Algebra Examples

h (x) = - {(x - 2)}^{2}

$h (x) = - {(x - 2)}^{2}$

Step 1

Find the properties of the given parabola.

Tap for more steps...

Step 1.1

Use the vertex form,

y = a {(x - h)}^{2} + k

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

a

$a$ ,

h

$h$ , and

k

$k$ .

a = - 1

$a = - 1$

h = 2

$h = 2$

k = 0

$k = 0$

Step 1.2

Since the value of

a

$a$ is negative, the parabola opens down.

Opens Down

Step 1.3

Find the vertex

(h, k)

$(h, k)$ .

(2, 0)

$(2, 0)$

Step 1.4

Find

p

$p$ , the distance from the vertex to the focus.

Tap for more steps...

Step 1.4.1

Find the distance from the vertex to a focus of the parabola by using the following formula.

\frac{1}{4 a}

$\frac{1}{4 a}$

Step 1.4.2

Substitute the value of

a

$a$ into the formula.

\frac{1}{4 \cdot - 1}

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

Step 1.4.3

Cancel the common factor of

1

$1$ and

- 1

$- 1$ .

Tap for more steps...

Step 1.4.3.1

Rewrite

1

$1$ as

- 1 (- 1)

$- 1 (- 1)$ .

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

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

Step 1.4.3.2

Move the negative in front of the fraction.

- \frac{1}{4}

$- \frac{1}{4}$

- \frac{1}{4}

$- \frac{1}{4}$

- \frac{1}{4}

$- \frac{1}{4}$

Step 1.5

Find the focus.

Tap for more steps...

Step 1.5.1

The focus of a parabola can be found by adding

p

$p$ to the y-coordinate

k

$k$ if the parabola opens up or down.

(h, k + p)

$(h, k + p)$

Step 1.5.2

Substitute the known values of

h

$h$ ,

p

$p$ , and

k

$k$ into the formula and simplify.

(2, - \frac{1}{4})

$(2, - \frac{1}{4})$

(2, - \frac{1}{4})

$(2, - \frac{1}{4})$

Step 1.6

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

x = 2

$x = 2$

Step 1.7

Find the directrix.

Tap for more steps...

Step 1.7.1

The directrix of a parabola is the horizontal line found by subtracting

p

$p$ from the y-coordinate

k

$k$ of the vertex if the parabola opens up or down.

y = k - p

$y = k - p$

Step 1.7.2

Substitute the known values of

p

$p$ and

k

$k$ into the formula and simplify.

y = \frac{1}{4}

$y = \frac{1}{4}$

y = \frac{1}{4}

$y = \frac{1}{4}$

Step 1.8

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

Direction: Opens Down

Vertex:

(2, 0)

$(2, 0)$

Focus:

(2, - \frac{1}{4})

$(2, - \frac{1}{4})$

Axis of Symmetry:

x = 2

$x = 2$

Directrix:

y = \frac{1}{4}

$y = \frac{1}{4}$

Direction: Opens Down

Vertex:

(2, 0)

$(2, 0)$

Focus:

(2, - \frac{1}{4})

$(2, - \frac{1}{4})$

Axis of Symmetry:

x = 2

$x = 2$

Directrix:

y = \frac{1}{4}

$y = \frac{1}{4}$

Step 2

Select a few

x

$x$ values, and plug them into the equation to find the corresponding

y

$y$ values. The

x

$x$ values should be selected around the vertex.

Tap for more steps...

Step 2.1

Replace the variable

x

$x$ with

1

$1$ in the expression.

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

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

Step 2.2

Simplify the result.

Tap for more steps...

Step 2.2.1

Simplify each term.

Tap for more steps...

Step 2.2.1.1

One to any power is one.

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

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

Step 2.2.1.2

Multiply

- 1

$- 1$ by

1

$1$ .

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

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

Step 2.2.1.3

Multiply

4

$4$ by

1

$1$ .

f (1) = - 1 + 4 - 4

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

f (1) = - 1 + 4 - 4

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

Step 2.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 2.2.2.1

Add

- 1

$- 1$ and

4

$4$ .

f (1) = 3 - 4

$f (1) = 3 - 4$

Step 2.2.2.2

Subtract

4

$4$ from

3

$3$ .

f (1) = - 1

$f (1) = - 1$

f (1) = - 1

$f (1) = - 1$

Step 2.2.3

The final answer is

- 1

$- 1$ .

- 1

$- 1$

- 1

$- 1$

Step 2.3

The

y

$y$ value at

x = 1

$x = 1$ is

- 1

$- 1$ .

y = - 1

$y = - 1$

Step 2.4

Replace the variable

x

$x$ with

0

$0$ in the expression.

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

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

Step 2.5

Simplify the result.

Tap for more steps...

Step 2.5.1

Simplify each term.

Tap for more steps...

Step 2.5.1.1

Raising

0

$0$ to any positive power yields

0

$0$ .

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

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

Step 2.5.1.2

Multiply

- 1

$- 1$ by

0

$0$ .

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

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

Step 2.5.1.3

Multiply

4

$4$ by

0

$0$ .

f (0) = 0 + 0 - 4

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

f (0) = 0 + 0 - 4

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

Step 2.5.2

Simplify by adding and subtracting.

Tap for more steps...

Step 2.5.2.1

Add

0

$0$ and

0

$0$ .

f (0) = 0 - 4

$f (0) = 0 - 4$

Step 2.5.2.2

Subtract

4

$4$ from

0

$0$ .

f (0) = - 4

$f (0) = - 4$

f (0) = - 4

$f (0) = - 4$

Step 2.5.3

The final answer is

- 4

$- 4$ .

- 4

$- 4$

- 4

$- 4$

Step 2.6

The

y

$y$ value at

x = 0

$x = 0$ is

- 4

$- 4$ .

y = - 4

$y = - 4$

Step 2.7

Replace the variable

x

$x$ with

3

$3$ in the expression.

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

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

Step 2.8

Simplify the result.

Tap for more steps...

Step 2.8.1

Simplify each term.

Tap for more steps...

Step 2.8.1.1

Raise

3

$3$ to the power of

2

$2$ .

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

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

Step 2.8.1.2

Multiply

- 1

$- 1$ by

9

$9$ .

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

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

Step 2.8.1.3

Multiply

4

$4$ by

3

$3$ .

f (3) = - 9 + 12 - 4

$f (3) = - 9 + 12 - 4$

f (3) = - 9 + 12 - 4

$f (3) = - 9 + 12 - 4$

Step 2.8.2

Simplify by adding and subtracting.

Tap for more steps...

Step 2.8.2.1

Add

- 9

$- 9$ and

12

$12$ .

f (3) = 3 - 4

$f (3) = 3 - 4$

Step 2.8.2.2

Subtract

4

$4$ from

3

$3$ .

f (3) = - 1

$f (3) = - 1$

f (3) = - 1

$f (3) = - 1$

Step 2.8.3

The final answer is

- 1

$- 1$ .

- 1

$- 1$

- 1

$- 1$

Step 2.9

The

y

$y$ value at

x = 3

$x = 3$ is

- 1

$- 1$ .

y = - 1

$y = - 1$

Step 2.10

Replace the variable

x

$x$ with

4

$4$ in the expression.

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

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

Step 2.11

Simplify the result.

Tap for more steps...

Step 2.11.1

Simplify each term.

Tap for more steps...

Step 2.11.1.1

Raise

4

$4$ to the power of

2

$2$ .

f (4) = - 1 \cdot 16 + 4 (4) - 4

$f (4) = - 1 \cdot 16 + 4 (4) - 4$

Step 2.11.1.2

Multiply

- 1

$- 1$ by

16

$16$ .

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

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

Step 2.11.1.3

Multiply

4

$4$ by

4

$4$ .

f (4) = - 16 + 16 - 4

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

f (4) = - 16 + 16 - 4

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

Step 2.11.2

Simplify by adding and subtracting.

Tap for more steps...

Step 2.11.2.1

Add

- 16

$- 16$ and

16

$16$ .

f (4) = 0 - 4

$f (4) = 0 - 4$

Step 2.11.2.2

Subtract

4

$4$ from

0

$0$ .

f (4) = - 4

$f (4) = - 4$

f (4) = - 4

$f (4) = - 4$

Step 2.11.3

The final answer is

- 4

$- 4$ .

- 4

$- 4$

- 4

$- 4$

Step 2.12

The

y

$y$ value at

x = 4

$x = 4$ is

- 4

$- 4$ .

y = - 4

$y = - 4$

Step 2.13

Graph the parabola using its properties and the selected points.

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

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

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

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

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex:

(2, 0)

$(2, 0)$

Focus:

(2, - \frac{1}{4})

$(2, - \frac{1}{4})$

Axis of Symmetry:

x = 2

$x = 2$

Directrix:

y = \frac{1}{4}

$y = \frac{1}{4}$

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

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

Step 4