Find the Axis of Symmetry y=5x^2-2x+3

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

Step 1

Rewrite the equation in vertex form.

Complete the square for $5 x^{2} - 2 x + 3$ .

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

$a = 5$

$b = - 2$

$c = 3$

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

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

Simplify the right side.

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

Factor $2$ out of $- 2$ .

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

Cancel the common factors.

Factor $2$ out of $2 \cdot 5$ .

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

Cancel the common factor.

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

Rewrite the expression.

$d = \frac{- 1}{5}$

Move the negative in front of the fraction.

$d = - \frac{1}{5}$

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

$e = 3 - \frac{{(- 2)}^{2}}{4 \cdot 5}$

Simplify the right side.

Simplify each term.

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

$e = 3 - \frac{4}{4 \cdot 5}$

Multiply $4$ by $5$ .

$e = 3 - \frac{4}{20}$

Cancel the common factor of $4$ and $20$ .

Factor $4$ out of $4$ .

$e = 3 - \frac{4 (1)}{20}$

Cancel the common factors.

Factor $4$ out of $20$ .

$e = 3 - \frac{4 \cdot 1}{4 \cdot 5}$

Cancel the common factor.

$e = 3 - \frac{4 \cdot 1}{4 \cdot 5}$

Rewrite the expression.

$e = 3 - \frac{1}{5}$

To write $3$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$e = 3 \cdot \frac{5}{5} - \frac{1}{5}$

Combine $3$ and $\frac{5}{5}$ .

$e = \frac{3 \cdot 5}{5} - \frac{1}{5}$

Combine the numerators over the common denominator.

$e = \frac{3 \cdot 5 - 1}{5}$

Simplify the numerator.

Multiply $3$ by $5$ .

$e = \frac{15 - 1}{5}$

Subtract $1$ from $15$ .

$e = \frac{14}{5}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $5 {(x - \frac{1}{5})}^{2} + \frac{14}{5}$ .

$5 {(x - \frac{1}{5})}^{2} + \frac{14}{5}$

Set $y$ equal to the new right side.

$y = 5 {(x - \frac{1}{5})}^{2} + \frac{14}{5}$

Step 2

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

$a = 5$

$h = \frac{1}{5}$

$k = \frac{14}{5}$

Step 3

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

Opens Up

Step 4

Find the vertex $(h, k)$ .

$(\frac{1}{5}, \frac{14}{5})$

Step 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.

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

Multiply $4$ by $5$ .

$\frac{1}{20}$

Step 6

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.

$(\frac{1}{5}, \frac{57}{20})$

Step 7

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

$x = \frac{1}{5}$

Step 8

Find the Axis of Symmetry y=5x^2-2x+3

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