Graph f(x)=9-(x-4)^2

$f (x) = 9 - {(x - 4)}^{2}$

Step 1

Find the properties of the given parabola.

Reorder $9$ and $- {(x - 4)}^{2}$ .

$y = - {(x - 4)}^{2} + 9$

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

$a = - 1$

$h = 4$

$k = 9$

Since the value of $a$ is negative, the parabola opens down.

Opens Down

Find the vertex $(h, k)$ .

$(4, 9)$

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 $1$ and $- 1$ .

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- 1 (- 1)}{4 \cdot - 1}$

Move the negative in front of the fraction.

$- \frac{1}{4}$

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.

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

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

$x = 4$

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 Down

Vertex: $(4, 9)$

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

Axis of Symmetry: $x = 4$

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

Direction: Opens Down

Vertex: $(4, 9)$

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

Axis of Symmetry: $x = 4$

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

Step 2

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 $3$ in the expression.

$f (3) = - {(3)}^{2} + 8 (3) - 7$

Simplify the result.

Simplify each term.

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

$f (3) = - 1 \cdot 9 + 8 (3) - 7$

Multiply $- 1$ by $9$ .

$f (3) = - 9 + 8 (3) - 7$

Multiply $8$ by $3$ .

$f (3) = - 9 + 24 - 7$

Simplify by adding and subtracting.

Add $- 9$ and $24$ .

$f (3) = 15 - 7$

Subtract $7$ from $15$ .

$f (3) = 8$

The final answer is $8$ .

$8$

The $y$ value at $x = 3$ is $8$ .

$y = 8$

Replace the variable $x$ with $2$ in the expression.

$f (2) = - {(2)}^{2} + 8 (2) - 7$

Simplify the result.

Simplify each term.

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

$f (2) = - 1 \cdot 4 + 8 (2) - 7$

Multiply $- 1$ by $4$ .

$f (2) = - 4 + 8 (2) - 7$

Multiply $8$ by $2$ .

$f (2) = - 4 + 16 - 7$

Simplify by adding and subtracting.

Add $- 4$ and $16$ .

$f (2) = 12 - 7$

Subtract $7$ from $12$ .

$f (2) = 5$

The final answer is $5$ .

$5$

The $y$ value at $x = 2$ is $5$ .

$y = 5$

Replace the variable $x$ with $5$ in the expression.

$f (5) = - {(5)}^{2} + 8 (5) - 7$

Simplify the result.

Simplify each term.

Raise $5$ to the power of $2$ .

$f (5) = - 1 \cdot 25 + 8 (5) - 7$

Multiply $- 1$ by $25$ .

$f (5) = - 25 + 8 (5) - 7$

Multiply $8$ by $5$ .

$f (5) = - 25 + 40 - 7$

Simplify by adding and subtracting.

Add $- 25$ and $40$ .

$f (5) = 15 - 7$

Subtract $7$ from $15$ .

$f (5) = 8$

The final answer is $8$ .

$8$

The $y$ value at $x = 5$ is $8$ .

$y = 8$

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

$f (6) = - {(6)}^{2} + 8 (6) - 7$

Simplify the result.

Simplify each term.

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

$f (6) = - 1 \cdot 36 + 8 (6) - 7$

Multiply $- 1$ by $36$ .

$f (6) = - 36 + 8 (6) - 7$

Multiply $8$ by $6$ .

$f (6) = - 36 + 48 - 7$

Simplify by adding and subtracting.

Add $- 36$ and $48$ .

$f (6) = 12 - 7$

Subtract $7$ from $12$ .

$f (6) = 5$

The final answer is $5$ .

$5$

The $y$ value at $x = 6$ is $5$ .

$y = 5$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 2 & 5 \\ 3 & 8 \\ 4 & 9 \\ 5 & 8 \\ 6 & 5 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(4, 9)$

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

Axis of Symmetry: $x = 4$

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

$\begin{array}{c} x & y \\ 2 & 5 \\ 3 & 8 \\ 4 & 9 \\ 5 & 8 \\ 6 & 5 \end{array}$

Step 4

Graph f(x)=9-(x-4)^2

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