Find the Axis of Symmetry y=7x^2-28x+29

$y = 7 x^{2} - 28 x + 29$

Step 1

Rewrite the equation in vertex form.

Complete the square for $7 x^{2} - 28 x + 29$ .

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

$a = 7$

$b = - 28$

$c = 29$

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

$d = \frac{- 28}{2 \cdot 7}$

Simplify the right side.

Cancel the common factor of $- 28$ and $2$ .

Factor $2$ out of $- 28$ .

$d = \frac{2 \cdot - 14}{2 \cdot 7}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 7$ .

$d = \frac{2 \cdot - 14}{2 (7)}$

Cancel the common factor.

$d = \frac{2 \cdot - 14}{2 \cdot 7}$

Rewrite the expression.

$d = \frac{- 14}{7}$

Cancel the common factor of $- 14$ and $7$ .

Factor $7$ out of $- 14$ .

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

Cancel the common factors.

Factor $7$ out of $7$ .

$d = \frac{7 \cdot - 2}{7 (1)}$

Cancel the common factor.

$d = \frac{7 \cdot - 2}{7 \cdot 1}$

Rewrite the expression.

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

Divide $- 2$ by $1$ .

$d = - 2$

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

$e = 29 - \frac{{(- 28)}^{2}}{4 \cdot 7}$

Simplify the right side.

Simplify each term.

Raise $- 28$ to the power of $2$ .

$e = 29 - \frac{784}{4 \cdot 7}$

Multiply $4$ by $7$ .

$e = 29 - \frac{784}{28}$

Divide $784$ by $28$ .

$e = 29 - 1 \cdot 28$

Multiply $- 1$ by $28$ .

$e = 29 - 28$

Subtract $28$ from $29$ .

$e = 1$

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

$7 {(x - 2)}^{2} + 1$

Set $y$ equal to the new right side.

$y = 7 {(x - 2)}^{2} + 1$

Step 2

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

$a = 7$

$h = 2$

$k = 1$

Step 3

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

Opens Up

Step 4

Find the vertex $(h, k)$ .

$(2, 1)$

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.

$\frac{1}{4 \cdot 7}$

Multiply $4$ by $7$ .

$\frac{1}{28}$

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.

$(2, \frac{29}{28})$

Step 7

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

$x = 2$

Step 8

Find the Axis of Symmetry y=7x^2-28x+29

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