Graph y=1/4(x-2)^2-4

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

Step 1

Find the properties of the given parabola.

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

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

$h = 2$

$k = - 4$

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

Opens Up

Find the vertex $(h, k)$ .

$(2, - 4)$

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

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

$\frac{1}{4 a}$

Substitute the value of $a$ into the formula.

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

Simplify.

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

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

Simplify by dividing numbers.

Divide $4$ by $4$ .

$\frac{1}{1}$

Divide $1$ by $1$ .

$1$

Find the focus.

The focus of a parabola can be found by adding $p$ to the y-coordinate $k$ if the parabola opens up or down.

$(h, k + p)$

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

$(2, - 3)$

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

$x = 2$

Find the directrix.

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.

$y = k - p$

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

$y = - 5$

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

Direction: Opens Up

Vertex: $(2, - 4)$

Focus: $(2, - 3)$

Axis of Symmetry: $x = 2$

Directrix: $y = - 5$

Direction: Opens Up

Vertex: $(2, - 4)$

Focus: $(2, - 3)$

Axis of Symmetry: $x = 2$

Directrix: $y = - 5$

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.

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

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

Simplify the result.

Find the common denominator.

Write $- (0)$ as a fraction with denominator $1$ .

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

Multiply $\frac{- (0)}{1}$ by $\frac{4}{4}$ .

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

Multiply $\frac{- (0)}{1}$ by $\frac{4}{4}$ .

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify each term.

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

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

Multiply $- (0) \cdot 4$ .

Multiply $- 1$ by $0$ .

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

Multiply $0$ by $4$ .

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

Multiply $- 3$ by $4$ .

$f (0) = \frac{0 + 0 - 12}{4}$

Simplify the expression.

Add $0$ and $0$ .

$f (0) = \frac{0 - 12}{4}$

Subtract $12$ from $0$ .

$f (0) = \frac{- 12}{4}$

Divide $- 12$ by $4$ .

$f (0) = - 3$

The final answer is $- 3$ .

$- 3$

The $y$ value at $x = 0$ is $- 3$ .

$y = - 3$

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

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

Simplify the result.

Find the common denominator.

Write $- (1)$ as a fraction with denominator $1$ .

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

Multiply $\frac{- (1)}{1}$ by $\frac{4}{4}$ .

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

Multiply $\frac{- (1)}{1}$ by $\frac{4}{4}$ .

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify each term.

One to any power is one.

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

Multiply $- 1$ by $4$ .

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

Multiply $- 3$ by $4$ .

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

Simplify the expression.

Subtract $4$ from $1$ .

$f (1) = \frac{- 3 - 12}{4}$

Subtract $12$ from $- 3$ .

$f (1) = \frac{- 15}{4}$

Move the negative in front of the fraction.

$f (1) = - \frac{15}{4}$

The final answer is $- \frac{15}{4}$ .

$- \frac{15}{4}$

The $y$ value at $x = 1$ is $- \frac{15}{4}$ .

$y = - \frac{15}{4}$

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

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

Simplify the result.

Find the common denominator.

Write $- (4)$ as a fraction with denominator $1$ .

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

Multiply $\frac{- (4)}{1}$ by $\frac{4}{4}$ .

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

Multiply $\frac{- (4)}{1}$ by $\frac{4}{4}$ .

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify each term.

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

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

Multiply $- (4) \cdot 4$ .

Multiply $- 1$ by $4$ .

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

Multiply $- 4$ by $4$ .

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

Multiply $- 3$ by $4$ .

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

Simplify the expression.

Subtract $16$ from $16$ .

$f (4) = \frac{0 - 12}{4}$

Subtract $12$ from $0$ .

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

Divide $- 12$ by $4$ .

$f (4) = - 3$

The final answer is $- 3$ .

$- 3$

The $y$ value at $x = 4$ is $- 3$ .

$y = - 3$

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

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

Simplify the result.

Find the common denominator.

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

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

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

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

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

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify each term.

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

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

Multiply $- (3) \cdot 4$ .

Multiply $- 1$ by $3$ .

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

Multiply $- 3$ by $4$ .

$f (3) = \frac{9 - 12 - 3 \cdot 4}{4}$

Multiply $- 3$ by $4$ .

$f (3) = \frac{9 - 12 - 12}{4}$

Simplify the expression.

Subtract $12$ from $9$ .

$f (3) = \frac{- 3 - 12}{4}$

Subtract $12$ from $- 3$ .

$f (3) = \frac{- 15}{4}$

Move the negative in front of the fraction.

$f (3) = - \frac{15}{4}$

The final answer is $- \frac{15}{4}$ .

$- \frac{15}{4}$

The $y$ value at $x = 3$ is $- \frac{15}{4}$ .

$y = - \frac{15}{4}$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 0 & - 3 \\ 1 & - \frac{15}{4} \\ 2 & - 4 \\ 3 & - \frac{15}{4} \\ 4 & - 3 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(2, - 4)$

Focus: $(2, - 3)$

Axis of Symmetry: $x = 2$

Directrix: $y = - 5$

$\begin{array}{c} x & y \\ 0 & - 3 \\ 1 & - \frac{15}{4} \\ 2 & - 4 \\ 3 & - \frac{15}{4} \\ 4 & - 3 \end{array}$

Step 4

Graph y=1/4(x-2)^2-4

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