.1.2.4.2

Cancel the common factor of $\frac{1}{4}$ .

Tap for more steps...

Step 2.1.2.4.2.1

Cancel the common factor.

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

Step 2.1.2.4.2.2

Rewrite the expression.

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

Step 2.1.2.5

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

Tap for more steps...

Step 2.1.2.5.1

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

$e = \frac{17}{16} - \frac{{(\frac{1}{4})}^{2}}{4 (\frac{1}{4})}$

Step 2.1.2.5.2

Simplify the right side.

Tap for more steps...

Step 2.1.2.5.2.1

Simplify each term.

Tap for more steps...

Step 2.1.2.5.2.1.1

Cancel the common factor of ${(\frac{1}{4})}^{2}$ and $\frac{1}{4}$ .

Tap for more steps...

Step 2.1.2.5.2.1.1.1

Factor $\frac{1}{4}$ out of ${(\frac{1}{4})}^{2}$ .

$e = \frac{17}{16} - \frac{\frac{1}{4} \cdot \frac{1}{4}}{4 (\frac{1}{4})}$

Step 2.1.2.5.2.1.1.2

Cancel the common factors.

Tap for more steps...

Step 2.1.2.5.2.1.1.2.1

Factor $\frac{1}{4}$ out of $4 (\frac{1}{4})$ .

$e = \frac{17}{16} - \frac{\frac{1}{4} \cdot \frac{1}{4}}{\frac{1}{4} \cdot 4}$

Step 2.1.2.5.2.1.1.2.2

Cancel the common factor.

$e = \frac{17}{16} - \frac{\frac{1}{4} \cdot \frac{1}{4}}{\frac{1}{4} \cdot 4}$

Step 2.1.2.5.2.1.1.2.3

Rewrite the expression.

$e = \frac{17}{16} - \frac{\frac{1}{4}}{4}$

Step 2.1.2.5.2.1.2

Multiply the numerator by the reciprocal of the denominator.

$e = \frac{17}{16} - (\frac{1}{4} \cdot \frac{1}{4})$

Step 2.1.2.5.2.1.3

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

Tap for more steps...

Step 2.1.2.5.2.1.3.1

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

$e = \frac{17}{16} - \frac{1}{4 \cdot 4}$

Step 2.1.2.5.2.1.3.2

Multiply $4$ by $4$ .

$e = \frac{17}{16} - \frac{1}{16}$

Step 2.1.2.5.2.2

Combine the numerators over the common denominator.

$e = \frac{17 - 1}{16}$

Step 2.1.2.5.2.3

Subtract $1$ from $17$ .

$e = \frac{16}{16}$

Step 2.1.2.5.2.4

Divide $16$ by $16$ .

$e = 1$

Step 2.1.2.6

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

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

Step 2.1.3

Set $y$ equal to the new right side.

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

Step 2.2

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

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

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

$k = 1$

Step 2.3

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

Opens Up

Step 2.4

Find the vertex $(h, k)$ .

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

Step 2.5

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

Tap for more steps...

Step 2.5.1

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

$\frac{1}{4 a}$

Step 2.5.2

Substitute the value of $a$ into the formula.

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

Step 2.5.3

Simplify.

Tap for more steps...

Step 2.5.3.1

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

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

Step 2.5.3.2

Simplify by dividing numbers.

Tap for more steps...

Step 2.5.3.2.1

Divide $4$ by $4$ .

$\frac{1}{1}$

Step 2.5.3.2.2

Divide $1$ by $1$ .

$1$

Step 2.6

Find the focus.

Tap for more steps...

Step 2.6.1

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 2.6.2

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

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

Step 2.7

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

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

Step 2.8

Find the directrix.

Tap for more steps...

Step 2.8.1

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 2.8.2

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

$y = 0$

Step 2.9

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

Direction: Opens Up

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

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

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

Directrix: $y = 0$

Direction: Opens Up

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

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

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

Directrix: $y = 0$

Step 3

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 3.1

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

$f (- 1) = \frac{4 {(- 1)}^{2} + 4 (- 1) + 17}{16}$

Step 3.2

Simplify the result.

Tap for more steps...

Step 3.2.1

Simplify the numerator.

Tap for more steps...

Step 3.2.1.1

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

$f (- 1) = \frac{4 \cdot 1 + 4 (- 1) + 17}{16}$

Step 3.2.1.2

Multiply $4$ by $1$ .

$f (- 1) = \frac{4 + 4 (- 1) + 17}{16}$

Step 3.2.1.3

Multiply $4$ by $- 1$ .

$f (- 1) = \frac{4 - 4 + 17}{16}$

Step 3.2.1.4

Subtract $4$ from $4$ .

$f (- 1) = \frac{0 + 17}{16}$

Step 3.2.1.5

Add $0$ and $17$ .

$f (- 1) = \frac{17}{16}$

Step 3.2.2

The final answer is $\frac{17}{16}$ .

$\frac{17}{16}$

Step 3.3

The $y$ value at $x = - 1$ is $\frac{17}{16}$ .

$y = \frac{17}{16}$

Step 3.4

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

$f (- 2) = \frac{4 {(- 2)}^{2} + 4 (- 2) + 17}{16}$

Step 3.5

Simplify the result.

Tap for more steps...

Step 3.5.1

Simplify the numerator.

Tap for more steps...

Step 3.5.1.1

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

$f (- 2) = \frac{4 \cdot 4 + 4 (- 2) + 17}{16}$

Step 3.5.1.2

Multiply $4$ by $4$ .

$f (- 2) = \frac{16 + 4 (- 2) + 17}{16}$

Step 3.5.1.3

Multiply $4$ by $- 2$ .

$f (- 2) = \frac{16 - 8 + 17}{16}$

Step 3.5.1.4

Subtract $8$ from $16$ .

$f (- 2) = \frac{8 + 17}{16}$

Step 3.5.1.5

Add $8$ and $17$ .

$f (- 2) = \frac{25}{16}$

Step 3.5.2

The final answer is $\frac{25}{16}$ .

$\frac{25}{16}$

Step 3.6

The $y$ value at $x = - 2$ is $\frac{25}{16}$ .

$y = \frac{25}{16}$

Step 3.7

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

$f (1) = \frac{4 {(1)}^{2} + 4 (1) + 17}{16}$

Step 3.8

Simplify the result.

Tap for more steps...

Step 3.8.1

Simplify the numerator.

Tap for more steps...

Step 3.8.1.1

One to any power is one.

$f (1) = \frac{4 \cdot 1 + 4 (1) + 17}{16}$

Step 3.8.1.2

Multiply $4$ by $1$ .

$f (1) = \frac{4 + 4 (1) + 17}{16}$

Step 3.8.1.3

Multiply $4$ by $1$ .

$f (1) = \frac{4 + 4 + 17}{16}$

Step 3.8.1.4

Add $4$ and $4$ .

$f (1) = \frac{8 + 17}{16}$

Step 3.8.1.5

Add $8$ and $17$ .

$f (1) = \frac{25}{16}$

Step 3.8.2

The final answer is $\frac{25}{16}$ .

$\frac{25}{16}$

Step 3.9

The $y$ value at $x = 1$ is $\frac{25}{16}$ .

$y = \frac{25}{16}$

Step 3.10

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

$f (2) = \frac{4 {(2)}^{2} + 4 (2) + 17}{16}$

Step 3.11

Simplify the result.

Tap for more steps...

Step 3.11.1

Simplify the numerator.

Tap for more steps...

Step 3.11.1.1

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

$f (2) = \frac{4 \cdot 4 + 4 (2) + 17}{16}$

Step 3.11.1.2

Multiply $4$ by $4$ .

$f (2) = \frac{16 + 4 (2) + 17}{16}$

Step 3.11.1.3

Multiply $4$ by $2$ .

$f (2) = \frac{16 + 8 + 17}{16}$

Step 3.11.1.4

Add $16$ and $8$ .

$f (2) = \frac{24 + 17}{16}$

Step 3.11.1.5

Add $24$ and $17$ .

$f (2) = \frac{41}{16}$

Step 3.11.2

The final answer is $\frac{41}{16}$ .

$\frac{41}{16}$

Step 3.12

The $y$ value at $x = 2$ is $\frac{41}{16}$ .

$y = \frac{41}{16}$

Step 3.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 2 & \frac{25}{16} \\ - 1 & \frac{17}{16} \\ - \frac{1}{2} & 1 \\ 1 & \frac{25}{16} \\ 2 & \frac{41}{16} \end{array}$

Step 4

Graph the parabola using its properties and the selected points.

Direction: Opens Up

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

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

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

Directrix: $y = 0$

$\begin{array}{c} x & y \\ - 2 & \frac{25}{16} \\ - 1 & \frac{17}{16} \\ - \frac{1}{2} & 1 \\ 1 & \frac{25}{16} \\ 2 & \frac{41}{16} \end{array}$

Step 5