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

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

Step 1

Find the properties of the given parabola.

Rewrite the equation in vertex form.

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

Simplify the expression.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Move $4$ to the left of $x$ .

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

Rewrite $- 1 x$ as $- x$ .

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

Multiply $- 1$ by $4$ .

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

Subtract $x$ from $4 x$ .

$x^{2} + 3 x - 4$

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

$a = 1$

$b = 3$

$c = - 4$

Consider the vertex form of a parabola.

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

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{3}{2 \cdot 1}$

Multiply $2$ by $1$ .

$d = \frac{3}{2}$

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = - 4 - \frac{3^{2}}{4 \cdot 1}$

Simplify the right side.

Simplify each term.

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

$e = - 4 - \frac{9}{4 \cdot 1}$

Multiply $4$ by $1$ .

$e = - 4 - \frac{9}{4}$

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$e = - 4 \cdot \frac{4}{4} - \frac{9}{4}$

Combine $- 4$ and $\frac{4}{4}$ .

$e = \frac{- 4 \cdot 4}{4} - \frac{9}{4}$

Combine the numerators over the common denominator.

$e = \frac{- 4 \cdot 4 - 9}{4}$

Simplify the numerator.

Multiply $- 4$ by $4$ .

$e = \frac{- 16 - 9}{4}$

Subtract $9$ from $- 16$ .

$e = \frac{- 25}{4}$

Move the negative in front of the fraction.

$e = - \frac{25}{4}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x + \frac{3}{2})}^{2} - \frac{25}{4}$ .

${(x + \frac{3}{2})}^{2} - \frac{25}{4}$

Set $y$ equal to the new right side.

$y = {(x + \frac{3}{2})}^{2} - \frac{25}{4}$

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

$a = 1$

$h = - \frac{3}{2}$

$k = - \frac{25}{4}$

Since the value of $a$ is positive, the parabola opens up.

Opens Up

Find the vertex $(h, k)$ .

$(- \frac{3}{2}, - \frac{25}{4})$

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 $1$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{1}{4}$

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.

$(- \frac{3}{2}, - 6)$

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

$x = - \frac{3}{2}$

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.

$y = - \frac{13}{2}$

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

Direction: Opens Up

Vertex: $(- \frac{3}{2}, - \frac{25}{4})$

Focus: $(- \frac{3}{2}, - 6)$

Axis of Symmetry: $x = - \frac{3}{2}$

Directrix: $y = - \frac{13}{2}$

Direction: Opens Up

Vertex: $(- \frac{3}{2}, - \frac{25}{4})$

Focus: $(- \frac{3}{2}, - 6)$

Axis of Symmetry: $x = - \frac{3}{2}$

Directrix: $y = - \frac{13}{2}$

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 $- 3$ in the expression.

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

Simplify the result.

Subtract $1$ from $- 3$ .

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

Add $- 3$ and $4$ .

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

Multiply $- 4$ by $1$ .

$f (- 3) = - 4$

The final answer is $- 4$ .

$- 4$

The $y$ value at $x = - 3$ is $- 4$ .

$y = - 4$

Replace the variable $x$ with $- 4$ in the expression.

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

Simplify the result.

Subtract $1$ from $- 4$ .

$f (- 4) = - 5 ((- 4) + 4)$

Add $- 4$ and $4$ .

$f (- 4) = - 5 \cdot 0$

Multiply $- 5$ by $0$ .

$f (- 4) = 0$

The final answer is $0$ .

$0$

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

$y = 0$

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

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

Simplify the result.

Subtract $1$ from $- 1$ .

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

Add $- 1$ and $4$ .

$f (- 1) = - 2 \cdot 3$

Multiply $- 2$ by $3$ .

$f (- 1) = - 6$

The final answer is $- 6$ .

$- 6$

The $y$ value at $x = - 1$ is $- 6$ .

$y = - 6$

Replace the variable $x$ with $0$ in the expression.

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

Simplify the result.

Subtract $1$ from $0$ .

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

Add $0$ and $4$ .

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

Multiply $- 1$ by $4$ .

$f (0) = - 4$

The final answer is $- 4$ .

$- 4$

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

$y = - 4$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 4 & 0 \\ - 3 & - 4 \\ - \frac{3}{2} & - \frac{25}{4} \\ - 1 & - 6 \\ 0 & - 4 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(- \frac{3}{2}, - \frac{25}{4})$

Focus: $(- \frac{3}{2}, - 6)$

Axis of Symmetry: $x = - \frac{3}{2}$

Directrix: $y = - \frac{13}{2}$

$\begin{array}{c} x & y \\ - 4 & 0 \\ - 3 & - 4 \\ - \frac{3}{2} & - \frac{25}{4} \\ - 1 & - 6 \\ 0 & - 4 \end{array}$

Step 4

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

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