Graph y=2x^2+4x+7

$y = 2 x^{2} + 4 x + 7$

Step 1

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Complete the square for $2 x^{2} + 4 x + 7$ .

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

$a = 2$

$b = 4$

$c = 7$

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

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

Simplify the right side.

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

Factor $2$ out of $4$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$d = 1$

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

$e = 7 - \frac{4^{2}}{4 \cdot 2}$

Simplify the right side.

Simplify each term.

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

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

$e = 7 - \frac{4 \cdot 4}{4 \cdot 2}$

Cancel the common factors.

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

$e = 7 - \frac{4 \cdot 4}{4 (2)}$

Cancel the common factor.

$e = 7 - \frac{4 \cdot 4}{4 \cdot 2}$

Rewrite the expression.

$e = 7 - \frac{4}{2}$

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

Factor $2$ out of $4$ .

$e = 7 - \frac{2 \cdot 2}{2}$

Cancel the common factors.

Factor $2$ out of $2$ .

$e = 7 - \frac{2 \cdot 2}{2 (1)}$

Cancel the common factor.

$e = 7 - \frac{2 \cdot 2}{2 \cdot 1}$

Rewrite the expression.

$e = 7 - \frac{2}{1}$

Divide $2$ by $1$ .

$e = 7 - 1 \cdot 2$

Multiply $- 1$ by $2$ .

$e = 7 - 2$

Subtract $2$ from $7$ .

$e = 5$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $2 {(x + 1)}^{2} + 5$ .

$2 {(x + 1)}^{2} + 5$

Set $y$ equal to the new right side.

$y = 2 {(x + 1)}^{2} + 5$

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

$a = 2$

$h = - 1$

$k = 5$

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

Opens Up

Find the vertex $(h, k)$ .

$(- 1, 5)$

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.

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

Multiply $4$ by $2$ .

$\frac{1}{8}$

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.

$(- 1, \frac{41}{8})$

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.

$y = \frac{39}{8}$

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

Direction: Opens Up

Vertex: $(- 1, 5)$

Focus: $(- 1, \frac{41}{8})$

Axis of Symmetry: $x = - 1$

Directrix: $y = \frac{39}{8}$

Direction: Opens Up

Vertex: $(- 1, 5)$

Focus: $(- 1, \frac{41}{8})$

Axis of Symmetry: $x = - 1$

Directrix: $y = \frac{39}{8}$

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 $- 2$ in the expression.

$f (- 2) = 2 {(- 2)}^{2} + 4 (- 2) + 7$

Simplify the result.

Simplify each term.

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

$f (- 2) = 2 \cdot 4 + 4 (- 2) + 7$

Multiply $2$ by $4$ .

$f (- 2) = 8 + 4 (- 2) + 7$

Multiply $4$ by $- 2$ .

$f (- 2) = 8 - 8 + 7$

Simplify by adding and subtracting.

Subtract $8$ from $8$ .

$f (- 2) = 0 + 7$

Add $0$ and $7$ .

$f (- 2) = 7$

The final answer is $7$ .

$7$

The $y$ value at $x = - 2$ is $7$ .

$y = 7$

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

$f (- 3) = 2 {(- 3)}^{2} + 4 (- 3) + 7$

Simplify the result.

Simplify each term.

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

$f (- 3) = 2 \cdot 9 + 4 (- 3) + 7$

Multiply $2$ by $9$ .

$f (- 3) = 18 + 4 (- 3) + 7$

Multiply $4$ by $- 3$ .

$f (- 3) = 18 - 12 + 7$

Simplify by adding and subtracting.

Subtract $12$ from $18$ .

$f (- 3) = 6 + 7$

Add $6$ and $7$ .

$f (- 3) = 13$

The final answer is $13$ .

$13$

The $y$ value at $x = - 3$ is $13$ .

$y = 13$

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

$f (0) = 2 {(0)}^{2} + 4 (0) + 7$

Simplify the result.

Simplify each term.

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

$f (0) = 2 \cdot 0 + 4 (0) + 7$

Multiply $2$ by $0$ .

$f (0) = 0 + 4 (0) + 7$

Multiply $4$ by $0$ .

$f (0) = 0 + 0 + 7$

Simplify by adding numbers.

Add $0$ and $0$ .

$f (0) = 0 + 7$

Add $0$ and $7$ .

$f (0) = 7$

The final answer is $7$ .

$7$

The $y$ value at $x = 0$ is $7$ .

$y = 7$

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

$f (1) = 2 {(1)}^{2} + 4 (1) + 7$

Simplify the result.

Simplify each term.

One to any power is one.

$f (1) = 2 \cdot 1 + 4 (1) + 7$

Multiply $2$ by $1$ .

$f (1) = 2 + 4 (1) + 7$

Multiply $4$ by $1$ .

$f (1) = 2 + 4 + 7$

Simplify by adding numbers.

Add $2$ and $4$ .

$f (1) = 6 + 7$

Add $6$ and $7$ .

$f (1) = 13$

The final answer is $13$ .

$13$

The $y$ value at $x = 1$ is $13$ .

$y = 13$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 3 & 13 \\ - 2 & 7 \\ - 1 & 5 \\ 0 & 7 \\ 1 & 13 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(- 1, 5)$

Focus: $(- 1, \frac{41}{8})$

Axis of Symmetry: $x = - 1$

Directrix: $y = \frac{39}{8}$

$\begin{array}{c} x & y \\ - 3 & 13 \\ - 2 & 7 \\ - 1 & 5 \\ 0 & 7 \\ 1 & 13 \end{array}$

Step 4

Graph y=2x^2+4x+7

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