Graph y=2x^2-4x+5

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

Step 1

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Complete the square for $2 x^{2} - 4 x + 5$ .

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

$a = 2$

$b = - 4$

$c = 5$

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

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

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 $- 2$ and $2$ .

Factor $2$ out of $- 2$ .

$d = \frac{2 \cdot - 1}{2}$

Cancel the common factors.

Factor $2$ out of $2$ .

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

Cancel the common factor.

$d = \frac{2 \cdot - 1}{2 \cdot 1}$

Rewrite the expression.

$d = \frac{- 1}{1}$

Divide $- 1$ 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 $e = c - \frac{b^{2}}{4 a}$ .

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

Simplify the right side.

Simplify each term.

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

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

$e = 5 - \frac{{(- 1 (4))}^{2}}{4 \cdot 2}$

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

$e = 5 - \frac{{(- 1)}^{2} \cdot 4^{2}}{4 \cdot 2}$

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

$e = 5 - \frac{1 \cdot 4^{2}}{4 \cdot 2}$

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

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

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

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

Factor $2$ out of $4$ .

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

Cancel the common factors.

Factor $2$ out of $2$ .

$e = 5 - \frac{2 \cdot 2}{2 (1)}$

Cancel the common factor.

$e = 5 - \frac{2 \cdot 2}{2 \cdot 1}$

Rewrite the expression.

$e = 5 - \frac{2}{1}$

Divide $2$ by $1$ .

$e = 5 - 1 \cdot 2$

Multiply $- 1$ by $2$ .

$e = 5 - 2$

Subtract $2$ from $5$ .

$e = 3$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $2 {(x - 1)}^{2} + 3$ .

$2 {(x - 1)}^{2} + 3$

Set $y$ equal to the new right side.

$y = 2 {(x - 1)}^{2} + 3$

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

$a = 2$

$h = 1$

$k = 3$

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

Opens Up

Find the vertex $(h, k)$ .

$(1, 3)$

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.

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

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.

$(1, \frac{25}{8})$

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.

$y = \frac{23}{8}$

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

Direction: Opens Up

Vertex: $(1, 3)$

Focus: $(1, \frac{25}{8})$

Axis of Symmetry: $x = 1$

Directrix: $y = \frac{23}{8}$

Direction: Opens Up

Vertex: $(1, 3)$

Focus: $(1, \frac{25}{8})$

Axis of Symmetry: $x = 1$

Directrix: $y = \frac{23}{8}$

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 $0$ in the expression.

$f (0) = 2 {(0)}^{2} - 4 \cdot 0 + 5$

Simplify the result.

Simplify each term.

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

$f (0) = 2 \cdot 0 - 4 \cdot 0 + 5$

Multiply $2$ by $0$ .

$f (0) = 0 - 4 \cdot 0 + 5$

Multiply $- 4$ by $0$ .

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

Simplify by adding numbers.

Add $0$ and $0$ .

$f (0) = 0 + 5$

Add $0$ and $5$ .

$f (0) = 5$

The final answer is $5$ .

$5$

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

$y = 5$

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

$f (- 1) = 2 {(- 1)}^{2} - 4 \cdot - 1 + 5$

Simplify the result.

Simplify each term.

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

$f (- 1) = 2 \cdot 1 - 4 \cdot - 1 + 5$

Multiply $2$ by $1$ .

$f (- 1) = 2 - 4 \cdot - 1 + 5$

Multiply $- 4$ by $- 1$ .

$f (- 1) = 2 + 4 + 5$

Simplify by adding numbers.

Add $2$ and $4$ .

$f (- 1) = 6 + 5$

Add $6$ and $5$ .

$f (- 1) = 11$

The final answer is $11$ .

$11$

The $y$ value at $x = - 1$ is $11$ .

$y = 11$

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

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

Simplify the result.

Simplify each term.

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

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

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

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

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

$f (2) = 2^{1 + 2} - 4 \cdot 2 + 5$

Add $1$ and $2$ .

$f (2) = 2^{3} - 4 \cdot 2 + 5$

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

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

Multiply $- 4$ by $2$ .

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

Simplify by adding and subtracting.

Subtract $8$ from $8$ .

$f (2) = 0 + 5$

Add $0$ and $5$ .

$f (2) = 5$

The final answer is $5$ .

$5$

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

$y = 5$

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

$f (3) = 2 {(3)}^{2} - 4 \cdot 3 + 5$

Simplify the result.

Simplify each term.

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

$f (3) = 2 \cdot 9 - 4 \cdot 3 + 5$

Multiply $2$ by $9$ .

$f (3) = 18 - 4 \cdot 3 + 5$

Multiply $- 4$ by $3$ .

$f (3) = 18 - 12 + 5$

Simplify by adding and subtracting.

Subtract $12$ from $18$ .

$f (3) = 6 + 5$

Add $6$ and $5$ .

$f (3) = 11$

The final answer is $11$ .

$11$

The $y$ value at $x = 3$ is $11$ .

$y = 11$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 1 & 11 \\ 0 & 5 \\ 1 & 3 \\ 2 & 5 \\ 3 & 11 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(1, 3)$

Focus: $(1, \frac{25}{8})$

Axis of Symmetry: $x = 1$

Directrix: $y = \frac{23}{8}$

$\begin{array}{c} x & y \\ - 1 & 11 \\ 0 & 5 \\ 1 & 3 \\ 2 & 5 \\ 3 & 11 \end{array}$

Step 4

Graph y=2x^2-4x+5

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