Algebra Examples

A (x) = - x (x - 80)

$A (x) = - x (x - 80)$

Step 1

Find the properties of the given parabola.

Tap for more steps...

Step 1.1

Rewrite the equation in vertex form.

Tap for more steps...

Step 1.1.1

Simplify

- x (x - 80)

$- x (x - 80)$ .

Tap for more steps...

Step 1.1.1.1

Apply the distributive property.

y = - x \cdot x - x \cdot - 80

$y = - x \cdot x - x \cdot - 80$

Step 1.1.1.2

Multiply

x

$x$ by

x

$x$ by adding the exponents.

Tap for more steps...

Step 1.1.1.2.1

Move

x

$x$ .

y = - (x \cdot x) - x \cdot - 80

$y = - (x \cdot x) - x \cdot - 80$

Step 1.1.1.2.2

Multiply

x

$x$ by

x

$x$ .

y = - x^{2} - x \cdot - 80

$y = - x^{2} - x \cdot - 80$

y = - x^{2} - x \cdot - 80

$y = - x^{2} - x \cdot - 80$

Step 1.1.1.3

Multiply

- 80

$- 80$ by

- 1

$- 1$ .

y = - x^{2} + 80 x

$y = - x^{2} + 80 x$

y = - x^{2} + 80 x

$y = - x^{2} + 80 x$

Step 1.1.2

Complete the square for

- x^{2} + 80 x

$- x^{2} + 80 x$ .

Tap for more steps...

Step 1.1.2.1

Use the form

a x^{2} + b x + c

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

a

$a$ ,

b

$b$ , and

c

$c$ .

a = - 1

$a = - 1$

b = 80

$b = 80$

c = 0

$c = 0$

Step 1.1.2.2

Consider the vertex form of a parabola.

a {(x + d)}^{2} + e

$a {(x + d)}^{2} + e$

Step 1.1.2.3

Find the value of

d

$d$ using the formula

d = \frac{b}{2 a}

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

Tap for more steps...

Step 1.1.2.3.1

Substitute the values of

a

$a$ and

b

$b$ into the formula

d = \frac{b}{2 a}

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

d = \frac{80}{2 \cdot - 1}

$d = \frac{80}{2 \cdot - 1}$

Step 1.1.2.3.2

Simplify the right side.

Tap for more steps...

Step 1.1.2.3.2.1

Cancel the common factor of

80

$80$ and

2

$2$ .

Tap for more steps...

Step 1.1.2.3.2.1.1

Factor

2

$2$ out of

80

$80$ .

d = \frac{2 \cdot 40}{2 \cdot - 1}

$d = \frac{2 \cdot 40}{2 \cdot - 1}$

Step 1.1.2.3.2.1.2

Move the negative one from the denominator of

\frac{40}{- 1}

$\frac{40}{- 1}$ .

d = - 1 \cdot 40

$d = - 1 \cdot 40$

d = - 1 \cdot 40

$d = - 1 \cdot 40$

Step 1.1.2.3.2.2

Multiply

- 1

$- 1$ by

40

$40$ .

d = - 40

$d = - 40$

d = - 40

$d = - 40$

d = - 40

$d = - 40$

Step 1.1.2.4

Find the value of

e

$e$ using the formula

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

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

Tap for more steps...

Step 1.1.2.4.1

Substitute the values of

c

$c$ ,

b

$b$ and

a

$a$ into the formula

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

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

e = 0 - \frac{80^{2}}{4 \cdot - 1}

$e = 0 - \frac{80^{2}}{4 \cdot - 1}$

Step 1.1.2.4.2

Simplify the right side.

Tap for more steps...

Step 1.1.2.4.2.1

Simplify each term.

Tap for more steps...

Step 1.1.2.4.2.1.1

Raise

80

$80$ to the power of

2

$2$ .

e = 0 - \frac{6400}{4 \cdot - 1}

$e = 0 - \frac{6400}{4 \cdot - 1}$

Step 1.1.2.4.2.1.2

Multiply

4

$4$ by

- 1

$- 1$ .

e = 0 - \frac{6400}{- 4}

$e = 0 - \frac{6400}{- 4}$

Step 1.1.2.4.2.1.3

Divide

6400

$6400$ by

- 4

$- 4$ .

e = 0 - - 1600

$e = 0 - - 1600$

Step 1.1.2.4.2.1.4

Multiply

- 1

$- 1$ by

- 1600

$- 1600$ .

e = 0 + 1600

$e = 0 + 1600$

e = 0 + 1600

$e = 0 + 1600$

Step 1.1.2.4.2.2

Add

0

$0$ and

1600

$1600$ .

e = 1600

$e = 1600$

e = 1600

$e = 1600$

e = 1600

$e = 1600$

Step 1.1.2.5

Substitute the values of

a

$a$ ,

d

$d$ , and

e

$e$ into the vertex form

- {(x - 40)}^{2} + 1600

$- {(x - 40)}^{2} + 1600$ .

- {(x - 40)}^{2} + 1600

$- {(x - 40)}^{2} + 1600$

- {(x - 40)}^{2} + 1600

$- {(x - 40)}^{2} + 1600$

Step 1.1.3

Set

y

$y$ equal to the new right side.

y = - {(x - 40)}^{2} + 1600

$y = - {(x - 40)}^{2} + 1600$

y = - {(x - 40)}^{2} + 1600

$y = - {(x - 40)}^{2} + 1600$

Step 1.2

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 = 40

$h = 40$

k = 1600

$k = 1600$

Step 1.3

Since the value of

a

$a$ is negative, the parabola opens down.

Opens Down

Step 1.4

Find the vertex

(h, k)

$(h, k)$ .

(40, 1600)

$(40, 1600)$

Step 1.5

Find

p

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

Tap for more steps...

Step 1.5.1

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 1.5.2

Substitute the value of

a

$a$ into the formula.

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

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

Step 1.5.3

Cancel the common factor of

1

$1$ and

- 1

$- 1$ .

Tap for more steps...

Step 1.5.3.1

Rewrite

1

$1$ as

- 1 (- 1)

$- 1 (- 1)$ .

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

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

Step 1.5.3.2

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 1.6

Find the focus.

Tap for more steps...

Step 1.6.1

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 1.6.2

Substitute the known values of

h

$h$ ,

p

$p$ , and

k

$k$ into the formula and simplify.

(40, \frac{6399}{4})

$(40, \frac{6399}{4})$

(40, \frac{6399}{4})

$(40, \frac{6399}{4})$

Step 1.7

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

x = 40

$x = 40$

Step 1.8

Find the directrix.

Tap for more steps...

Step 1.8.1

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 1.8.2

Substitute the known values of

p

$p$ and

k

$k$ into the formula and simplify.

y = \frac{6401}{4}

$y = \frac{6401}{4}$

y = \frac{6401}{4}

$y = \frac{6401}{4}$

Step 1.9

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

Direction: Opens Down

Vertex:

(40, 1600)

$(40, 1600)$

Focus:

(40, \frac{6399}{4})

$(40, \frac{6399}{4})$

Axis of Symmetry:

x = 40

$x = 40$

Directrix:

y = \frac{6401}{4}

$y = \frac{6401}{4}$

Direction: Opens Down

Vertex:

(40, 1600)

$(40, 1600)$

Focus:

(40, \frac{6399}{4})

$(40, \frac{6399}{4})$

Axis of Symmetry:

x = 40

$x = 40$

Directrix:

y = \frac{6401}{4}

$y = \frac{6401}{4}$

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

39

$39$ in the expression.

f (39) = - {(39)}^{2} + 80 (39)

$f (39) = - {(39)}^{2} + 80 (39)$

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

Raise

39

$39$ to the power of

2

$2$ .

f (39) = - 1 \cdot 1521 + 80 (39)

$f (39) = - 1 \cdot 1521 + 80 (39)$

Step 2.2.1.2

Multiply

- 1

$- 1$ by

1521

$1521$ .

f (39) = - 1521 + 80 (39)

$f (39) = - 1521 + 80 (39)$

Step 2.2.1.3

Multiply

80

$80$ by

39

$39$ .

f (39) = - 1521 + 3120

$f (39) = - 1521 + 3120$

f (39) = - 1521 + 3120

$f (39) = - 1521 + 3120$

Step 2.2.2

Add

- 1521

$- 1521$ and

3120

$3120$ .

f (39) = 1599

$f (39) = 1599$

Step 2.2.3

The final answer is

1599

$1599$ .

1599

$1599$

1599

$1599$

Step 2.3

The

y

$y$ value at

x = 39

$x = 39$ is

1599

$1599$ .

y = 1599

$y = 1599$

Step 2.4

Replace the variable

x

$x$ with

38

$38$ in the expression.

f (38) = - {(38)}^{2} + 80 (38)

$f (38) = - {(38)}^{2} + 80 (38)$

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

Raise

38

$38$ to the power of

2

$2$ .

f (38) = - 1 \cdot 1444 + 80 (38)

$f (38) = - 1 \cdot 1444 + 80 (38)$

Step 2.5.1.2

Multiply

- 1

$- 1$ by

1444

$1444$ .

f (38) = - 1444 + 80 (38)

$f (38) = - 1444 + 80 (38)$

Step 2.5.1.3

Multiply

80

$80$ by

38

$38$ .

f (38) = - 1444 + 3040

$f (38) = - 1444 + 3040$

f (38) = - 1444 + 3040

$f (38) = - 1444 + 3040$

Step 2.5.2

Add

- 1444

$- 1444$ and

3040

$3040$ .

f (38) = 1596

$f (38) = 1596$

Step 2.5.3

The final answer is

1596

$1596$ .

1596

$1596$

1596

$1596$

Step 2.6

The

y

$y$ value at

x = 38

$x = 38$ is

1596

$1596$ .

y = 1596

$y = 1596$

Step 2.7

Replace the variable

x

$x$ with

41

$41$ in the expression.

f (41) = - {(41)}^{2} + 80 (41)

$f (41) = - {(41)}^{2} + 80 (41)$

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

41

$41$ to the power of

2

$2$ .

f (41) = - 1 \cdot 1681 + 80 (41)

$f (41) = - 1 \cdot 1681 + 80 (41)$

Step 2.8.1.2

Multiply

- 1

$- 1$ by

1681

$1681$ .

f (41) = - 1681 + 80 (41)

$f (41) = - 1681 + 80 (41)$

Step 2.8.1.3

Multiply

80

$80$ by

41

$41$ .

f (41) = - 1681 + 3280

$f (41) = - 1681 + 3280$

f (41) = - 1681 + 3280

$f (41) = - 1681 + 3280$

Step 2.8.2

Add

- 1681

$- 1681$ and

3280

$3280$ .

f (41) = 1599

$f (41) = 1599$

Step 2.8.3

The final answer is

1599

$1599$ .

1599

$1599$

1599

$1599$

Step 2.9

The

y

$y$ value at

x = 41

$x = 41$ is

1599

$1599$ .

y = 1599

$y = 1599$

Step 2.10

Replace the variable

x

$x$ with

42

$42$ in the expression.

f (42) = - {(42)}^{2} + 80 (42)

$f (42) = - {(42)}^{2} + 80 (42)$

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

42

$42$ to the power of

2

$2$ .

f (42) = - 1 \cdot 1764 + 80 (42)

$f (42) = - 1 \cdot 1764 + 80 (42)$

Step 2.11.1.2

Multiply

- 1

$- 1$ by

1764

$1764$ .

f (42) = - 1764 + 80 (42)

$f (42) = - 1764 + 80 (42)$

Step 2.11.1.3

Multiply

80

$80$ by

42

$42$ .

f (42) = - 1764 + 3360

$f (42) = - 1764 + 3360$

f (42) = - 1764 + 3360

$f (42) = - 1764 + 3360$

Step 2.11.2

Add

- 1764

$- 1764$ and

3360

$3360$ .

f (42) = 1596

$f (42) = 1596$

Step 2.11.3

The final answer is

1596

$1596$ .

1596

$1596$

1596

$1596$

Step 2.12

The

y

$y$ value at

x = 42

$x = 42$ is

1596

$1596$ .

y = 1596

$y = 1596$

Step 2.13

Graph the parabola using its properties and the selected points.

\begin{array}{c} x & y \\ 38 & 1596 \\ 39 & 1599 \\ 40 & 1600 \\ 41 & 1599 \\ 42 & 1596 \end{array}

$\begin{array}{c} x & y \\ 38 & 1596 \\ 39 & 1599 \\ 40 & 1600 \\ 41 & 1599 \\ 42 & 1596 \end{array}$

\begin{array}{c} x & y \\ 38 & 1596 \\ 39 & 1599 \\ 40 & 1600 \\ 41 & 1599 \\ 42 & 1596 \end{array}

$\begin{array}{c} x & y \\ 38 & 1596 \\ 39 & 1599 \\ 40 & 1600 \\ 41 & 1599 \\ 42 & 1596 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex:

(40, 1600)

$(40, 1600)$

Focus:

(40, \frac{6399}{4})

$(40, \frac{6399}{4})$

Axis of Symmetry:

x = 40

$x = 40$

Directrix:

y = \frac{6401}{4}

$y = \frac{6401}{4}$

\begin{array}{c} x & y \\ 38 & 1596 \\ 39 & 1599 \\ 40 & 1600 \\ 41 & 1599 \\ 42 & 1596 \end{array}

$\begin{array}{c} x & y \\ 38 & 1596 \\ 39 & 1599 \\ 40 & 1600 \\ 41 & 1599 \\ 42 & 1596 \end{array}$

Step 4