Graph y=3/4(x+1)(x-3)

$y = \frac{3}{4} (x + 1) (x - 3)$

Solve for $y$ .

Remove parentheses.

$y = \frac{3}{4} \cdot ((x + 1) (x - 3))$

Remove parentheses.

$y = \frac{3}{4} \cdot ((x + 1) (x - 3))$

Simplify $\frac{3}{4} \cdot ((x + 1) (x - 3))$ .

Expand $(x + 1) (x - 3)$ using the FOIL Method.

Apply the distributive property.

$y = \frac{3}{4} \cdot (x (x - 3) + 1 (x - 3))$

Apply the distributive property.

$y = \frac{3}{4} \cdot (x \cdot x + x \cdot - 3 + 1 (x - 3))$

Apply the distributive property.

$y = \frac{3}{4} \cdot (x \cdot x + x \cdot - 3 + 1 x + 1 \cdot - 3)$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$y = \frac{3}{4} \cdot (x^{2} + x \cdot - 3 + 1 x + 1 \cdot - 3)$

Move $- 3$ to the left of $x$ .

$y = \frac{3}{4} \cdot (x^{2} - 3 \cdot x + 1 x + 1 \cdot - 3)$

Multiply $x$ by $1$ .

$y = \frac{3}{4} \cdot (x^{2} - 3 x + x + 1 \cdot - 3)$

Multiply $- 3$ by $1$ .

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

Add $- 3 x$ and $x$ .

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

Apply the distributive property.

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

Simplify.

Combine $\frac{3}{4}$ and $x^{2}$ .

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

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

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

Factor $2$ out of $- 2 x$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Combine $\frac{3}{2}$ and $x$ .

$y = \frac{3 x^{2}}{4} - \frac{3 x}{2} + \frac{3}{4} \cdot - 3$

Multiply $\frac{3}{4} \cdot - 3$ .

Combine $\frac{3}{4}$ and $- 3$ .

$y = \frac{3 x^{2}}{4} - \frac{3 x}{2} + \frac{3 \cdot - 3}{4}$

Multiply $3$ by $- 3$ .

$y = \frac{3 x^{2}}{4} - \frac{3 x}{2} + \frac{- 9}{4}$

Move the negative in front of the fraction.

$y = \frac{3 x^{2}}{4} - \frac{3 x}{2} - \frac{9}{4}$

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Complete the square for $\frac{3 x^{2}}{4} - \frac{3 x}{2} - \frac{9}{4}$ .

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

$a = \frac{3}{4}$

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

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

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

$d = \frac{- \frac{3}{2}}{2 (\frac{3}{4})}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$d = - \frac{3}{2} \cdot \frac{1}{2 (\frac{3}{4})}$

Combine $2$ and $\frac{3}{4}$ .

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

Multiply $2$ by $3$ .

$d = - \frac{3}{2} \cdot \frac{1}{\frac{6}{4}}$

Cancel the common factor of $6$ and $4$ .

Factor $2$ out of $6$ .

$d = - \frac{3}{2} \cdot \frac{1}{\frac{2 (3)}{4}}$

Cancel the common factors.

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply the numerator by the reciprocal of the denominator.

$d = - \frac{3}{2} (1 (\frac{2}{3}))$

Multiply $\frac{2}{3}$ by $1$ .

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

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{3}{2}$ into the numerator.

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

Factor $3$ out of $- 3$ .

$d = \frac{3 (- 1)}{2} \cdot \frac{2}{3}$

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$d = - 1$

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

$e = - \frac{9}{4} - \frac{{(- \frac{3}{2})}^{2}}{4 (\frac{3}{4})}$

Simplify the right side.

Simplify each term.

Simplify the numerator.

Apply the product rule to $- \frac{3}{2}$ .

$e = - \frac{9}{4} - \frac{{(- 1)}^{2} {(\frac{3}{2})}^{2}}{4 (\frac{3}{4})}$

Raise $- 1$ to the power of $2$ .

$e = - \frac{9}{4} - \frac{1 {(\frac{3}{2})}^{2}}{4 (\frac{3}{4})}$

Apply the product rule to $\frac{3}{2}$ .

$e = - \frac{9}{4} - \frac{1 \frac{3^{2}}{2^{2}}}{4 (\frac{3}{4})}$

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

$e = - \frac{9}{4} - \frac{1 \frac{9}{2^{2}}}{4 (\frac{3}{4})}$

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

$e = - \frac{9}{4} - \frac{1 (\frac{9}{4})}{4 (\frac{3}{4})}$

Multiply $\frac{9}{4}$ by $1$ .

$e = - \frac{9}{4} - \frac{\frac{9}{4}}{4 (\frac{3}{4})}$

Combine $4$ and $\frac{3}{4}$ .

$e = - \frac{9}{4} - \frac{\frac{9}{4}}{\frac{4 \cdot 3}{4}}$

Multiply $4$ by $3$ .

$e = - \frac{9}{4} - \frac{\frac{9}{4}}{\frac{12}{4}}$

Divide $12$ by $4$ .

$e = - \frac{9}{4} - \frac{\frac{9}{4}}{3}$

Multiply the numerator by the reciprocal of the denominator.

$e = - \frac{9}{4} - (\frac{9}{4} \cdot \frac{1}{3})$

Cancel the common factor of $3$ .

Factor $3$ out of $9$ .

$e = - \frac{9}{4} - (\frac{3 (3)}{4} \cdot \frac{1}{3})$

Cancel the common factor.

$e = - \frac{9}{4} - (\frac{3 \cdot 3}{4} \cdot \frac{1}{3})$

Rewrite the expression.

$e = - \frac{9}{4} - \frac{3}{4}$

Combine the numerators over the common denominator.

$e = \frac{- 9 - 3}{4}$

Subtract $3$ from $- 9$ .

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

Divide $- 12$ by $4$ .

$e = - 3$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{3}{4} {(x - 1)}^{2} - 3$ .

$\frac{3}{4} {(x - 1)}^{2} - 3$

Set $y$ equal to the new right side.

$y = \frac{3}{4} \cdot {(x - 1)}^{2} - 3$

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

$a = \frac{3}{4}$

$h = 1$

$k = - 3$

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

Opens Up

Find the vertex $(h, k)$ .

$(1, - 3)$

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.

$\frac{1}{4 \cdot \frac{3}{4}}$

Simplify.

Combine $4$ and $\frac{3}{4}$ .

$\frac{1}{\frac{4 \cdot 3}{4}}$

Simplify the expression.

Multiply $4$ by $3$ .

$\frac{1}{\frac{12}{4}}$

Divide $12$ by $4$ .

$\frac{1}{3}$

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.

$(1, - \frac{8}{3})$

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

$x = 1$

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.

$y = - \frac{10}{3}$

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

Direction: Opens Up

Vertex: $(1, - 3)$

Focus: $(1, - \frac{8}{3})$

Axis of Symmetry: $x = 1$

Directrix: $y = - \frac{10}{3}$

Direction: Opens Up

Vertex: $(1, - 3)$

Focus: $(1, - \frac{8}{3})$

Axis of Symmetry: $x = 1$

Directrix: $y = - \frac{10}{3}$

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 $- 1$ in the expression.

$f (- 1) = \frac{3 {(- 1)}^{2}}{4} - \frac{3 (- 1)}{2} - \frac{9}{4}$

Simplify the result.

Combine the numerators over the common denominator.

$f (- 1) = \frac{3 {(- 1)}^{2} - 9}{4} + \frac{- 3 \cdot - 1}{2}$

Simplify each term.

Raise $- 1$ to the power of $2$ .

$f (- 1) = \frac{3 \cdot 1 - 9}{4} + \frac{- 3 \cdot - 1}{2}$

Multiply $3$ by $1$ .

$f (- 1) = \frac{3 - 9}{4} + \frac{- 3 \cdot - 1}{2}$

Simplify the expression.

Subtract $9$ from $3$ .

$f (- 1) = \frac{- 6}{4} + \frac{- 3 \cdot - 1}{2}$

Multiply $- 3$ by $- 1$ .

$f (- 1) = \frac{- 6}{4} + \frac{3}{2}$

Simplify each term.

Cancel the common factor of $- 6$ and $4$ .

Factor $2$ out of $- 6$ .

$f (- 1) = \frac{2 (- 3)}{4} + \frac{3}{2}$

Cancel the common factors.

Factor $2$ out of $4$ .

$f (- 1) = \frac{2 \cdot - 3}{2 \cdot 2} + \frac{3}{2}$

Cancel the common factor.

$f (- 1) = \frac{2 \cdot - 3}{2 \cdot 2} + \frac{3}{2}$

Rewrite the expression.

$f (- 1) = \frac{- 3}{2} + \frac{3}{2}$

Move the negative in front of the fraction.

$f (- 1) = - \frac{3}{2} + \frac{3}{2}$

Combine fractions.

Combine the numerators over the common denominator.

$f (- 1) = \frac{- 3 + 3}{2}$

Simplify the expression.

Add $- 3$ and $3$ .

$f (- 1) = \frac{0}{2}$

Divide $0$ by $2$ .

$f (- 1) = 0$

The final answer is $0$ .

$0$

The $y$ value at $x = - 1$ is $0$ .

$y = 0$

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

$f (0) = \frac{3 {(0)}^{2}}{4} - \frac{3 (0)}{2} - \frac{9}{4}$

Simplify the result.

Combine the numerators over the common denominator.

$f (0) = \frac{3 {(0)}^{2} - 9}{4} + \frac{- 3 \cdot 0}{2}$

Simplify each term.

Raising $0$ to any positive power yields $0$ .

$f (0) = \frac{3 \cdot 0 - 9}{4} + \frac{- 3 \cdot 0}{2}$

Multiply $3$ by $0$ .

$f (0) = \frac{0 - 9}{4} + \frac{- 3 \cdot 0}{2}$

Simplify the expression.

Subtract $9$ from $0$ .

$f (0) = \frac{- 9}{4} + \frac{- 3 \cdot 0}{2}$

Multiply $- 3$ by $0$ .

$f (0) = \frac{- 9}{4} + \frac{0}{2}$

Simplify each term.

Move the negative in front of the fraction.

$f (0) = - \frac{9}{4} + \frac{0}{2}$

Divide $0$ by $2$ .

$f (0) = - \frac{9}{4} + 0$

Add $- \frac{9}{4}$ and $0$ .

$f (0) = - \frac{9}{4}$

The final answer is $- \frac{9}{4}$ .

$- \frac{9}{4}$

The $y$ value at $x = 0$ is $- \frac{9}{4}$ .

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

Replace the variable $x$ with $3$ in the expression.

$f (3) = \frac{3 {(3)}^{2}}{4} - \frac{3 (3)}{2} - \frac{9}{4}$

Simplify the result.

Combine the numerators over the common denominator.

$f (3) = \frac{3 {(3)}^{2} - 9}{4} + \frac{- 3 \cdot 3}{2}$

Simplify each term.

Multiply $3$ by ${(3)}^{2}$ by adding the exponents.

Multiply $3$ by ${(3)}^{2}$ .

Raise $3$ to the power of $1$ .

$f (3) = \frac{3 {(3)}^{2} - 9}{4} + \frac{- 3 \cdot 3}{2}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (3) = \frac{3^{1 + 2} - 9}{4} + \frac{- 3 \cdot 3}{2}$

Add $1$ and $2$ .

$f (3) = \frac{3^{3} - 9}{4} + \frac{- 3 \cdot 3}{2}$

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

$f (3) = \frac{27 - 9}{4} + \frac{- 3 \cdot 3}{2}$

Simplify the expression.

Subtract $9$ from $27$ .

$f (3) = \frac{18}{4} + \frac{- 3 \cdot 3}{2}$

Multiply $- 3$ by $3$ .

$f (3) = \frac{18}{4} + \frac{- 9}{2}$

Simplify each term.

Cancel the common factor of $18$ and $4$ .

Factor $2$ out of $18$ .

$f (3) = \frac{2 (9)}{4} + \frac{- 9}{2}$

Cancel the common factors.

Factor $2$ out of $4$ .

$f (3) = \frac{2 \cdot 9}{2 \cdot 2} + \frac{- 9}{2}$

Cancel the common factor.

$f (3) = \frac{2 \cdot 9}{2 \cdot 2} + \frac{- 9}{2}$

Rewrite the expression.

$f (3) = \frac{9}{2} + \frac{- 9}{2}$

Move the negative in front of the fraction.

$f (3) = \frac{9}{2} - \frac{9}{2}$

Combine fractions.

Combine the numerators over the common denominator.

$f (3) = \frac{9 - 9}{2}$

Simplify the expression.

Subtract $9$ from $9$ .

$f (3) = \frac{0}{2}$

Divide $0$ by $2$ .

$f (3) = 0$

The final answer is $0$ .

$0$

The $y$ value at $x = 3$ is $0$ .

$y = 0$

Replace the variable $x$ with $2$ in the expression.

$f (2) = \frac{3 {(2)}^{2}}{4} - \frac{3 (2)}{2} - \frac{9}{4}$

Simplify the result.

Combine the numerators over the common denominator.

$f (2) = \frac{3 {(2)}^{2} - 9}{4} + \frac{- 3 \cdot 2}{2}$

Simplify each term.

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

$f (2) = \frac{3 \cdot 4 - 9}{4} + \frac{- 3 \cdot 2}{2}$

Multiply $3$ by $4$ .

$f (2) = \frac{12 - 9}{4} + \frac{- 3 \cdot 2}{2}$

Simplify the expression.

Subtract $9$ from $12$ .

$f (2) = \frac{3}{4} + \frac{- 3 \cdot 2}{2}$

Multiply $- 3$ by $2$ .

$f (2) = \frac{3}{4} + \frac{- 6}{2}$

Divide $- 6$ by $2$ .

$f (2) = \frac{3}{4} - 3$

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$f (2) = \frac{3}{4} - 3 \cdot \frac{4}{4}$

Combine $- 3$ and $\frac{4}{4}$ .

$f (2) = \frac{3}{4} + \frac{- 3 \cdot 4}{4}$

Combine the numerators over the common denominator.

$f (2) = \frac{3 - 3 \cdot 4}{4}$

Simplify the numerator.

Multiply $- 3$ by $4$ .

$f (2) = \frac{3 - 12}{4}$

Subtract $12$ from $3$ .

$f (2) = \frac{- 9}{4}$

Move the negative in front of the fraction.

$f (2) = - \frac{9}{4}$

The final answer is $- \frac{9}{4}$ .

$- \frac{9}{4}$

The $y$ value at $x = 2$ is $- \frac{9}{4}$ .

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

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 1 & 0 \\ 0 & - \frac{9}{4} \\ 1 & - 3 \\ 2 & - \frac{9}{4} \\ 3 & 0 \end{array}$

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(1, - 3)$

Focus: $(1, - \frac{8}{3})$

Axis of Symmetry: $x = 1$

Directrix: $y = - \frac{10}{3}$

$\begin{array}{c} x & y \\ - 1 & 0 \\ 0 & - \frac{9}{4} \\ 1 & - 3 \\ 2 & - \frac{9}{4} \\ 3 & 0 \end{array}$

Graph y=3/4(x+1)(x-3)

Related Questions

Support LCMGCF.com by disabling your adblocker.