Graph y=3/2x^2-4x

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

Find the properties of the given parabola.

Rewrite the equation in vertex form.

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

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

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

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

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

$b = - 4$

$c = 0$

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

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

Simplify the right side.

Cancel the common factor of $- 4$ and $2$ .

Factor $2$ out of $- 4$ .

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

Cancel the common factors.

Cancel the common factor.

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

Rewrite the expression.

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

Multiply the numerator by the reciprocal of the denominator.

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

Multiply $- 2 (\frac{2}{3})$ .

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

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

Multiply $- 2$ by $2$ .

$d = \frac{- 4}{3}$

Move the negative in front of the fraction.

$d = - \frac{4}{3}$

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

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

Simplify the right side.

Simplify each term.

Cancel the common factor of ${(- 4)}^{2}$ and $4$ .

Rewrite $- 4$ as $- 1 (4)$ .

$e = 0 - \frac{{(- 1 (4))}^{2}}{4 (\frac{3}{2})}$

Apply the product rule to $- 1 (4)$ .

$e = 0 - \frac{{(- 1)}^{2} \cdot 4^{2}}{4 (\frac{3}{2})}$

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

$e = 0 - \frac{1 \cdot 4^{2}}{4 (\frac{3}{2})}$

Multiply $4^{2}$ by $1$ .

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

Factor $4$ out of $4^{2}$ .

$e = 0 - \frac{4 \cdot 4}{4 (\frac{3}{2})}$

Cancel the common factors.

Cancel the common factor.

$e = 0 - \frac{4 \cdot 4}{4 (\frac{3}{2})}$

Rewrite the expression.

$e = 0 - \frac{4}{\frac{3}{2}}$

Multiply the numerator by the reciprocal of the denominator.

$e = 0 - (4 (\frac{2}{3}))$

Multiply $4 (\frac{2}{3})$ .

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

$e = 0 - \frac{4 \cdot 2}{3}$

Multiply $4$ by $2$ .

$e = 0 - \frac{8}{3}$

Subtract $\frac{8}{3}$ from $0$ .

$e = - \frac{8}{3}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{3}{2} {(x - \frac{4}{3})}^{2} - \frac{8}{3}$ .

$\frac{3}{2} {(x - \frac{4}{3})}^{2} - \frac{8}{3}$

Set $y$ equal to the new right side.

$y = \frac{3}{2} \cdot {(x - \frac{4}{3})}^{2} - \frac{8}{3}$

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

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

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

$k = - \frac{8}{3}$

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

Opens Up

Find the vertex $(h, k)$ .

$(\frac{4}{3}, - \frac{8}{3})$

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.

$\frac{1}{4 \cdot \frac{3}{2}}$

Simplify.

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

$\frac{1}{\frac{4 \cdot 3}{2}}$

Simplify the expression.

Multiply $4$ by $3$ .

$\frac{1}{\frac{12}{2}}$

Divide $12$ by $2$ .

$\frac{1}{6}$

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.

$(\frac{4}{3}, - \frac{5}{2})$

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

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

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.

$y = - \frac{17}{6}$

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

Direction: Opens Up

Vertex: $(\frac{4}{3}, - \frac{8}{3})$

Focus: $(\frac{4}{3}, - \frac{5}{2})$

Axis of Symmetry: $x = \frac{4}{3}$

Directrix: $y = - \frac{17}{6}$

Direction: Opens Up

Vertex: $(\frac{4}{3}, - \frac{8}{3})$

Focus: $(\frac{4}{3}, - \frac{5}{2})$

Axis of Symmetry: $x = \frac{4}{3}$

Directrix: $y = - \frac{17}{6}$

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 $0$ in the expression.

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

Simplify the result.

Simplify each term.

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

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

Multiply $3$ by $0$ .

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

Divide $0$ by $2$ .

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

Multiply $- 4$ by $0$ .

$f (0) = 0 + 0$

Add $0$ and $0$ .

$f (0) = 0$

The final answer is $0$ .

$0$

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

$y = 0$

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

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

Simplify the result.

Simplify each term.

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

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

Multiply $3$ by $1$ .

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

Multiply $- 4$ by $- 1$ .

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

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $4$ by $2$ .

$f (- 1) = \frac{3 + 8}{2}$

Add $3$ and $8$ .

$f (- 1) = \frac{11}{2}$

The final answer is $\frac{11}{2}$ .

$\frac{11}{2}$

The $y$ value at $x = - 1$ is $\frac{11}{2}$ .

$y = \frac{11}{2}$

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

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

Simplify the result.

Simplify each term.

Cancel the common factor of ${(2)}^{2}$ and $2$ .

Factor $2$ out of $3 {(2)}^{2}$ .

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

Cancel the common factors.

Factor $2$ out of $2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $3 \cdot 2$ by $1$ .

$f (2) = 3 \cdot 2 - 4 \cdot 2$

Multiply $3$ by $2$ .

$f (2) = 6 - 4 \cdot 2$

Multiply $- 4$ by $2$ .

$f (2) = 6 - 8$

Subtract $8$ from $6$ .

$f (2) = - 2$

The final answer is $- 2$ .

$- 2$

The $y$ value at $x = 2$ is $- 2$ .

$y = - 2$

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

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

Simplify the result.

Simplify each term.

Raise $4$ to the power of $2$ .

$f (4) = \frac{3 \cdot 16}{2} - 4 \cdot 4$

Multiply $3$ by $16$ .

$f (4) = \frac{48}{2} - 4 \cdot 4$

Divide $48$ by $2$ .

$f (4) = 24 - 4 \cdot 4$

Multiply $- 4$ by $4$ .

$f (4) = 24 - 16$

Subtract $16$ from $24$ .

$f (4) = 8$

The final answer is $8$ .

$8$

The $y$ value at $x = 4$ is $8$ .

$y = 8$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 1 & \frac{11}{2} \\ 0 & 0 \\ \frac{4}{3} & - \frac{8}{3} \\ 2 & - 2 \\ 4 & 8 \end{array}$

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(\frac{4}{3}, - \frac{8}{3})$

Focus: $(\frac{4}{3}, - \frac{5}{2})$

Axis of Symmetry: $x = \frac{4}{3}$

Directrix: $y = - \frac{17}{6}$

$\begin{array}{c} x & y \\ - 1 & \frac{11}{2} \\ 0 & 0 \\ \frac{4}{3} & - \frac{8}{3} \\ 2 & - 2 \\ 4 & 8 \end{array}$

Graph y=3/2x^2-4x

Related Questions

Support LCMGCF.com by disabling your adblocker.