Graph f(x)=3/2(x-4)^2+3

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

Step 1

Find the properties of the given parabola.

Isolate $y$ to the left side of the equation.

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

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

Reorder terms.

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

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

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

$h = 4$

$k = 3$

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

Opens Up

Find the vertex $(h, k)$ .

$(4, 3)$

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.

$(4, \frac{19}{6})$

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.

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

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

Direction: Opens Up

Vertex: $(4, 3)$

Focus: $(4, \frac{19}{6})$

Axis of Symmetry: $x = 4$

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

Direction: Opens Up

Vertex: $(4, 3)$

Focus: $(4, \frac{19}{6})$

Axis of Symmetry: $x = 4$

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

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 $2$ in the expression.

$f (2) = \frac{3 {(2)}^{2}}{2} - 12 \cdot 2 + 27$

Simplify the result.

Simplify each term.

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

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

$f (2) = \frac{2 (3 \cdot 2)}{2} - 12 \cdot 2 + 27$

Cancel the common factors.

Factor $2$ out of $2$ .

$f (2) = \frac{2 (3 \cdot 2)}{2 (1)} - 12 \cdot 2 + 27$

Cancel the common factor.

$f (2) = \frac{2 (3 \cdot 2)}{2 \cdot 1} - 12 \cdot 2 + 27$

Rewrite the expression.

$f (2) = \frac{3 \cdot 2}{1} - 12 \cdot 2 + 27$

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

$f (2) = 3 \cdot 2 - 12 \cdot 2 + 27$

Multiply $3$ by $2$ .

$f (2) = 6 - 12 \cdot 2 + 27$

Multiply $- 12$ by $2$ .

$f (2) = 6 - 24 + 27$

Simplify by adding and subtracting.

Subtract $24$ from $6$ .

$f (2) = - 18 + 27$

Add $- 18$ and $27$ .

$f (2) = 9$

The final answer is $9$ .

$9$

The $y$ value at $x = 2$ is $9$ .

$y = 9$

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

$f (3) = \frac{3 {(3)}^{2}}{2} - 12 \cdot 3 + 27$

Simplify the result.

Simplify each term.

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

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

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

$f (3) = \frac{3 {(3)}^{2}}{2} - 12 \cdot 3 + 27$

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

$f (3) = \frac{3^{1 + 2}}{2} - 12 \cdot 3 + 27$

Add $1$ and $2$ .

$f (3) = \frac{3^{3}}{2} - 12 \cdot 3 + 27$

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

$f (3) = \frac{27}{2} - 12 \cdot 3 + 27$

Multiply $- 12$ by $3$ .

$f (3) = \frac{27}{2} - 36 + 27$

Find the common denominator.

Write $- 36$ as a fraction with denominator $1$ .

$f (3) = \frac{27}{2} + \frac{- 36}{1} + 27$

Multiply $\frac{- 36}{1}$ by $\frac{2}{2}$ .

$f (3) = \frac{27}{2} + \frac{- 36}{1} \cdot \frac{2}{2} + 27$

Multiply $\frac{- 36}{1}$ by $\frac{2}{2}$ .

$f (3) = \frac{27}{2} + \frac{- 36 \cdot 2}{2} + 27$

Write $27$ as a fraction with denominator $1$ .

$f (3) = \frac{27}{2} + \frac{- 36 \cdot 2}{2} + \frac{27}{1}$

Multiply $\frac{27}{1}$ by $\frac{2}{2}$ .

$f (3) = \frac{27}{2} + \frac{- 36 \cdot 2}{2} + \frac{27}{1} \cdot \frac{2}{2}$

Multiply $\frac{27}{1}$ by $\frac{2}{2}$ .

$f (3) = \frac{27}{2} + \frac{- 36 \cdot 2}{2} + \frac{27 \cdot 2}{2}$

Combine the numerators over the common denominator.

$f (3) = \frac{27 - 36 \cdot 2 + 27 \cdot 2}{2}$

Simplify each term.

Multiply $- 36$ by $2$ .

$f (3) = \frac{27 - 72 + 27 \cdot 2}{2}$

Multiply $27$ by $2$ .

$f (3) = \frac{27 - 72 + 54}{2}$

Simplify by adding and subtracting.

Subtract $72$ from $27$ .

$f (3) = \frac{- 45 + 54}{2}$

Add $- 45$ and $54$ .

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

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.

$f (6) = \frac{3 {(6)}^{2}}{2} - 12 \cdot 6 + 27$

Simplify the result.

Simplify each term.

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

$f (6) = \frac{3 \cdot 36}{2} - 12 \cdot 6 + 27$

Multiply $3$ by $36$ .

$f (6) = \frac{108}{2} - 12 \cdot 6 + 27$

Divide $108$ by $2$ .

$f (6) = 54 - 12 \cdot 6 + 27$

Multiply $- 12$ by $6$ .

$f (6) = 54 - 72 + 27$

Simplify by adding and subtracting.

Subtract $72$ from $54$ .

$f (6) = - 18 + 27$

Add $- 18$ and $27$ .

$f (6) = 9$

The final answer is $9$ .

$9$

The $y$ value at $x = 6$ is $9$ .

$y = 9$

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

$f (5) = \frac{3 {(5)}^{2}}{2} - 12 \cdot 5 + 27$

Simplify the result.

Simplify each term.

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

$f (5) = \frac{3 \cdot 25}{2} - 12 \cdot 5 + 27$

Multiply $3$ by $25$ .

$f (5) = \frac{75}{2} - 12 \cdot 5 + 27$

Multiply $- 12$ by $5$ .

$f (5) = \frac{75}{2} - 60 + 27$

Find the common denominator.

Write $- 60$ as a fraction with denominator $1$ .

$f (5) = \frac{75}{2} + \frac{- 60}{1} + 27$

Multiply $\frac{- 60}{1}$ by $\frac{2}{2}$ .

$f (5) = \frac{75}{2} + \frac{- 60}{1} \cdot \frac{2}{2} + 27$

Multiply $\frac{- 60}{1}$ by $\frac{2}{2}$ .

$f (5) = \frac{75}{2} + \frac{- 60 \cdot 2}{2} + 27$

Write $27$ as a fraction with denominator $1$ .

$f (5) = \frac{75}{2} + \frac{- 60 \cdot 2}{2} + \frac{27}{1}$

Multiply $\frac{27}{1}$ by $\frac{2}{2}$ .

$f (5) = \frac{75}{2} + \frac{- 60 \cdot 2}{2} + \frac{27}{1} \cdot \frac{2}{2}$

Multiply $\frac{27}{1}$ by $\frac{2}{2}$ .

$f (5) = \frac{75}{2} + \frac{- 60 \cdot 2}{2} + \frac{27 \cdot 2}{2}$

Combine the numerators over the common denominator.

$f (5) = \frac{75 - 60 \cdot 2 + 27 \cdot 2}{2}$

Simplify each term.

Multiply $- 60$ by $2$ .

$f (5) = \frac{75 - 120 + 27 \cdot 2}{2}$

Multiply $27$ by $2$ .

$f (5) = \frac{75 - 120 + 54}{2}$

Simplify by adding and subtracting.

Subtract $120$ from $75$ .

$f (5) = \frac{- 45 + 54}{2}$

Add $- 45$ and $54$ .

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

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.

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

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(4, 3)$

Focus: $(4, \frac{19}{6})$

Axis of Symmetry: $x = 4$

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

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

Step 4

Graph f(x)=3/2(x-4)^2+3

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