Graph (x-5)^2=y+9

${(x - 5)}^{2} = y + 9$

Solve for $y$ .

Rewrite the equation as $y + 9 = {(x - 5)}^{2}$ .

$y + 9 = {(x - 5)}^{2}$

Simplify ${(x - 5)}^{2}$ .

Rewrite.

$y + 9 = 0 + 0 + {(x - 5)}^{2}$

Rewrite ${(x - 5)}^{2}$ as $(x - 5) (x - 5)$ .

$y + 9 = (x - 5) (x - 5)$

Expand $(x - 5) (x - 5)$ using the FOIL Method.

Apply the distributive property.

$y + 9 = x (x - 5) - 5 (x - 5)$

Apply the distributive property.

$y + 9 = x \cdot x + x \cdot - 5 - 5 (x - 5)$

Apply the distributive property.

$y + 9 = x \cdot x + x \cdot - 5 - 5 x - 5 \cdot - 5$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$y + 9 = x^{2} + x \cdot - 5 - 5 x - 5 \cdot - 5$

Move $- 5$ to the left of $x$ .

$y + 9 = x^{2} - 5 \cdot x - 5 x - 5 \cdot - 5$

Multiply $- 5$ by $- 5$ .

$y + 9 = x^{2} - 5 x - 5 x + 25$

Subtract $5 x$ from $- 5 x$ .

$y + 9 = x^{2} - 10 x + 25$

Move all terms not containing $y$ to the right side of the equation.

Subtract $9$ from both sides of the equation.

$y = x^{2} - 10 x + 25 - 9$

Subtract $9$ from $25$ .

$y = x^{2} - 10 x + 16$

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Complete the square for $x^{2} - 10 x + 16$ .

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

$a = 1$

$b = - 10$

$c = 16$

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

$d = \frac{- 10}{2 \cdot 1}$

Cancel the common factor of $- 10$ and $2$ .

Factor $2$ out of $- 10$ .

$d = \frac{2 \cdot - 5}{2 \cdot 1}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 1$ .

$d = \frac{2 \cdot - 5}{2 (1)}$

Cancel the common factor.

$d = \frac{2 \cdot - 5}{2 \cdot 1}$

Rewrite the expression.

$d = \frac{- 5}{1}$

Divide $- 5$ by $1$ .

$d = - 5$

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

$e = 16 - \frac{{(- 10)}^{2}}{4 \cdot 1}$

Simplify the right side.

Simplify each term.

Raise $- 10$ to the power of $2$ .

$e = 16 - \frac{100}{4 \cdot 1}$

Multiply $4$ by $1$ .

$e = 16 - \frac{100}{4}$

Divide $100$ by $4$ .

$e = 16 - 1 \cdot 25$

Multiply $- 1$ by $25$ .

$e = 16 - 25$

Subtract $25$ from $16$ .

$e = - 9$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x - 5)}^{2} - 9$ .

${(x - 5)}^{2} - 9$

Set $y$ equal to the new right side.

$y = {(x - 5)}^{2} - 9$

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

$a = 1$

$h = 5$

$k = - 9$

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

Opens Up

Find the vertex $(h, k)$ .

$(5, - 9)$

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.

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

Cancel the common factor of $1$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{1}{4}$

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.

$(5, - \frac{35}{4})$

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

$x = 5$

Find the directrix.

The directrix of a parabola is the horizontal line found by subtracting $p$ from the y-coordinate $k$ of the vertex if the parabola opens up or down.

$y = k - p$

Substitute the known values of $p$ and $k$ into the formula and simplify.

$y = - \frac{37}{4}$

Use the properties of the parabola to analyze and graph the parabola.

Direction: Opens Up

Vertex: $(5, - 9)$

Focus: $(5, - \frac{35}{4})$

Axis of Symmetry: $x = 5$

Directrix: $y = - \frac{37}{4}$

Direction: Opens Up

Vertex: $(5, - 9)$

Focus: $(5, - \frac{35}{4})$

Axis of Symmetry: $x = 5$

Directrix: $y = - \frac{37}{4}$

Select a few $x$ values, and plug them into the equation to find the corresponding $y$ values. The $x$ values should be selected around the vertex.

Replace the variable $x$ with $4$ in the expression.

$f (4) = {(4)}^{2} - 10 \cdot 4 + 16$

Simplify the result.

Simplify each term.

Raise $4$ to the power of $2$ .

$f (4) = 16 - 10 \cdot 4 + 16$

Multiply $- 10$ by $4$ .

$f (4) = 16 - 40 + 16$

Simplify by adding and subtracting.

Subtract $40$ from $16$ .

$f (4) = - 24 + 16$

Add $- 24$ and $16$ .

$f (4) = - 8$

The final answer is $- 8$ .

$- 8$

The $y$ value at $x = 4$ is $- 8$ .

$y = - 8$

Replace the variable $x$ with $3$ in the expression.

$f (3) = {(3)}^{2} - 10 \cdot 3 + 16$

Simplify the result.

Simplify each term.

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

$f (3) = 9 - 10 \cdot 3 + 16$

Multiply $- 10$ by $3$ .

$f (3) = 9 - 30 + 16$

Simplify by adding and subtracting.

Subtract $30$ from $9$ .

$f (3) = - 21 + 16$

Add $- 21$ and $16$ .

$f (3) = - 5$

The final answer is $- 5$ .

$- 5$

The $y$ value at $x = 3$ is $- 5$ .

$y = - 5$

Replace the variable $x$ with $6$ in the expression.

$f (6) = {(6)}^{2} - 10 \cdot 6 + 16$

Simplify the result.

Simplify each term.

Raise $6$ to the power of $2$ .

$f (6) = 36 - 10 \cdot 6 + 16$

Multiply $- 10$ by $6$ .

$f (6) = 36 - 60 + 16$

Simplify by adding and subtracting.

Subtract $60$ from $36$ .

$f (6) = - 24 + 16$

Add $- 24$ and $16$ .

$f (6) = - 8$

The final answer is $- 8$ .

$- 8$

The $y$ value at $x = 6$ is $- 8$ .

$y = - 8$

Replace the variable $x$ with $7$ in the expression.

$f (7) = {(7)}^{2} - 10 \cdot 7 + 16$

Simplify the result.

Simplify each term.

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

$f (7) = 49 - 10 \cdot 7 + 16$

Multiply $- 10$ by $7$ .

$f (7) = 49 - 70 + 16$

Simplify by adding and subtracting.

Subtract $70$ from $49$ .

$f (7) = - 21 + 16$

Add $- 21$ and $16$ .

$f (7) = - 5$

The final answer is $- 5$ .

$- 5$

The $y$ value at $x = 7$ is $- 5$ .

$y = - 5$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 3 & - 5 \\ 4 & - 8 \\ 5 & - 9 \\ 6 & - 8 \\ 7 & - 5 \end{array}$

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(5, - 9)$

Focus: $(5, - \frac{35}{4})$

Axis of Symmetry: $x = 5$

Directrix: $y = - \frac{37}{4}$

$\begin{array}{c} x & y \\ 3 & - 5 \\ 4 & - 8 \\ 5 & - 9 \\ 6 & - 8 \\ 7 & - 5 \end{array}$

Graph (x-5)^2=y+9

Related Questions

Support LCMGCF.com by disabling your adblocker.