Algebra Examples

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

Step 1

Find the properties of the given parabola.

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Step 1.1

Rewrite the equation in vertex form.

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Step 1.1.1

Complete the square for $- 4 (x - 3) (x - 1)$ .

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Step 1.1.1.1

Simplify the expression.

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Step 1.1.1.1.1

Apply the distributive property.

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

Step 1.1.1.1.2

Multiply $- 4$ by $- 3$ .

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

Step 1.1.1.1.3

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

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Step 1.1.1.1.3.1

Apply the distributive property.

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

Step 1.1.1.1.3.2

Apply the distributive property.

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

Step 1.1.1.1.3.3

Apply the distributive property.

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

Step 1.1.1.1.4

Simplify and combine like terms.

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Step 1.1.1.1.4.1

Simplify each term.

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Step 1.1.1.1.4.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 1.1.1.1.4.1.1.1

Move $x$ .

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

Step 1.1.1.1.4.1.1.2

Multiply $x$ by $x$ .

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

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

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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.

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Step 1.1.1.4.2.1

Cancel the common factor of $16$ and $2$ .

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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.

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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 $- 4$ .

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

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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.

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Step 1.1.1.5.2.1

Simplify each term.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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