Graph y=-1/2x^2+4x-3

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

Step 1

Simplify each term.

Move the negative in front of the fraction.

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

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

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

Step 2

Find the properties of the given parabola.

Rewrite the equation in vertex form.

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

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

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

$b = 4$

$c = - 3$

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

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

Simplify the right side.

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

Factor $2$ out of $4$ .

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

Cancel the common factors.

Cancel the common factor.

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

Rewrite the expression.

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

Multiply the numerator by the reciprocal of the denominator.

$d = 2 (- 1 \cdot 2)$

Multiply $2 (- 1 \cdot 2)$ .

Multiply $- 1$ by $2$ .

$d = 2 \cdot - 2$

Multiply $2$ by $- 2$ .

$d = - 4$

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

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

Simplify the right side.

Simplify each term.

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

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

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

Cancel the common factors.

Cancel the common factor.

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

Rewrite the expression.

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

Multiply the numerator by the reciprocal of the denominator.

$e = - 3 - (4 (- 1 \cdot 2))$

Multiply $- (4 (- 1 \cdot 2))$ .

Multiply $- 1$ by $2$ .

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

Multiply $4$ by $- 2$ .

$e = - 3 - - 8$

Multiply $- 1$ by $- 8$ .

$e = - 3 + 8$

Add $- 3$ and $8$ .

$e = 5$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{2} {(x - 4)}^{2} + 5$ .

$- \frac{1}{2} {(x - 4)}^{2} + 5$

Set $y$ equal to the new right side.

$y = - \frac{1}{2} \cdot {(x - 4)}^{2} + 5$

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

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

$h = 4$

$k = 5$

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

Opens Down

Find the vertex $(h, k)$ .

$(4, 5)$

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.

$(4, \frac{9}{2})$

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

$x = 4$

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.

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

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

Direction: Opens Down

Vertex: $(4, 5)$

Focus: $(4, \frac{9}{2})$

Axis of Symmetry: $x = 4$

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

Direction: Opens Down

Vertex: $(4, 5)$

Focus: $(4, \frac{9}{2})$

Axis of Symmetry: $x = 4$

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

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 $2$ in the expression.

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

Simplify the result.

Simplify each term.

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

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

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

Cancel the common factors.

Factor $2$ out of $2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $2$ by $1$ .

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

Multiply $- 1$ by $2$ .

$f (2) = - 2 + 4 (2) - 3$

Multiply $4$ by $2$ .

$f (2) = - 2 + 8 - 3$

Simplify by adding and subtracting.

Add $- 2$ and $8$ .

$f (2) = 6 - 3$

Subtract $3$ from $6$ .

$f (2) = 3$

The final answer is $3$ .

$3$

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

$y = 3$

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

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

Simplify the result.

Simplify each term.

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

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

Multiply $4$ by $3$ .

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

Find the common denominator.

Write $12$ as a fraction with denominator $1$ .

$f (3) = - \frac{9}{2} + \frac{12}{1} - 3$

Multiply $\frac{12}{1}$ by $\frac{2}{2}$ .

$f (3) = - \frac{9}{2} + \frac{12}{1} \cdot \frac{2}{2} - 3$

Multiply $\frac{12}{1}$ by $\frac{2}{2}$ .

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

Write $- 3$ as a fraction with denominator $1$ .

$f (3) = - \frac{9}{2} + \frac{12 \cdot 2}{2} + \frac{- 3}{1}$

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

$f (3) = - \frac{9}{2} + \frac{12 \cdot 2}{2} + \frac{- 3}{1} \cdot \frac{2}{2}$

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

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

Combine the numerators over the common denominator.

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

Simplify each term.

Multiply $12$ by $2$ .

$f (3) = \frac{- 9 + 24 - 3 \cdot 2}{2}$

Multiply $- 3$ by $2$ .

$f (3) = \frac{- 9 + 24 - 6}{2}$

Simplify by adding and subtracting.

Add $- 9$ and $24$ .

$f (3) = \frac{15 - 6}{2}$

Subtract $6$ from $15$ .

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

The final answer is $\frac{9}{2}$ .

$\frac{9}{2}$

The $y$ value at $x = 3$ is $\frac{9}{2}$ .

$y = \frac{9}{2}$

Replace the variable $x$ with $6$ in the expression.

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

Simplify the result.

Simplify each term.

Raise $6$ to the power of $2$ .

$f (6) = - \frac{36}{2} + 4 (6) - 3$

Divide $36$ by $2$ .

$f (6) = - 1 \cdot 18 + 4 (6) - 3$

Multiply $- 1$ by $18$ .

$f (6) = - 18 + 4 (6) - 3$

Multiply $4$ by $6$ .

$f (6) = - 18 + 24 - 3$

Simplify by adding and subtracting.

Add $- 18$ and $24$ .

$f (6) = 6 - 3$

Subtract $3$ from $6$ .

$f (6) = 3$

The final answer is $3$ .

$3$

The $y$ value at $x = 6$ is $3$ .

$y = 3$

Replace the variable $x$ with $5$ in the expression.

$f (5) = - \frac{{(5)}^{2}}{2} + 4 (5) - 3$

Simplify the result.

Simplify each term.

Raise $5$ to the power of $2$ .

$f (5) = - \frac{25}{2} + 4 (5) - 3$

Multiply $4$ by $5$ .

$f (5) = - \frac{25}{2} + 20 - 3$

Find the common denominator.

Write $20$ as a fraction with denominator $1$ .

$f (5) = - \frac{25}{2} + \frac{20}{1} - 3$

Multiply $\frac{20}{1}$ by $\frac{2}{2}$ .

$f (5) = - \frac{25}{2} + \frac{20}{1} \cdot \frac{2}{2} - 3$

Multiply $\frac{20}{1}$ by $\frac{2}{2}$ .

$f (5) = - \frac{25}{2} + \frac{20 \cdot 2}{2} - 3$

Write $- 3$ as a fraction with denominator $1$ .

$f (5) = - \frac{25}{2} + \frac{20 \cdot 2}{2} + \frac{- 3}{1}$

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

$f (5) = - \frac{25}{2} + \frac{20 \cdot 2}{2} + \frac{- 3}{1} \cdot \frac{2}{2}$

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

$f (5) = - \frac{25}{2} + \frac{20 \cdot 2}{2} + \frac{- 3 \cdot 2}{2}$

Combine the numerators over the common denominator.

$f (5) = \frac{- 25 + 20 \cdot 2 - 3 \cdot 2}{2}$

Simplify each term.

Multiply $20$ by $2$ .

$f (5) = \frac{- 25 + 40 - 3 \cdot 2}{2}$

Multiply $- 3$ by $2$ .

$f (5) = \frac{- 25 + 40 - 6}{2}$

Simplify by adding and subtracting.

Add $- 25$ and $40$ .

$f (5) = \frac{15 - 6}{2}$

Subtract $6$ from $15$ .

$f (5) = \frac{9}{2}$

The final answer is $\frac{9}{2}$ .

$\frac{9}{2}$

The $y$ value at $x = 5$ is $\frac{9}{2}$ .

$y = \frac{9}{2}$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 2 & 3 \\ 3 & \frac{9}{2} \\ 4 & 5 \\ 5 & \frac{9}{2} \\ 6 & 3 \end{array}$

Step 4

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(4, 5)$

Focus: $(4, \frac{9}{2})$

Axis of Symmetry: $x = 4$

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

$\begin{array}{c} x & y \\ 2 & 3 \\ 3 & \frac{9}{2} \\ 4 & 5 \\ 5 & \frac{9}{2} \\ 6 & 3 \end{array}$

Step 5

Graph y=-1/2x^2+4x-3

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