Find the Axis of Symmetry y=6x^2+84x-77

$y = 6 x^{2} + 84 x - 77$

Step 1

Rewrite the equation in vertex form.

Complete the square for $6 x^{2} + 84 x - 77$ .

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

$a = 6$

$b = 84$

$c = - 77$

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

$d = \frac{84}{2 \cdot 6}$

Simplify the right side.

Cancel the common factor of $84$ and $2$ .

Factor $2$ out of $84$ .

$d = \frac{2 \cdot 42}{2 \cdot 6}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 6$ .

$d = \frac{2 \cdot 42}{2 (6)}$

Cancel the common factor.

$d = \frac{2 \cdot 42}{2 \cdot 6}$

Rewrite the expression.

$d = \frac{42}{6}$

Cancel the common factor of $42$ and $6$ .

Factor $6$ out of $42$ .

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

Cancel the common factors.

Factor $6$ out of $6$ .

$d = \frac{6 \cdot 7}{6 (1)}$

Cancel the common factor.

$d = \frac{6 \cdot 7}{6 \cdot 1}$

Rewrite the expression.

$d = \frac{7}{1}$

Divide $7$ by $1$ .

$d = 7$

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

$e = - 77 - \frac{84^{2}}{4 \cdot 6}$

Simplify the right side.

Simplify each term.

Raise $84$ to the power of $2$ .

$e = - 77 - \frac{7056}{4 \cdot 6}$

Multiply $4$ by $6$ .

$e = - 77 - \frac{7056}{24}$

Divide $7056$ by $24$ .

$e = - 77 - 1 \cdot 294$

Multiply $- 1$ by $294$ .

$e = - 77 - 294$

Subtract $294$ from $- 77$ .

$e = - 371$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $6 {(x + 7)}^{2} - 371$ .

$6 {(x + 7)}^{2} - 371$

Set $y$ equal to the new right side.

$y = 6 {(x + 7)}^{2} - 371$

Step 2

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

$a = 6$

$h = - 7$

$k = - 371$

Step 3

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

Opens Up

Step 4

Find the vertex $(h, k)$ .

$(- 7, - 371)$

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.

$\frac{1}{4 \cdot 6}$

Multiply $4$ by $6$ .

$\frac{1}{24}$

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.

$(- 7, - \frac{8903}{24})$

Step 7

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

$x = - 7$

Step 8

Find the Axis of Symmetry y=6x^2+84x-77

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