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

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

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 $k$ .

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

$h = - 2$

$k = 1$

Since the value of $a$ is negative, the parabola opens down.

Opens Down

Find the vertex $(h, k)$ .

$(- 2, 1)$

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.

Cancel the common factor of $1$ and $- 1$ .

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

$\frac{- 1 (- 1)}{4 \cdot (- \frac{1}{4})}$

Move the negative in front of the fraction.

$- \frac{1}{4 (\frac{1}{4})}$

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

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

Divide $4$ by $4$ .

$- \frac{1}{1}$

Cancel the common factor of $1$ .

Cancel the common factor.

$- \frac{1}{1}$

Rewrite the expression.

$- 1 \cdot 1$

Multiply $- 1$ 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.

$(- 2, 0)$

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.

$y = 2$

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

Direction: Opens Down

Vertex: $(- 2, 1)$

Focus: $(- 2, 0)$

Axis of Symmetry: $x = - 2$

Directrix: $y = 2$

Direction: Opens Down

Vertex: $(- 2, 1)$

Focus: $(- 2, 0)$

Axis of Symmetry: $x = - 2$

Directrix: $y = 2$

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 $- 4$ in the expression.

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

Simplify the result.

Simplify each term.

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

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

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

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

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

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

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

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

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

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

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

Cancel the common factors.

Factor $4$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $4$ by $1$ .

$f (- 4) = - 1 \cdot 4 - (- 4)$

Multiply $- 1$ by $4$ .

$f (- 4) = - 4 - (- 4)$

Multiply $- 1$ by $- 4$ .

$f (- 4) = - 4 + 4$

Add $- 4$ and $4$ .

$f (- 4) = 0$

The final answer is $0$ .

$0$

The $y$ value at $x = - 4$ is $0$ .

$y = 0$

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

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

Simplify the result.

Simplify each term.

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

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

Multiply $- 1$ by $- 3$ .

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

To write $3$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Combine $3$ and $\frac{4}{4}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $3$ by $4$ .

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

Add $- 9$ and $12$ .

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

The final answer is $\frac{3}{4}$ .

$\frac{3}{4}$

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

$y = \frac{3}{4}$

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

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

Simplify the result.

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

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

Divide $0$ by $4$ .

$f (0) = - 0 - (0)$

Multiply $- 1$ by $0$ .

$f (0) = 0 - (0)$

Subtract $0$ from $0$ .

$f (0) = 0$

The final answer is $0$ .

$0$

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

$y = 0$

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

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

Simplify the result.

Simplify each term.

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

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

Multiply $- 1$ by $- 1$ .

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

Simplify the expression.

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

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

Combine the numerators over the common denominator.

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

Add $- 1$ and $4$ .

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

The final answer is $\frac{3}{4}$ .

$\frac{3}{4}$

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

$y = \frac{3}{4}$

Graph the parabola using its properties and the selected points.

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

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(- 2, 1)$

Focus: $(- 2, 0)$

Axis of Symmetry: $x = - 2$

Directrix: $y = 2$

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

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

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