Calculus Examples

$y = 17 x - 6 x^{2}$

Step 1

Find the properties of the given parabola.

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Step 1.1

Rewrite the equation in vertex form.

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Step 1.1.1

Reorder $17 x$ and $- 6 x^{2}$ .

$y = - 6 x^{2} + 17 x$

Step 1.1.2

Complete the square for $- 6 x^{2} + 17 x$ .

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Step 1.1.2.1

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

$a = - 6$

$b = 17$

$c = 0$

Step 1.1.2.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 1.1.2.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 1.1.2.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{17}{2 \cdot - 6}$

Step 1.1.2.3.2

Simplify the right side.

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Step 1.1.2.3.2.1

Multiply $2$ by $- 6$ .

$d = \frac{17}{- 12}$

Step 1.1.2.3.2.2

Move the negative in front of the fraction.

$d = - \frac{17}{12}$

Step 1.1.2.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 1.1.2.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{17^{2}}{4 \cdot - 6}$

Step 1.1.2.4.2

Simplify the right side.

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Step 1.1.2.4.2.1

Simplify each term.

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Step 1.1.2.4.2.1.1

Raise $17$ to the power of $2$ .

$e = 0 - \frac{289}{4 \cdot - 6}$

Step 1.1.2.4.2.1.2

Multiply $4$ by $- 6$ .

$e = 0 - \frac{289}{- 24}$

Step 1.1.2.4.2.1.3

Move the negative in front of the fraction.

$e = 0 - - \frac{289}{24}$

Step 1.1.2.4.2.1.4

Multiply $- - \frac{289}{24}$ .

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Step 1.1.2.4.2.1.4.1

Multiply $- 1$ by $- 1$ .

$e = 0 + 1 (\frac{289}{24})$

Step 1.1.2.4.2.1.4.2

Multiply $\frac{289}{24}$ by $1$ .

$e = 0 + \frac{289}{24}$

Step 1.1.2.4.2.2

Add $0$ and $\frac{289}{24}$ .

$e = \frac{289}{24}$

Step 1.1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 6 {(x - \frac{17}{12})}^{2} + \frac{289}{24}$ .

$- 6 {(x - \frac{17}{12})}^{2} + \frac{289}{24}$

Step 1.1.3

Set $y$ equal to the new right side.

$y = - 6 {(x - \frac{17}{12})}^{2} + \frac{289}{24}$

Step 1.2

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

$a = - 6$

$h = \frac{17}{12}$

$k = \frac{289}{24}$

Step 1.3

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

Opens Down

Step 1.4

Find the vertex $(h, k)$ .

$(\frac{17}{12}, \frac{289}{24})$

Step 1.5

Find $p$ , the distance from the vertex to the focus.

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Step 1.5.1

Find the distance from the vertex to a focus of the parabola by using the following formula.

$\frac{1}{4 a}$

Step 1.5.2

Substitute the value of $a$ into the formula.

$\frac{1}{4 \cdot - 6}$

Step 1.5.3

Simplify.

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Step 1.5.3.1

Multiply $4$ by $- 6$ .

$\frac{1}{- 24}$

Step 1.5.3.2

Move the negative in front of the fraction.

$- \frac{1}{24}$

Step 1.6

Find the focus.

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Step 1.6.1

The focus of a parabola can be found by adding $p$ to the y-coordinate $k$ if the parabola opens up or down.

$(h, k + p)$

Step 1.6.2

Substitute the known values of $h$ , $p$ , and $k$ into the formula and simplify.

$(\frac{17}{12}, 12)$

Step 1.7

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

$x = \frac{17}{12}$

Step 1.8

Find the directrix.

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Step 1.8.1

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.

$y = k - p$

Step 1.8.2

Substitute the known values of $p$ and $k$ into the formula and simplify.

$y = \frac{145}{12}$

Step 1.9

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

Direction: Opens Down

Vertex: $(\frac{17}{12}, \frac{289}{24})$

Focus: $(\frac{17}{12}, 12)$

Axis of Symmetry: $x = \frac{17}{12}$

Directrix: $y = \frac{145}{12}$

Direction: Opens Down

Vertex: $(\frac{17}{12}, \frac{289}{24})$

Focus: $(\frac{17}{12}, 12)$

Axis of Symmetry: $x = \frac{17}{12}$

Directrix: $y = \frac{145}{12}$

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.

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Step 2.1

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

$f (0) = - 6 {(0)}^{2} + 17 (0)$

Step 2.2

Simplify the result.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

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

$f (0) = - 6 \cdot 0 + 17 (0)$

Step 2.2.1.2

Multiply $- 6$ by $0$ .

$f (0) = 0 + 17 (0)$

Step 2.2.1.3

Multiply $17$ by $0$ .

$f (0) = 0 + 0$

Step 2.2.2

Add $0$ and $0$ .

$f (0) = 0$

Step 2.2.3

The final answer is $0$ .

$0$

Step 2.3

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

$y = 0$

Step 2.4

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

$f (2) = - 6 {(2)}^{2} + 17 (2)$

Step 2.5

Simplify the result.

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Step 2.5.1

Simplify each term.

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Step 2.5.1.1

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

$f (2) = - 6 \cdot 4 + 17 (2)$

Step 2.5.1.2

Multiply $- 6$ by $4$ .

$f (2) = - 24 + 17 (2)$

Step 2.5.1.3

Multiply $17$ by $2$ .

$f (2) = - 24 + 34$

Step 2.5.2

Add $- 24$ and $34$ .

$f (2) = 10$

Step 2.5.3

The final answer is $10$ .

$10$

Step 2.6

The $y$ value at $x = 2$ is $10$ .

$y = 10$

Step 2.7

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

$f (3) = - 6 {(3)}^{2} + 17 (3)$

Step 2.8

Simplify the result.

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Step 2.8.1

Simplify each term.

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Step 2.8.1.1

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

$f (3) = - 6 \cdot 9 + 17 (3)$

Step 2.8.1.2

Multiply $- 6$ by $9$ .

$f (3) = - 54 + 17 (3)$

Step 2.8.1.3

Multiply $17$ by $3$ .

$f (3) = - 54 + 51$

Step 2.8.2

Add $- 54$ and $51$ .

$f (3) = - 3$

Step 2.8.3

The final answer is $- 3$ .

$- 3$

Step 2.9

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

$y = - 3$

Step 2.10

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 0 & 0 \\ \frac{17}{12} & \frac{289}{24} \\ 2 & 10 \\ 3 & - 3 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(\frac{17}{12}, \frac{289}{24})$

Focus: $(\frac{17}{12}, 12)$

Axis of Symmetry: $x = \frac{17}{12}$

Directrix: $y = \frac{145}{12}$

$\begin{array}{c} x & y \\ 0 & 0 \\ \frac{17}{12} & \frac{289}{24} \\ 2 & 10 \\ 3 & - 3 \end{array}$

Step 4