Trigonometry Examples

$h (x) \sqrt{x^{2}} - 5 x + 4$

Step 1

Find the $y$ value at $x = - 1$ .

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Step 1.1

Replace the variable $x$ with $- 1$ in the expression.

$f (- 1) = \frac{5}{- 1} - \frac{4}{{(- 1)}^{2}}$

Step 1.2

Simplify the result.

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Divide $5$ by $- 1$ .

$f (- 1) = - 5 - \frac{4}{{(- 1)}^{2}}$

Step 1.2.1.2

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

$f (- 1) = - 5 - \frac{4}{1}$

Step 1.2.1.3

Divide $4$ by $1$ .

$f (- 1) = - 5 - 1 \cdot 4$

Step 1.2.1.4

Multiply $- 1$ by $4$ .

$f (- 1) = - 5 - 4$

Step 1.2.2

Subtract $4$ from $- 5$ .

$f (- 1) = - 9$

Step 1.2.3

The final answer is $- 9$ .

$- 9$

Step 1.3

The $y$ value at $x = - 1$ is $- 9$ .

$y = - 9$

Step 2

Find the $y$ value at $x = - 3$ .

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Step 2.1

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

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

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

Move the negative in front of the fraction.

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

Step 2.2.1.2

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

$f (- 3) = - \frac{5}{3} - \frac{4}{9}$

Step 2.2.2

To write $- \frac{5}{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (- 3) = - \frac{5}{3} \cdot \frac{3}{3} - \frac{4}{9}$

Step 2.2.3

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.2.3.1

Multiply $\frac{5}{3}$ by $\frac{3}{3}$ .

$f (- 3) = - \frac{5 \cdot 3}{3 \cdot 3} - \frac{4}{9}$

Step 2.2.3.2

Multiply $3$ by $3$ .

$f (- 3) = - \frac{5 \cdot 3}{9} - \frac{4}{9}$

Step 2.2.4

Combine the numerators over the common denominator.

$f (- 3) = \frac{- 5 \cdot 3 - 4}{9}$

Step 2.2.5

Simplify the numerator.

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Step 2.2.5.1

Multiply $- 5$ by $3$ .

$f (- 3) = \frac{- 15 - 4}{9}$

Step 2.2.5.2

Subtract $4$ from $- 15$ .

$f (- 3) = \frac{- 19}{9}$

Step 2.2.6

Move the negative in front of the fraction.

$f (- 3) = - \frac{19}{9}$

Step 2.2.7

The final answer is $- \frac{19}{9}$ .

$- \frac{19}{9}$

Step 2.3

The $y$ value at $x = - 3$ is $- \frac{19}{9}$ .

$y = - \frac{19}{9}$

Step 3

Find the $y$ value at $x = - 2$ .

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Step 3.1

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

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

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Move the negative in front of the fraction.

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

Step 3.2.1.2

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

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

Step 3.2.1.3

Cancel the common factor of $4$ .

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Step 3.2.1.3.1

Cancel the common factor.

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

Step 3.2.1.3.2

Rewrite the expression.

$f (- 2) = - \frac{5}{2} - 1 \cdot 1$

Step 3.2.1.4

Multiply $- 1$ by $1$ .

$f (- 2) = - \frac{5}{2} - 1$

Step 3.2.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (- 2) = - \frac{5}{2} - 1 \cdot \frac{2}{2}$

Step 3.2.3

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

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

Step 3.2.4

Combine the numerators over the common denominator.

$f (- 2) = \frac{- 5 - 1 \cdot 2}{2}$

Step 3.2.5

Simplify the numerator.

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Step 3.2.5.1

Multiply $- 1$ by $2$ .

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

Step 3.2.5.2

Subtract $2$ from $- 5$ .

$f (- 2) = \frac{- 7}{2}$

Step 3.2.6

Move the negative in front of the fraction.

$f (- 2) = - \frac{7}{2}$

Step 3.2.7

The final answer is $- \frac{7}{2}$ .

$- \frac{7}{2}$

Step 3.3

The $y$ value at $x = - 2$ is $- \frac{7}{2}$ .

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

Step 4

Find the $y$ value at $x = 1$ .

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Step 4.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = \frac{5}{1} - \frac{4}{{(1)}^{2}}$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Divide $5$ by $1$ .

$f (1) = 5 - \frac{4}{{(1)}^{2}}$

Step 4.2.1.2

One to any power is one.

$f (1) = 5 - \frac{4}{1}$

Step 4.2.1.3

Divide $4$ by $1$ .

$f (1) = 5 - 1 \cdot 4$

Step 4.2.1.4

Multiply $- 1$ by $4$ .

$f (1) = 5 - 4$

Step 4.2.2

Subtract $4$ from $5$ .

$f (1) = 1$

Step 4.2.3

The final answer is $1$ .

$1$

Step 4.3

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

$y = 1$

Step 5

Find the $y$ value at $x = 2$ .

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Step 5.1

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

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

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

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

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

Step 5.2.1.2

Cancel the common factor of $4$ .

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Step 5.2.1.2.1

Cancel the common factor.

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

Step 5.2.1.2.2

Rewrite the expression.

$f (2) = \frac{5}{2} - 1 \cdot 1$

Step 5.2.1.3

Multiply $- 1$ by $1$ .

$f (2) = \frac{5}{2} - 1$

Step 5.2.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (2) = \frac{5}{2} - 1 \cdot \frac{2}{2}$

Step 5.2.3

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

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

Step 5.2.4

Combine the numerators over the common denominator.

$f (2) = \frac{5 - 1 \cdot 2}{2}$

Step 5.2.5

Simplify the numerator.

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Step 5.2.5.1

Multiply $- 1$ by $2$ .

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

Step 5.2.5.2

Subtract $2$ from $5$ .

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

Step 5.2.6

The final answer is $\frac{3}{2}$ .

$\frac{3}{2}$

Step 5.3

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

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

Step 6

List the points to graph.

$(- 1, - 9), (- 3, - 2.\overline{1}), (- 2, - 3.5), (1, 1), (4, 1), (2, 1.5)$

Step 7

Select a few points to graph.

$\begin{array}{c} x & y \\ - 3 & - 2.111 \\ - 2 & - 3.5 \\ - 1 & - 9 \\ 1 & 1 \\ 2 & 1.5 \\ 4 & 1 \end{array}$

Step 8