Graph y=-5(x+4)^2+3

$y = - 5 {(x + 4)}^{2} + 3$

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 $k$ .

$a = - 5$

$h = - 4$

$k = 3$

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

Opens Down

Find the vertex $(h, k)$ .

$(- 4, 3)$

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.

$\frac{1}{4 \cdot - 5}$

Simplify.

Multiply $4$ by $- 5$ .

$\frac{1}{- 20}$

Move the negative in front of the fraction.

$- \frac{1}{20}$

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.

$(- 4, \frac{59}{20})$

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

$x = - 4$

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.

$y = \frac{61}{20}$

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

Direction: Opens Down

Vertex: $(- 4, 3)$

Focus: $(- 4, \frac{59}{20})$

Axis of Symmetry: $x = - 4$

Directrix: $y = \frac{61}{20}$

Direction: Opens Down

Vertex: $(- 4, 3)$

Focus: $(- 4, \frac{59}{20})$

Axis of Symmetry: $x = - 4$

Directrix: $y = \frac{61}{20}$

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 $- 5$ in the expression.

$f (- 5) = - 5 {(- 5)}^{2} - 40 \cdot - 5 - 77$

Simplify the result.

Simplify each term.

Multiply $- 5$ by ${(- 5)}^{2}$ by adding the exponents.

Multiply $- 5$ by ${(- 5)}^{2}$ .

Raise $- 5$ to the power of $1$ .

$f (- 5) = (- 5) {(- 5)}^{2} - 40 \cdot - 5 - 77$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (- 5) = {(- 5)}^{1 + 2} - 40 \cdot - 5 - 77$

Add $1$ and $2$ .

$f (- 5) = {(- 5)}^{3} - 40 \cdot - 5 - 77$

Raise $- 5$ to the power of $3$ .

$f (- 5) = - 125 - 40 \cdot - 5 - 77$

Multiply $- 40$ by $- 5$ .

$f (- 5) = - 125 + 200 - 77$

Simplify by adding and subtracting.

Add $- 125$ and $200$ .

$f (- 5) = 75 - 77$

Subtract $77$ from $75$ .

$f (- 5) = - 2$

The final answer is $- 2$ .

$- 2$

The $y$ value at $x = - 5$ is $- 2$ .

$y = - 2$

Replace the variable $x$ with $- 6$ in the expression.

$f (- 6) = - 5 {(- 6)}^{2} - 40 \cdot - 6 - 77$

Simplify the result.

Simplify each term.

Raise $- 6$ to the power of $2$ .

$f (- 6) = - 5 \cdot 36 - 40 \cdot - 6 - 77$

Multiply $- 5$ by $36$ .

$f (- 6) = - 180 - 40 \cdot - 6 - 77$

Multiply $- 40$ by $- 6$ .

$f (- 6) = - 180 + 240 - 77$

Simplify by adding and subtracting.

Add $- 180$ and $240$ .

$f (- 6) = 60 - 77$

Subtract $77$ from $60$ .

$f (- 6) = - 17$

The final answer is $- 17$ .

$- 17$

The $y$ value at $x = - 6$ is $- 17$ .

$y = - 17$

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

$f (- 3) = - 5 {(- 3)}^{2} - 40 \cdot - 3 - 77$

Simplify the result.

Simplify each term.

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

$f (- 3) = - 5 \cdot 9 - 40 \cdot - 3 - 77$

Multiply $- 5$ by $9$ .

$f (- 3) = - 45 - 40 \cdot - 3 - 77$

Multiply $- 40$ by $- 3$ .

$f (- 3) = - 45 + 120 - 77$

Simplify by adding and subtracting.

Add $- 45$ and $120$ .

$f (- 3) = 75 - 77$

Subtract $77$ from $75$ .

$f (- 3) = - 2$

The final answer is $- 2$ .

$- 2$

The $y$ value at $x = - 3$ is $- 2$ .

$y = - 2$

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

$f (- 2) = - 5 {(- 2)}^{2} - 40 \cdot - 2 - 77$

Simplify the result.

Simplify each term.

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

$f (- 2) = - 5 \cdot 4 - 40 \cdot - 2 - 77$

Multiply $- 5$ by $4$ .

$f (- 2) = - 20 - 40 \cdot - 2 - 77$

Multiply $- 40$ by $- 2$ .

$f (- 2) = - 20 + 80 - 77$

Simplify by adding and subtracting.

Add $- 20$ and $80$ .

$f (- 2) = 60 - 77$

Subtract $77$ from $60$ .

$f (- 2) = - 17$

The final answer is $- 17$ .

$- 17$

The $y$ value at $x = - 2$ is $- 17$ .

$y = - 17$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 6 & - 17 \\ - 5 & - 2 \\ - 4 & 3 \\ - 3 & - 2 \\ - 2 & - 17 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(- 4, 3)$

Focus: $(- 4, \frac{59}{20})$

Axis of Symmetry: $x = - 4$

Directrix: $y = \frac{61}{20}$

$\begin{array}{c} x & y \\ - 6 & - 17 \\ - 5 & - 2 \\ - 4 & 3 \\ - 3 & - 2 \\ - 2 & - 17 \end{array}$

Step 4

Graph y=-5(x+4)^2+3

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