Calculus Examples

$x^{2} - 5 x + 3$

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

Complete the square for $x^{2} - 5 x + 3$ .

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Step 1.1.1.1

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

$a = 1$

$b = - 5$

$c = 3$

Step 1.1.1.2

Consider the vertex form of a parabola.

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

Step 1.1.1.3

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

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Step 1.1.1.3.1

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

$d = \frac{- 5}{2 \cdot 1}$

Step 1.1.1.3.2

Simplify the right side.

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Step 1.1.1.3.2.1

Multiply $2$ by $1$ .

$d = \frac{- 5}{2}$

Step 1.1.1.3.2.2

Move the negative in front of the fraction.

$d = - \frac{5}{2}$

Step 1.1.1.4

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

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Step 1.1.1.4.1

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

$e = 3 - \frac{{(- 5)}^{2}}{4 \cdot 1}$

Step 1.1.1.4.2

Simplify the right side.

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Step 1.1.1.4.2.1

Simplify each term.

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Step 1.1.1.4.2.1.1

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

$e = 3 - \frac{25}{4 \cdot 1}$

Step 1.1.1.4.2.1.2

Multiply $4$ by $1$ .

$e = 3 - \frac{25}{4}$

Step 1.1.1.4.2.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$e = 3 \cdot \frac{4}{4} - \frac{25}{4}$

Step 1.1.1.4.2.3

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

$e = \frac{3 \cdot 4}{4} - \frac{25}{4}$

Step 1.1.1.4.2.4

Combine the numerators over the common denominator.

$e = \frac{3 \cdot 4 - 25}{4}$

Step 1.1.1.4.2.5

Simplify the numerator.

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Step 1.1.1.4.2.5.1

Multiply $3$ by $4$ .

$e = \frac{12 - 25}{4}$

Step 1.1.1.4.2.5.2

Subtract $25$ from $12$ .

$e = \frac{- 13}{4}$

Step 1.1.1.4.2.6

Move the negative in front of the fraction.

$e = - \frac{13}{4}$

Step 1.1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x - \frac{5}{2})}^{2} - \frac{13}{4}$ .

${(x - \frac{5}{2})}^{2} - \frac{13}{4}$

Step 1.1.2

Set $y$ equal to the new right side.

$y = {(x - \frac{5}{2})}^{2} - \frac{13}{4}$

Step 1.2

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

$a = 1$

$h = \frac{5}{2}$

$k = - \frac{13}{4}$

Step 1.3

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

Opens Up

Step 1.4

Find the vertex $(h, k)$ .

$(\frac{5}{2}, - \frac{13}{4})$

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.

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

Step 1.5.3

Cancel the common factor of $1$ .

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Step 1.5.3.1

Cancel the common factor.

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

Step 1.5.3.2

Rewrite the expression.

$\frac{1}{4}$

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.

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

Step 1.7

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

$x = \frac{5}{2}$

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.

$y = - \frac{7}{2}$

Step 1.9

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

Direction: Opens Up

Vertex: $(\frac{5}{2}, - \frac{13}{4})$

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

Axis of Symmetry: $x = \frac{5}{2}$

Directrix: $y = - \frac{7}{2}$

Direction: Opens Up

Vertex: $(\frac{5}{2}, - \frac{13}{4})$

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

Axis of Symmetry: $x = \frac{5}{2}$

Directrix: $y = - \frac{7}{2}$

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 $1$ in the expression.

$f (1) = {(1)}^{2} - 5 \cdot 1 + 3$

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

One to any power is one.

$f (1) = 1 - 5 \cdot 1 + 3$

Step 2.2.1.2

Multiply $- 5$ by $1$ .

$f (1) = 1 - 5 + 3$

Step 2.2.2

Simplify by adding and subtracting.

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Step 2.2.2.1

Subtract $5$ from $1$ .

$f (1) = - 4 + 3$

Step 2.2.2.2

Add $- 4$ and $3$ .

$f (1) = - 1$

Step 2.2.3

The final answer is $- 1$ .

$- 1$

Step 2.3

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

$y = - 1$

Step 2.4

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

$f (0) = {(0)}^{2} - 5 \cdot 0 + 3$

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

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

$f (0) = 0 - 5 \cdot 0 + 3$

Step 2.5.1.2

Multiply $- 5$ by $0$ .

$f (0) = 0 + 0 + 3$

Step 2.5.2

Simplify by adding numbers.

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Step 2.5.2.1

Add $0$ and $0$ .

$f (0) = 0 + 3$

Step 2.5.2.2

Add $0$ and $3$ .

$f (0) = 3$

Step 2.5.3

The final answer is $3$ .

$3$

Step 2.6

The $y$ value at $x = 0$ is $3$ .

$y = 3$

Step 2.7

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

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

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 $2$ .

$f (3) = 9 - 5 \cdot 3 + 3$

Step 2.8.1.2

Multiply $- 5$ by $3$ .

$f (3) = 9 - 15 + 3$

Step 2.8.2

Simplify by adding and subtracting.

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Step 2.8.2.1

Subtract $15$ from $9$ .

$f (3) = - 6 + 3$

Step 2.8.2.2

Add $- 6$ and $3$ .

$f (3) = - 3$

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

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

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

Step 2.11

Simplify the result.

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Step 2.11.1

Simplify each term.

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Step 2.11.1.1

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

$f (4) = 16 - 5 \cdot 4 + 3$

Step 2.11.1.2

Multiply $- 5$ by $4$ .

$f (4) = 16 - 20 + 3$

Step 2.11.2

Simplify by adding and subtracting.

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Step 2.11.2.1

Subtract $20$ from $16$ .

$f (4) = - 4 + 3$

Step 2.11.2.2

Add $- 4$ and $3$ .

$f (4) = - 1$

Step 2.11.3

The final answer is $- 1$ .

$- 1$

Step 2.12

The $y$ value at $x = 4$ is $- 1$ .

$y = - 1$

Step 2.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 0 & 3 \\ 1 & - 1 \\ \frac{5}{2} & - \frac{13}{4} \\ 3 & - 3 \\ 4 & - 1 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(\frac{5}{2}, - \frac{13}{4})$

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

Axis of Symmetry: $x = \frac{5}{2}$

Directrix: $y = - \frac{7}{2}$

$\begin{array}{c} x & y \\ 0 & 3 \\ 1 & - 1 \\ \frac{5}{2} & - \frac{13}{4} \\ 3 & - 3 \\ 4 & - 1 \end{array}$

Step 4

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