Find the Axis of Symmetry y-5=1/16(x-3)^2

$y - 5 = \frac{1}{16} {(x - 3)}^{2}$

Step 1

Rewrite the equation in vertex form.

Isolate $y$ to the left side of the equation.

Simplify $\frac{1}{16} {(x - 3)}^{2}$ .

Rewrite.

$y - 5 = 0 + 0 + \frac{1}{16} {(x - 3)}^{2}$

Simplify by adding zeros.

$y - 5 = \frac{1}{16} {(x - 3)}^{2}$

Combine $\frac{1}{16}$ and ${(x - 3)}^{2}$ .

$y - 5 = \frac{{(x - 3)}^{2}}{16}$

Add $5$ to both sides of the equation.

$y = \frac{{(x - 3)}^{2}}{16} + 5$

Reorder terms.

$y = \frac{1}{16} \cdot {(x - 3)}^{2} + 5$

Complete the square for $\frac{1}{16} \cdot {(x - 3)}^{2} + 5$ .

Simplify the expression.

Simplify each term.

Rewrite ${(x - 3)}^{2}$ as $(x - 3) (x - 3)$ .

$\frac{1}{16} \cdot ((x - 3) (x - 3)) + 5$

Expand $(x - 3) (x - 3)$ using the FOIL Method.

Apply the distributive property.

$\frac{1}{16} \cdot (x (x - 3) - 3 (x - 3)) + 5$

Apply the distributive property.

$\frac{1}{16} \cdot (x \cdot x + x \cdot - 3 - 3 (x - 3)) + 5$

Apply the distributive property.

$\frac{1}{16} \cdot (x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3) + 5$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$\frac{1}{16} \cdot (x^{2} + x \cdot - 3 - 3 x - 3 \cdot - 3) + 5$

Move $- 3$ to the left of $x$ .

$\frac{1}{16} \cdot (x^{2} - 3 \cdot x - 3 x - 3 \cdot - 3) + 5$

Multiply $- 3$ by $- 3$ .

$\frac{1}{16} \cdot (x^{2} - 3 x - 3 x + 9) + 5$

Subtract $3 x$ from $- 3 x$ .

$\frac{1}{16} \cdot (x^{2} - 6 x + 9) + 5$

Apply the distributive property.

$\frac{1}{16} x^{2} + \frac{1}{16} (- 6 x) + \frac{1}{16} \cdot 9 + 5$

Simplify.

Combine $\frac{1}{16}$ and $x^{2}$ .

$\frac{x^{2}}{16} + \frac{1}{16} (- 6 x) + \frac{1}{16} \cdot 9 + 5$

Cancel the common factor of $2$ .

Factor $2$ out of $16$ .

$\frac{x^{2}}{16} + \frac{1}{2 (8)} (- 6 x) + \frac{1}{16} \cdot 9 + 5$

Factor $2$ out of $- 6 x$ .

$\frac{x^{2}}{16} + \frac{1}{2 (8)} (2 (- 3 x)) + \frac{1}{16} \cdot 9 + 5$

Cancel the common factor.

$\frac{x^{2}}{16} + \frac{1}{2 \cdot 8} (2 (- 3 x)) + \frac{1}{16} \cdot 9 + 5$

Rewrite the expression.

$\frac{x^{2}}{16} + \frac{1}{8} (- 3 x) + \frac{1}{16} \cdot 9 + 5$

Combine $- 3$ and $\frac{1}{8}$ .

$\frac{x^{2}}{16} + \frac{- 3}{8} x + \frac{1}{16} \cdot 9 + 5$

Combine $\frac{- 3}{8}$ and $x$ .

$\frac{x^{2}}{16} + \frac{- 3 x}{8} + \frac{1}{16} \cdot 9 + 5$

Combine $\frac{1}{16}$ and $9$ .

$\frac{x^{2}}{16} + \frac{- 3 x}{8} + \frac{9}{16} + 5$

Move the negative in front of the fraction.

$\frac{x^{2}}{16} - \frac{3 x}{8} + \frac{9}{16} + 5$

To write $5$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$\frac{x^{2}}{16} - \frac{3 x}{8} + \frac{9}{16} + 5 \cdot \frac{16}{16}$

Combine $5$ and $\frac{16}{16}$ .

$\frac{x^{2}}{16} - \frac{3 x}{8} + \frac{9}{16} + \frac{5 \cdot 16}{16}$

Combine the numerators over the common denominator.

$\frac{x^{2}}{16} - \frac{3 x}{8} + \frac{9 + 5 \cdot 16}{16}$

Simplify the numerator.

Multiply $5$ by $16$ .

$\frac{x^{2}}{16} - \frac{3 x}{8} + \frac{9 + 80}{16}$

Add $9$ and $80$ .

$\frac{x^{2}}{16} - \frac{3 x}{8} + \frac{89}{16}$

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

$a = \frac{1}{16}$

$b = - \frac{3}{8}$

$c = \frac{89}{16}$

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

$d = \frac{- \frac{3}{8}}{2 (\frac{1}{16})}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$d = - \frac{3}{8} \cdot \frac{1}{2 (\frac{1}{16})}$

Combine $2$ and $\frac{1}{16}$ .

$d = - \frac{3}{8} \cdot \frac{1}{\frac{2}{16}}$

Cancel the common factor of $2$ and $16$ .

Factor $2$ out of $2$ .

$d = - \frac{3}{8} \cdot \frac{1}{\frac{2 (1)}{16}}$

Cancel the common factors.

Factor $2$ out of $16$ .

$d = - \frac{3}{8} \cdot \frac{1}{\frac{2 \cdot 1}{2 \cdot 8}}$

Cancel the common factor.

$d = - \frac{3}{8} \cdot \frac{1}{\frac{2 \cdot 1}{2 \cdot 8}}$

Rewrite the expression.

$d = - \frac{3}{8} \cdot \frac{1}{\frac{1}{8}}$

Multiply the numerator by the reciprocal of the denominator.

$d = - \frac{3}{8} (1 \cdot 8)$

Cancel the common factor of $8$ .

Move the leading negative in $- \frac{3}{8}$ into the numerator.

$d = \frac{- 3}{8} (1 \cdot 8)$

Factor $8$ out of $1 \cdot 8$ .

$d = \frac{- 3}{8} (8 \cdot 1)$

Cancel the common factor.

$d = \frac{- 3}{8} (8 \cdot 1)$

Rewrite the expression.

$d = - 3$

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

$e = \frac{89}{16} - \frac{{(- \frac{3}{8})}^{2}}{4 (\frac{1}{16})}$

Simplify the right side.

Simplify each term.

Simplify the numerator.

Apply the product rule to $- \frac{3}{8}$ .

$e = \frac{89}{16} - \frac{{(- 1)}^{2} {(\frac{3}{8})}^{2}}{4 (\frac{1}{16})}$

Raise $- 1$ to the power of $2$ .

$e = \frac{89}{16} - \frac{1 {(\frac{3}{8})}^{2}}{4 (\frac{1}{16})}$

Apply the product rule to $\frac{3}{8}$ .

$e = \frac{89}{16} - \frac{1 \frac{3^{2}}{8^{2}}}{4 (\frac{1}{16})}$

Raise $3$ to the power of $2$ .

$e = \frac{89}{16} - \frac{1 \frac{9}{8^{2}}}{4 (\frac{1}{16})}$

Raise $8$ to the power of $2$ .

$e = \frac{89}{16} - \frac{1 (\frac{9}{64})}{4 (\frac{1}{16})}$

Multiply $\frac{9}{64}$ by $1$ .

$e = \frac{89}{16} - \frac{\frac{9}{64}}{4 (\frac{1}{16})}$

Combine $4$ and $\frac{1}{16}$ .

$e = \frac{89}{16} - \frac{\frac{9}{64}}{\frac{4}{16}}$

Cancel the common factor of $4$ and $16$ .

Factor $4$ out of $4$ .

$e = \frac{89}{16} - \frac{\frac{9}{64}}{\frac{4 (1)}{16}}$

Cancel the common factors.

Factor $4$ out of $16$ .

$e = \frac{89}{16} - \frac{\frac{9}{64}}{\frac{4 \cdot 1}{4 \cdot 4}}$

Cancel the common factor.

$e = \frac{89}{16} - \frac{\frac{9}{64}}{\frac{4 \cdot 1}{4 \cdot 4}}$

Rewrite the expression.

$e = \frac{89}{16} - \frac{\frac{9}{64}}{\frac{1}{4}}$

Multiply the numerator by the reciprocal of the denominator.

$e = \frac{89}{16} - (\frac{9}{64} \cdot 4)$

Cancel the common factor of $4$ .

Factor $4$ out of $64$ .

$e = \frac{89}{16} - (\frac{9}{4 (16)} \cdot 4)$

Cancel the common factor.

$e = \frac{89}{16} - (\frac{9}{4 \cdot 16} \cdot 4)$

Rewrite the expression.

$e = \frac{89}{16} - \frac{9}{16}$

Combine the numerators over the common denominator.

$e = \frac{89 - 9}{16}$

Subtract $9$ from $89$ .

$e = \frac{80}{16}$

Divide $80$ by $16$ .

$e = 5$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{16} {(x - 3)}^{2} + 5$ .

$\frac{1}{16} {(x - 3)}^{2} + 5$

Set $y$ equal to the new right side.

$y = \frac{1}{16} \cdot {(x - 3)}^{2} + 5$

Step 2

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

$a = \frac{1}{16}$

$h = 3$

$k = 5$

Step 3

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

Opens Up

Step 4

Find the vertex $(h, k)$ .

$(3, 5)$

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.

$\frac{1}{4 \cdot \frac{1}{16}}$

Simplify.

Combine $4$ and $\frac{1}{16}$ .

$\frac{1}{\frac{4}{16}}$

Cancel the common factor of $4$ and $16$ .

Factor $4$ out of $4$ .

$\frac{1}{\frac{4 (1)}{16}}$

Cancel the common factors.

Factor $4$ out of $16$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{1}{\frac{1}{4}}$

Multiply the numerator by the reciprocal of the denominator.

$1 \cdot 4$

Multiply $4$ by $1$ .

$4$

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.

$(3, 9)$

Step 7

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

$x = 3$

Step 8

Find the Axis of Symmetry y-5=1/16(x-3)^2

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