Algebra Examples

- 4 (x - 3) (x - 1)

$- 4 (x - 3) (x - 1)$

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

Complete the square for

- 4 (x - 3) (x - 1)

$- 4 (x - 3) (x - 1)$ .

Tap for more steps...

Step 1.1.1.1

Simplify the expression.

Tap for more steps...

Step 1.1.1.1.1

Apply the distributive property.

(- 4 x - 4 \cdot - 3) (x - 1)

$(- 4 x - 4 \cdot - 3) (x - 1)$

Step 1.1.1.1.2

Multiply

- 4

$- 4$ by

- 3

$- 3$ .

(- 4 x + 12) (x - 1)

$(- 4 x + 12) (x - 1)$

Step 1.1.1.1.3

Expand

(- 4 x + 12) (x - 1)

$(- 4 x + 12) (x - 1)$ using the FOIL Method.

Tap for more steps...

Step 1.1.1.1.3.1

Apply the distributive property.

- 4 x (x - 1) + 12 (x - 1)

$- 4 x (x - 1) + 12 (x - 1)$

Step 1.1.1.1.3.2

Apply the distributive property.

- 4 x \cdot x - 4 x \cdot - 1 + 12 (x - 1)

$- 4 x \cdot x - 4 x \cdot - 1 + 12 (x - 1)$

Step 1.1.1.1.3.3

Apply the distributive property.

- 4 x \cdot x - 4 x \cdot - 1 + 12 x + 12 \cdot - 1

$- 4 x \cdot x - 4 x \cdot - 1 + 12 x + 12 \cdot - 1$

- 4 x \cdot x - 4 x \cdot - 1 + 12 x + 12 \cdot - 1

$- 4 x \cdot x - 4 x \cdot - 1 + 12 x + 12 \cdot - 1$

Step 1.1.1.1.4

Simplify and combine like terms.

Tap for more steps...

Step 1.1.1.1.4.1

Simplify each term.

Tap for more steps...

Step 1.1.1.1.4.1.1

Multiply

x

$x$ by

x

$x$ by adding the exponents.

Tap for more steps...

Step 1.1.1.1.4.1.1.1

Move

x

$x$ .

- 4 (x \cdot x) - 4 x \cdot - 1 + 12 x + 12 \cdot - 1

$- 4 (x \cdot x) - 4 x \cdot - 1 + 12 x + 12 \cdot - 1$

Step 1.1.1.1.4.1.1.2

Multiply

x

$x$ by

x

$x$ .

- 4 x^{2} - 4 x \cdot - 1 + 12 x + 12 \cdot - 1

$- 4 x^{2} - 4 x \cdot - 1 + 12 x + 12 \cdot - 1$

Step 1.1.1.1.4.1.2

Multiply

$- 1$ by

$- 4$ .

$- 4 x^{2} + 4 x + 12 x + 12 \cdot - 1$

Step 1.1.1.1.4.1.3

Multiply

$12$ by

$- 1$ .

$- 4 x^{2} + 4 x + 12 x - 12$

Step 1.1.1.1.4.2

Add

$4 x$ and

$12 x$ .

$- 4 x^{2} + 16 x - 12$

Step 1.1.1.2

Use the form

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

$a$ ,

$b$ , and

$c$ .

$a = - 4$

$b = 16$

$c = - 12$

Step 1.1.1.3

Consider the vertex form of a parabola.

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

Step 1.1.1.4

Find the value of

$d$ using the formula

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

Tap for more steps...

Step 1.1.1.4.1

Substitute the values of

$a$ and

$b$ into the formula

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

$d = \frac{16}{2 \cdot - 4}$

Step 1.1.1.4.2

Simplify the right side.

Tap for more steps...

Step 1.1.1.4.2.1

Cancel the common factor of

$16$ and

$2$ .

Tap for more steps...

Step 1.1.1.4.2.1.1

Factor

$2$ out of

$16$ .

$d = \frac{2 \cdot 8}{2 \cdot - 4}$

Step 1.1.1.4.2.1.2

Cancel the common factors.

Tap for more steps...

Step 1.1.1.4.2.1.2.1

Factor

$2$ out of

$2 \cdot - 4$ .

$d = \frac{2 \cdot 8}{2 (- 4)}$

Step 1.1.1.4.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot 8}{2 \cdot - 4}$

Step 1.1.1.4.2.1.2.3

Rewrite the expression.

$d = \frac{8}{- 4}$

Step 1.1.1.4.2.2

Cancel the common factor of

$8$ and

$- 4$ .

Tap for more steps...

Step 1.1.1.4.2.2.1

Factor

$4$ out of

$8$ .

$d = \frac{4 (2)}{- 4}$

Step 1.1.1.4.2.2.2

Move the negative one from the denominator of

$\frac{2}{- 1}$ .

$d = - 1 \cdot 2$

Step 1.1.1.4.2.3

Multiply

$- 1$ by

$2$ .

$d = - 2$

Step 1.1.1.5

Find the value of

$e$ using the formula

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

Tap for more steps...

Step 1.1.1.5.1

Substitute the values of

$c$ ,

$b$ and

$a$ into the formula

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

$e = - 12 - \frac{16^{2}}{4 \cdot - 4}$

Step 1.1.1.5.2

Simplify the right side.

Tap for more steps...

Step 1.1.1.5.2.1

Simplify each term.

Tap for more steps...

Step 1.1.1.5.2.1.1

Raise

$16$ to the power of

$2$ .

$e = - 12 - \frac{256}{4 \cdot - 4}$

Step 1.1.1.5.2.1.2

Multiply

$4$ by

$- 4$ .

$e = - 12 - \frac{256}{- 16}$

Step 1.1.1.5.2.1.3

Divide

$256$ by

$- 16$ .

$e = - 12 - - 16$

Step 1.1.1.5.2.1.4

Multiply

$- 1$ by

$- 16$ .

$e = - 12 + 16$

Step 1.1.1.5.2.2

Add

$- 12$ and

$16$ .

$e = 4$

Step 1.1.1.6

Substitute the values of

$a$ ,

$d$ , and

$e$ into the vertex form

$- 4 {(x - 2)}^{2} + 4$ .

$- 4 {(x - 2)}^{2} + 4$

Step 1.1.2

Set

$y$ equal to the new right side.

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

Step 1.2

Use the vertex form,

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

$a$ ,

$h$ , and

$k$ .

$a = - 4$

$h = 2$

$k = 4$

Step 1.3

Since the value of

$a$ is negative, the parabola opens down.

Opens Down

Step 1.4

Find the vertex

$(h, k)$ .

$(2, 4)$

Step 1.5

Find

$p$ , 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.

$\frac{1}{4 a}$

Step 1.5.2

Substitute the value of

$a$ into the formula.

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

Step 1.5.3

Simplify.

Tap for more steps...

Step 1.5.3.1

Multiply

$4$ by

$- 4$ .

$\frac{1}{- 16}$

Step 1.5.3.2

Move the negative in front of the fraction.

$- \frac{1}{16}$

Step 1.6

Find the focus.

Tap for more steps...

Step 1.6.1

The focus of a parabola can be found by adding

$p$ to the y-coordinate

$k$ if the parabola opens up or down.

$(h, k + p)$

Step 1.6.2

Substitute the known values of

$h$ ,

$p$ , and

$k$ into the formula and simplify.

$(2, \frac{63}{16})$

Step 1.7

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

$x = 2$

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

$p$ from the y-coordinate

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

$y = k - p$

Step 1.8.2

Substitute the known values of

$p$ and

$k$ into the formula and simplify.

$y = \frac{65}{16}$

Step 1.9

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

Direction: Opens Down

Vertex:

$(2, 4)$

Focus:

$(2, \frac{63}{16})$

Axis of Symmetry:

$x = 2$

Directrix:

$y = \frac{65}{16}$

Direction: Opens Down

Vertex:

$(2, 4)$

Focus:

$(2, \frac{63}{16})$

Axis of Symmetry:

$x = 2$

Directrix:

$y = \frac{65}{16}$

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.

Tap for more steps...

Step 2.1

Replace the variable

$x$ with

$1$ in the expression.

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

Step 2.2

Simplify the result.

Tap for more steps...

Step 2.2.1

Subtract

$3$ from

$1$ .

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

Step 2.2.2

Multiply

$- 4$ by

$- 2$ .

$f (1) = 8 ((1) - 1)$

Step 2.2.3

Subtract

$1$ from

$1$ .

$f (1) = 8 \cdot 0$

Step 2.2.4

Multiply

$8$ by

$0$ .

$f (1) = 0$

Step 2.2.5

The final answer is

$0$ .

$0$

Step 2.3

The

$y$ value at

$x = 1$ is

$0$ .

$y = 0$

Step 2.4

Replace the variable

$x$ with

$0$ in the expression.

$f (0) = - 4 ((0) - 3) ((0) - 1)$

Step 2.5

Simplify the result.

Tap for more steps...

Step 2.5.1

Subtract

$3$ from

$0$ .

$f (0) = - 4 \cdot (- 3 ((0) - 1))$

Step 2.5.2

Multiply

$- 4$ by

$- 3$ .

$f (0) = 12 ((0) - 1)$

Step 2.5.3

Subtract

$1$ from

$0$ .

$f (0) = 12 \cdot - 1$

Step 2.5.4

Multiply

$12$ by

$- 1$ .

$f (0) = - 12$

Step 2.5.5

The final answer is

$- 12$ .

$- 12$

Step 2.6

The

$y$ value at

$x = 0$ is

$- 12$ .

$y = - 12$

Step 2.7

Replace the variable

$x$ with

$3$ in the expression.

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

Step 2.8

Simplify the result.

Tap for more steps...

Step 2.8.1

Subtract

$3$ from

$3$ .

$f (3) = - 4 \cdot (0 ((3) - 1))$

Step 2.8.2

Multiply

$- 4$ by

$0$ .

$f (3) = 0 ((3) - 1)$

Step 2.8.3

Subtract

$1$ from

$3$ .

$f (3) = 0 \cdot 2$

Step 2.8.4

Multiply

$0$ by

$2$ .

$f (3) = 0$

Step 2.8.5

The final answer is

$0$ .

$0$

Step 2.9

The

$y$ value at

$x = 3$ is

$0$ .

$y = 0$

Step 2.10

Replace the variable

$x$ with

$4$ in the expression.

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

Step 2.11

Simplify the result.

Tap for more steps...

Step 2.11.1

Subtract

$3$ from

$4$ .

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

Step 2.11.2

Multiply

$- 4$ by

$1$ .

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

Step 2.11.3

Subtract

$1$ from

$4$ .

$f (4) = - 4 \cdot 3$

Step 2.11.4

Multiply

$- 4$ by

$3$ .

$f (4) = - 12$

Step 2.11.5

The final answer is

$- 12$ .

$- 12$

Step 2.12

The

$y$ value at

$x = 4$ is

$- 12$ .

$y = - 12$

Step 2.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 0 & - 12 \\ 1 & 0 \\ 2 & 4 \\ 3 & 0 \\ 4 & - 12 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex:

$(2, 4)$

Focus:

$(2, \frac{63}{16})$

Axis of Symmetry:

$x = 2$

Directrix:

$y = \frac{65}{16}$

$\begin{array}{c} x & y \\ 0 & - 12 \\ 1 & 0 \\ 2 & 4 \\ 3 & 0 \\ 4 & - 12 \end{array}$

Step 4