Find the Axis of Symmetry f(x)=3x^2-14x-5

$f (x) = 3 x^{2} - 14 x - 5$

Step 1

Write $f (x) = 3 x^{2} - 14 x - 5$ as an equation.

$y = 3 x^{2} - 14 x - 5$

Step 2

Rewrite the equation in vertex form.

Complete the square for $3 x^{2} - 14 x - 5$ .

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

$a = 3$

$b = - 14$

$c = - 5$

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

$d = \frac{- 14}{2 \cdot 3}$

Simplify the right side.

Cancel the common factor of $- 14$ and $2$ .

Factor $2$ out of $- 14$ .

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

Cancel the common factors.

Factor $2$ out of $2 \cdot 3$ .

$d = \frac{2 \cdot - 7}{2 (3)}$

Cancel the common factor.

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

Rewrite the expression.

$d = \frac{- 7}{3}$

Move the negative in front of the fraction.

$d = - \frac{7}{3}$

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

$e = - 5 - \frac{{(- 14)}^{2}}{4 \cdot 3}$

Simplify the right side.

Simplify each term.

Raise $- 14$ to the power of $2$ .

$e = - 5 - \frac{196}{4 \cdot 3}$

Multiply $4$ by $3$ .

$e = - 5 - \frac{196}{12}$

Cancel the common factor of $196$ and $12$ .

Factor $4$ out of $196$ .

$e = - 5 - \frac{4 (49)}{12}$

Cancel the common factors.

Factor $4$ out of $12$ .

$e = - 5 - \frac{4 \cdot 49}{4 \cdot 3}$

Cancel the common factor.

$e = - 5 - \frac{4 \cdot 49}{4 \cdot 3}$

Rewrite the expression.

$e = - 5 - \frac{49}{3}$

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

$e = - 5 \cdot \frac{3}{3} - \frac{49}{3}$

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

$e = \frac{- 5 \cdot 3}{3} - \frac{49}{3}$

Combine the numerators over the common denominator.

$e = \frac{- 5 \cdot 3 - 49}{3}$

Simplify the numerator.

Multiply $- 5$ by $3$ .

$e = \frac{- 15 - 49}{3}$

Subtract $49$ from $- 15$ .

$e = \frac{- 64}{3}$

Move the negative in front of the fraction.

$e = - \frac{64}{3}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $3 {(x - \frac{7}{3})}^{2} - \frac{64}{3}$ .

$3 {(x - \frac{7}{3})}^{2} - \frac{64}{3}$

Set $y$ equal to the new right side.

$y = 3 {(x - \frac{7}{3})}^{2} - \frac{64}{3}$

Step 3

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

$a = 3$

$h = \frac{7}{3}$

$k = - \frac{64}{3}$

Step 4

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

Opens Up

Step 5

Find the vertex $(h, k)$ .

$(\frac{7}{3}, - \frac{64}{3})$

Step 6

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 7

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.

$(\frac{7}{3}, - \frac{85}{4})$

Step 8

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

$x = \frac{7}{3}$

Step 9

Find the Axis of Symmetry f(x)=3x^2-14x-5

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