Graph f(x)=-(x-1)^2-3

$f (x) = - {(x - 1)}^{2} - 3$

Step 1

Find the properties of the given parabola.

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

$a = - 1$

$h = 1$

$k = - 3$

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

Opens Down

Find the vertex $(h, k)$ .

$(1, - 3)$

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.

$(1, - \frac{13}{4})$

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

$x = 1$

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.

$y = - \frac{11}{4}$

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

Direction: Opens Down

Vertex: $(1, - 3)$

Focus: $(1, - \frac{13}{4})$

Axis of Symmetry: $x = 1$

Directrix: $y = - \frac{11}{4}$

Direction: Opens Down

Vertex: $(1, - 3)$

Focus: $(1, - \frac{13}{4})$

Axis of Symmetry: $x = 1$

Directrix: $y = - \frac{11}{4}$

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 $0$ in the expression.

$f (0) = - {(0)}^{2} + 2 (0) - 4$

Simplify the result.

Simplify each term.

Raising $0$ to any positive power yields $0$ .

$f (0) = - 0 + 2 (0) - 4$

Multiply $- 1$ by $0$ .

$f (0) = 0 + 2 (0) - 4$

Multiply $2$ by $0$ .

$f (0) = 0 + 0 - 4$

Simplify by adding and subtracting.

Add $0$ and $0$ .

$f (0) = 0 - 4$

Subtract $4$ from $0$ .

$f (0) = - 4$

The final answer is $- 4$ .

$- 4$

The $y$ value at $x = 0$ is $- 4$ .

$y = - 4$

Replace the variable $x$ with $- 1$ in the expression.

$f (- 1) = - {(- 1)}^{2} + 2 (- 1) - 4$

Simplify the result.

Simplify each term.

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

Multiply $- 1$ by ${(- 1)}^{2}$ .

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

$f (- 1) = (- 1) {(- 1)}^{2} + 2 (- 1) - 4$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (- 1) = {(- 1)}^{1 + 2} + 2 (- 1) - 4$

Add $1$ and $2$ .

$f (- 1) = {(- 1)}^{3} + 2 (- 1) - 4$

Raise $- 1$ to the power of $3$ .

$f (- 1) = - 1 + 2 (- 1) - 4$

Multiply $2$ by $- 1$ .

$f (- 1) = - 1 - 2 - 4$

Simplify by subtracting numbers.

Subtract $2$ from $- 1$ .

$f (- 1) = - 3 - 4$

Subtract $4$ from $- 3$ .

$f (- 1) = - 7$

The final answer is $- 7$ .

$- 7$

The $y$ value at $x = - 1$ is $- 7$ .

$y = - 7$

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

$f (2) = - {(2)}^{2} + 2 (2) - 4$

Simplify the result.

Simplify each term.

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

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

Multiply $- 1$ by $4$ .

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

Multiply $2$ by $2$ .

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

Simplify by adding and subtracting.

Add $- 4$ and $4$ .

$f (2) = 0 - 4$

Subtract $4$ from $0$ .

$f (2) = - 4$

The final answer is $- 4$ .

$- 4$

The $y$ value at $x = 2$ is $- 4$ .

$y = - 4$

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

$f (3) = - {(3)}^{2} + 2 (3) - 4$

Simplify the result.

Simplify each term.

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

$f (3) = - 1 \cdot 9 + 2 (3) - 4$

Multiply $- 1$ by $9$ .

$f (3) = - 9 + 2 (3) - 4$

Multiply $2$ by $3$ .

$f (3) = - 9 + 6 - 4$

Simplify by adding and subtracting.

Add $- 9$ and $6$ .

$f (3) = - 3 - 4$

Subtract $4$ from $- 3$ .

$f (3) = - 7$

The final answer is $- 7$ .

$- 7$

The $y$ value at $x = 3$ is $- 7$ .

$y = - 7$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 1 & - 7 \\ 0 & - 4 \\ 1 & - 3 \\ 2 & - 4 \\ 3 & - 7 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(1, - 3)$

Focus: $(1, - \frac{13}{4})$

Axis of Symmetry: $x = 1$

Directrix: $y = - \frac{11}{4}$

$\begin{array}{c} x & y \\ - 1 & - 7 \\ 0 & - 4 \\ 1 & - 3 \\ 2 & - 4 \\ 3 & - 7 \end{array}$

Step 4

Graph f(x)=-(x-1)^2-3

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