Graph y=-1/2x^2-3/2x+2

$y = - \frac{1}{2} x^{2} - \frac{3}{2} x + 2$

Step 1

Simplify each term.

Combine $x^{2}$ and $\frac{1}{2}$ .

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

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

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

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

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

Step 2

Find the properties of the given parabola.

Rewrite the equation in vertex form.

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

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

$a = - \frac{1}{2}$

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

$c = 2$

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

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

Simplify the right side.

Dividing two negative values results in a positive value.

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

Multiply the numerator by the reciprocal of the denominator.

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

Combine $2$ and $\frac{1}{2}$ .

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

Divide $2$ by $2$ .

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

Cancel the common factor of $1$ .

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $\frac{3}{2}$ 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 $e = c - \frac{b^{2}}{4 a}$ .

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

Simplify the right side.

Simplify each term.

Simplify the numerator.

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

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

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

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

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

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

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

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

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

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

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

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

Simplify the denominator.

Multiply $- 1$ by $4$ .

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

Combine $- 4$ and $\frac{1}{2}$ .

$e = 2 - \frac{\frac{9}{4}}{\frac{- 4}{2}}$

Divide $- 4$ by $2$ .

$e = 2 - \frac{\frac{9}{4}}{- 2}$

Multiply the numerator by the reciprocal of the denominator.

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

Move the negative in front of the fraction.

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

Multiply $\frac{9}{4} (- \frac{1}{2})$ .

Multiply $\frac{9}{4}$ by $\frac{1}{2}$ .

$e = 2 - - \frac{9}{4 \cdot 2}$

Multiply $4$ by $2$ .

$e = 2 - - \frac{9}{8}$

Multiply $- - \frac{9}{8}$ .

Multiply $- 1$ by $- 1$ .

$e = 2 + 1 (\frac{9}{8})$

Multiply $\frac{9}{8}$ by $1$ .

$e = 2 + \frac{9}{8}$

To write $2$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$e = 2 \cdot \frac{8}{8} + \frac{9}{8}$

Combine $2$ and $\frac{8}{8}$ .

$e = \frac{2 \cdot 8}{8} + \frac{9}{8}$

Combine the numerators over the common denominator.

$e = \frac{2 \cdot 8 + 9}{8}$

Simplify the numerator.

Multiply $2$ by $8$ .

$e = \frac{16 + 9}{8}$

Add $16$ and $9$ .

$e = \frac{25}{8}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{2} {(x + \frac{3}{2})}^{2} + \frac{25}{8}$ .

$- \frac{1}{2} {(x + \frac{3}{2})}^{2} + \frac{25}{8}$

Set $y$ equal to the new right side.

$y = - \frac{1}{2} \cdot {(x + \frac{3}{2})}^{2} + \frac{25}{8}$

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

$a = - \frac{1}{2}$

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

$k = \frac{25}{8}$

Since the value of $a$ is negative, the parabola opens down.

Opens Down

Find the vertex $(h, k)$ .

$(- \frac{3}{2}, \frac{25}{8})$

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.

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

Simplify.

Cancel the common factor of $1$ and $- 1$ .

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

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

Move the negative in front of the fraction.

$- \frac{1}{4 (\frac{1}{2})}$

Combine $4$ and $\frac{1}{2}$ .

$- \frac{1}{\frac{4}{2}}$

Divide $4$ by $2$ .

$- \frac{1}{2}$

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.

$(- \frac{3}{2}, \frac{21}{8})$

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.

$y = \frac{29}{8}$

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

Direction: Opens Down

Vertex: $(- \frac{3}{2}, \frac{25}{8})$

Focus: $(- \frac{3}{2}, \frac{21}{8})$

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

Directrix: $y = \frac{29}{8}$

Direction: Opens Down

Vertex: $(- \frac{3}{2}, \frac{25}{8})$

Focus: $(- \frac{3}{2}, \frac{21}{8})$

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

Directrix: $y = \frac{29}{8}$

Step 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 $- 3$ in the expression.

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

Simplify the result.

Combine the numerators over the common denominator.

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

Simplify each term.

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

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

Multiply $- 1$ by $9$ .

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

Multiply $- 3$ by $- 3$ .

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

Simplify the expression.

Add $- 9$ and $9$ .

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

Divide $0$ by $2$ .

$f (- 3) = 2 + 0$

Add $2$ and $0$ .

$f (- 3) = 2$

The final answer is $2$ .

$2$

The $y$ value at $x = - 3$ is $2$ .

$y = 2$

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

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

Simplify the result.

Combine the numerators over the common denominator.

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

Simplify each term.

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

$f (- 4) = 2 + \frac{- 1 \cdot 16 - 3 \cdot - 4}{2}$

Multiply $- 1$ by $16$ .

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

Multiply $- 3$ by $- 4$ .

$f (- 4) = 2 + \frac{- 16 + 12}{2}$

Simplify the expression.

Add $- 16$ and $12$ .

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

Divide $- 4$ by $2$ .

$f (- 4) = 2 - 2$

Subtract $2$ from $2$ .

$f (- 4) = 0$

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.

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

Simplify the result.

Combine the numerators over the common denominator.

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

Simplify each term.

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

Multiply $- 1$ by ${(- 1)}^{2}$ .

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

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

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

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

Add $1$ and $2$ .

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

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

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

Multiply $- 3$ by $- 1$ .

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

Simplify the expression.

Add $- 1$ and $3$ .

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

Divide $2$ by $2$ .

$f (- 1) = 2 + 1$

Add $2$ and $1$ .

$f (- 1) = 3$

The final answer is $3$ .

$3$

The $y$ value at $x = - 1$ is $3$ .

$y = 3$

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

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

Simplify the result.

Combine the numerators over the common denominator.

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

Simplify each term.

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

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

Multiply $- 1$ by $0$ .

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

Multiply $- 3$ by $0$ .

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

Simplify the expression.

Add $0$ and $0$ .

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

Divide $0$ by $2$ .

$f (0) = 2 + 0$

Add $2$ and $0$ .

$f (0) = 2$

The final answer is $2$ .

$2$

The $y$ value at $x = 0$ is $2$ .

$y = 2$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 4 & 0 \\ - 3 & 2 \\ - \frac{3}{2} & \frac{25}{8} \\ - 1 & 3 \\ 0 & 2 \end{array}$

Step 4

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(- \frac{3}{2}, \frac{25}{8})$

Focus: $(- \frac{3}{2}, \frac{21}{8})$

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

Directrix: $y = \frac{29}{8}$

$\begin{array}{c} x & y \\ - 4 & 0 \\ - 3 & 2 \\ - \frac{3}{2} & \frac{25}{8} \\ - 1 & 3 \\ 0 & 2 \end{array}$

Step 5

Graph y=-1/2x^2-3/2x+2

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