Graph y=-4x^2+8x+2

$y = - 4 x^{2} + 8 x + 2$

Step 1

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Complete the square for $- 4 x^{2} + 8 x + 2$ .

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

$a = - 4$

$b = 8$

$c = 2$

Consider the vertex form of a parabola.

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

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

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

$d = \frac{8}{2 \cdot - 4}$

Simplify the right side.

Cancel the common factor of $8$ and $2$ .

Factor $2$ out of $8$ .

$d = \frac{2 \cdot 4}{2 \cdot - 4}$

Cancel the common factors.

Factor $2$ out of $2 \cdot - 4$ .

$d = \frac{2 \cdot 4}{2 (- 4)}$

Cancel the common factor.

$d = \frac{2 \cdot 4}{2 \cdot - 4}$

Rewrite the expression.

$d = \frac{4}{- 4}$

Cancel the common factor of $4$ and $- 4$ .

Factor $4$ out of $4$ .

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

Move the negative one from the denominator of $\frac{1}{- 1}$ .

$d = - 1 \cdot 1$

Multiply $- 1$ by $1$ .

$d = - 1$

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

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

$e = 2 - \frac{8^{2}}{4 \cdot - 4}$

Simplify the right side.

Simplify each term.

Raise $8$ to the power of $2$ .

$e = 2 - \frac{64}{4 \cdot - 4}$

Multiply $4$ by $- 4$ .

$e = 2 - \frac{64}{- 16}$

Divide $64$ by $- 16$ .

$e = 2 - - 4$

Multiply $- 1$ by $- 4$ .

$e = 2 + 4$

Add $2$ and $4$ .

$e = 6$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 4 {(x - 1)}^{2} + 6$ .

$- 4 {(x - 1)}^{2} + 6$

Set $y$ equal to the new right side.

$y = - 4 {(x - 1)}^{2} + 6$

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

$a = - 4$

$h = 1$

$k = 6$

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

Opens Down

Find the vertex $(h, k)$ .

$(1, 6)$

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.

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

Simplify.

Multiply $4$ by $- 4$ .

$\frac{1}{- 16}$

Move the negative in front of the fraction.

$- \frac{1}{16}$

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.

$(1, \frac{95}{16})$

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

$x = 1$

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.

$y = \frac{97}{16}$

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

Direction: Opens Down

Vertex: $(1, 6)$

Focus: $(1, \frac{95}{16})$

Axis of Symmetry: $x = 1$

Directrix: $y = \frac{97}{16}$

Direction: Opens Down

Vertex: $(1, 6)$

Focus: $(1, \frac{95}{16})$

Axis of Symmetry: $x = 1$

Directrix: $y = \frac{97}{16}$

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.

$f (0) = - 4 {(0)}^{2} + 8 (0) + 2$

Simplify the result.

Simplify each term.

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

$f (0) = - 4 \cdot 0 + 8 (0) + 2$

Multiply $- 4$ by $0$ .

$f (0) = 0 + 8 (0) + 2$

Multiply $8$ by $0$ .

$f (0) = 0 + 0 + 2$

Simplify by adding numbers.

Add $0$ and $0$ .

$f (0) = 0 + 2$

Add $0$ and $2$ .

$f (0) = 2$

The final answer is $2$ .

$2$

The $y$ value at $x = 0$ is $2$ .

$y = 2$

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

$f (- 1) = - 4 {(- 1)}^{2} + 8 (- 1) + 2$

Simplify the result.

Simplify each term.

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

$f (- 1) = - 4 \cdot 1 + 8 (- 1) + 2$

Multiply $- 4$ by $1$ .

$f (- 1) = - 4 + 8 (- 1) + 2$

Multiply $8$ by $- 1$ .

$f (- 1) = - 4 - 8 + 2$

Simplify by adding and subtracting.

Subtract $8$ from $- 4$ .

$f (- 1) = - 12 + 2$

Add $- 12$ and $2$ .

$f (- 1) = - 10$

The final answer is $- 10$ .

$- 10$

The $y$ value at $x = - 1$ is $- 10$ .

$y = - 10$

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

$f (2) = - 4 {(2)}^{2} + 8 (2) + 2$

Simplify the result.

Simplify each term.

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

$f (2) = - 4 \cdot 4 + 8 (2) + 2$

Multiply $- 4$ by $4$ .

$f (2) = - 16 + 8 (2) + 2$

Multiply $8$ by $2$ .

$f (2) = - 16 + 16 + 2$

Simplify by adding numbers.

Add $- 16$ and $16$ .

$f (2) = 0 + 2$

Add $0$ and $2$ .

$f (2) = 2$

The final answer is $2$ .

$2$

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

$y = 2$

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

$f (3) = - 4 {(3)}^{2} + 8 (3) + 2$

Simplify the result.

Simplify each term.

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

$f (3) = - 4 \cdot 9 + 8 (3) + 2$

Multiply $- 4$ by $9$ .

$f (3) = - 36 + 8 (3) + 2$

Multiply $8$ by $3$ .

$f (3) = - 36 + 24 + 2$

Simplify by adding numbers.

Add $- 36$ and $24$ .

$f (3) = - 12 + 2$

Add $- 12$ and $2$ .

$f (3) = - 10$

The final answer is $- 10$ .

$- 10$

The $y$ value at $x = 3$ is $- 10$ .

$y = - 10$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 1 & - 10 \\ 0 & 2 \\ 1 & 6 \\ 2 & 2 \\ 3 & - 10 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(1, 6)$

Focus: $(1, \frac{95}{16})$

Axis of Symmetry: $x = 1$

Directrix: $y = \frac{97}{16}$

$\begin{array}{c} x & y \\ - 1 & - 10 \\ 0 & 2 \\ 1 & 6 \\ 2 & 2 \\ 3 & - 10 \end{array}$

Step 4

Graph y=-4x^2+8x+2

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