Graph y=-2x^2+8x-5

$y = - 2 x^{2} + 8 x - 5$

Step 1

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Complete the square for $- 2 x^{2} + 8 x - 5$ .

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

$a = - 2$

$b = 8$

$c = - 5$

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 $d = \frac{b}{2 a}$ .

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

Simplify the right side.

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

Factor $2$ out of $8$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $4$ and $- 2$ .

Factor $2$ out of $4$ .

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

Move the negative one from the denominator of $\frac{2}{- 1}$ .

$d = - 1 \cdot 2$

Multiply $- 1$ by $2$ .

$d = - 2$

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 $e = c - \frac{b^{2}}{4 a}$ .

$e = - 5 - \frac{8^{2}}{4 \cdot - 2}$

Simplify the right side.

Simplify each term.

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

$e = - 5 - \frac{64}{4 \cdot - 2}$

Multiply $4$ by $- 2$ .

$e = - 5 - \frac{64}{- 8}$

Divide $64$ by $- 8$ .

$e = - 5 - - 8$

Multiply $- 1$ by $- 8$ .

$e = - 5 + 8$

Add $- 5$ and $8$ .

$e = 3$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 2 {(x - 2)}^{2} + 3$ .

$- 2 {(x - 2)}^{2} + 3$

Set $y$ equal to the new right side.

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

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

$a = - 2$

$h = 2$

$k = 3$

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

Opens Down

Find the vertex $(h, k)$ .

$(2, 3)$

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.

$\frac{1}{4 \cdot - 2}$

Simplify.

Multiply $4$ by $- 2$ .

$\frac{1}{- 8}$

Move the negative in front of the fraction.

$- \frac{1}{8}$

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.

$(2, \frac{23}{8})$

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.

$y = \frac{25}{8}$

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

Direction: Opens Down

Vertex: $(2, 3)$

Focus: $(2, \frac{23}{8})$

Axis of Symmetry: $x = 2$

Directrix: $y = \frac{25}{8}$

Direction: Opens Down

Vertex: $(2, 3)$

Focus: $(2, \frac{23}{8})$

Axis of Symmetry: $x = 2$

Directrix: $y = \frac{25}{8}$

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 $1$ in the expression.

$f (1) = - 2 {(1)}^{2} + 8 (1) - 5$

Simplify the result.

Simplify each term.

One to any power is one.

$f (1) = - 2 \cdot 1 + 8 (1) - 5$

Multiply $- 2$ by $1$ .

$f (1) = - 2 + 8 (1) - 5$

Multiply $8$ by $1$ .

$f (1) = - 2 + 8 - 5$

Simplify by adding and subtracting.

Add $- 2$ and $8$ .

$f (1) = 6 - 5$

Subtract $5$ from $6$ .

$f (1) = 1$

The final answer is $1$ .

$1$

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

$y = 1$

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

$f (0) = - 2 {(0)}^{2} + 8 (0) - 5$

Simplify the result.

Simplify each term.

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

$f (0) = - 2 \cdot 0 + 8 (0) - 5$

Multiply $- 2$ by $0$ .

$f (0) = 0 + 8 (0) - 5$

Multiply $8$ by $0$ .

$f (0) = 0 + 0 - 5$

Simplify by adding and subtracting.

Add $0$ and $0$ .

$f (0) = 0 - 5$

Subtract $5$ from $0$ .

$f (0) = - 5$

The final answer is $- 5$ .

$- 5$

The $y$ value at $x = 0$ is $- 5$ .

$y = - 5$

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

$f (3) = - 2 {(3)}^{2} + 8 (3) - 5$

Simplify the result.

Simplify each term.

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

$f (3) = - 2 \cdot 9 + 8 (3) - 5$

Multiply $- 2$ by $9$ .

$f (3) = - 18 + 8 (3) - 5$

Multiply $8$ by $3$ .

$f (3) = - 18 + 24 - 5$

Simplify by adding and subtracting.

Add $- 18$ and $24$ .

$f (3) = 6 - 5$

Subtract $5$ from $6$ .

$f (3) = 1$

The final answer is $1$ .

$1$

The $y$ value at $x = 3$ is $1$ .

$y = 1$

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

$f (4) = - 2 {(4)}^{2} + 8 (4) - 5$

Simplify the result.

Simplify each term.

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

$f (4) = - 2 \cdot 16 + 8 (4) - 5$

Multiply $- 2$ by $16$ .

$f (4) = - 32 + 8 (4) - 5$

Multiply $8$ by $4$ .

$f (4) = - 32 + 32 - 5$

Simplify by adding and subtracting.

Add $- 32$ and $32$ .

$f (4) = 0 - 5$

Subtract $5$ from $0$ .

$f (4) = - 5$

The final answer is $- 5$ .

$- 5$

The $y$ value at $x = 4$ is $- 5$ .

$y = - 5$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 0 & - 5 \\ 1 & 1 \\ 2 & 3 \\ 3 & 1 \\ 4 & - 5 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(2, 3)$

Focus: $(2, \frac{23}{8})$

Axis of Symmetry: $x = 2$

Directrix: $y = \frac{25}{8}$

$\begin{array}{c} x & y \\ 0 & - 5 \\ 1 & 1 \\ 2 & 3 \\ 3 & 1 \\ 4 & - 5 \end{array}$

Step 4

Graph y=-2x^2+8x-5

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