Find the Axis of Symmetry y=3x^2+7x+4

$y = 3 x^{2} + 7 x + 4$

Step 1

Rewrite the equation in vertex form.

Complete the square for $3 x^{2} + 7 x + 4$ .

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

$a = 3$

$b = 7$

$c = 4$

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

$d = \frac{7}{2 \cdot 3}$

Multiply $2$ by $3$ .

$d = \frac{7}{6}$

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

$e = 4 - \frac{7^{2}}{4 \cdot 3}$

Simplify the right side.

Simplify each term.

Raise $7$ to the power of $2$ .

$e = 4 - \frac{49}{4 \cdot 3}$

Multiply $4$ by $3$ .

$e = 4 - \frac{49}{12}$

To write $4$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$e = 4 \cdot \frac{12}{12} - \frac{49}{12}$

Combine $4$ and $\frac{12}{12}$ .

$e = \frac{4 \cdot 12}{12} - \frac{49}{12}$

Combine the numerators over the common denominator.

$e = \frac{4 \cdot 12 - 49}{12}$

Simplify the numerator.

Multiply $4$ by $12$ .

$e = \frac{48 - 49}{12}$

Subtract $49$ from $48$ .

$e = \frac{- 1}{12}$

Move the negative in front of the fraction.

$e = - \frac{1}{12}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $3 {(x + \frac{7}{6})}^{2} - \frac{1}{12}$ .

$3 {(x + \frac{7}{6})}^{2} - \frac{1}{12}$

Set $y$ equal to the new right side.

$y = 3 {(x + \frac{7}{6})}^{2} - \frac{1}{12}$

Step 2

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

$a = 3$

$h = - \frac{7}{6}$

$k = - \frac{1}{12}$

Step 3

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

Opens Up

Step 4

Find the vertex $(h, k)$ .

$(- \frac{7}{6}, - \frac{1}{12})$

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.

$\frac{1}{4 \cdot 3}$

Multiply $4$ by $3$ .

$\frac{1}{12}$

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.

$(- \frac{7}{6}, 0)$

Step 7

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

$x = - \frac{7}{6}$

Step 8

Find the Axis of Symmetry y=3x^2+7x+4

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