Trigonometry Examples

$319 \sqrt{x} + 11 y + 187 = 0$

Step 1

Solve for $y$ .

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Step 1.1

Move all terms not containing $y$ to the right side of the equation.

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Step 1.1.1

Subtract $319 \sqrt{x}$ from both sides of the equation.

$11 y + 187 = - 319 \sqrt{x}$

Step 1.1.2

Subtract $187$ from both sides of the equation.

$11 y = - 319 \sqrt{x} - 187$

Step 1.2

Divide each term in $11 y = - 319 \sqrt{x} - 187$ by $11$ and simplify.

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Step 1.2.1

Divide each term in $11 y = - 319 \sqrt{x} - 187$ by $11$ .

$\frac{11 y}{11} = \frac{- 319 \sqrt{x}}{11} + \frac{- 187}{11}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $11$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{11 y}{11} = \frac{- 319 \sqrt{x}}{11} + \frac{- 187}{11}$

Step 1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 319 \sqrt{x}}{11} + \frac{- 187}{11}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Cancel the common factor of $- 319$ and $11$ .

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Step 1.2.3.1.1.1

Factor $11$ out of $- 319 \sqrt{x}$ .

$y = \frac{11 (- 29 \sqrt{x})}{11} + \frac{- 187}{11}$

Step 1.2.3.1.1.2

Cancel the common factors.

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Step 1.2.3.1.1.2.1

Factor $11$ out of $11$ .

$y = \frac{11 (- 29 \sqrt{x})}{11 (1)} + \frac{- 187}{11}$

Step 1.2.3.1.1.2.2

Cancel the common factor.

$y = \frac{11 (- 29 \sqrt{x})}{11 \cdot 1} + \frac{- 187}{11}$

Step 1.2.3.1.1.2.3

Rewrite the expression.

$y = \frac{- 29 \sqrt{x}}{1} + \frac{- 187}{11}$

Step 1.2.3.1.1.2.4

Divide $- 29 \sqrt{x}$ by $1$ .

$y = - 29 \sqrt{x} + \frac{- 187}{11}$

Step 1.2.3.1.2

Divide $- 187$ by $11$ .

$y = - 29 \sqrt{x} - 17$

Step 2

Find the domain for $319 \sqrt{x} + 11 y + 187 = 0$ so that a list of $x$ values can be picked to find a list of points, which will help graphing the radical.

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Step 2.1

Set the radicand in $\sqrt{x}$ greater than or equal to $0$ to find where the expression is defined.

$x \geq 0$

Step 2.2

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${x | x \geq 0}$

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${x | x \geq 0}$

Step 3

To find the radical expression end point, substitute the $x$ value $0$ , which is the least value in the domain, into $f (x) = - 29 \sqrt{x} - 17$ .

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Step 3.1

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

$f (0) = - 29 \sqrt{0} - 17$

Step 3.2

Simplify the result.

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Step 3.2.1

Remove parentheses.

$f (0) = - 29 \sqrt{0} - 17$

Step 3.2.2

Simplify each term.

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Step 3.2.2.1

Rewrite $0$ as $0^{2}$ .

$f (0) = - 29 \sqrt{0^{2}} - 17$

Step 3.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$f (0) = - 29 \cdot 0 - 17$

Step 3.2.2.3

Multiply $- 29$ by $0$ .

$f (0) = 0 - 17$

Step 3.2.3

Subtract $17$ from $0$ .

$f (0) = - 17$

Step 3.2.4

The final answer is $- 17$ .

$- 17$

Step 4

The radical expression end point is $(0, - 17)$ .

$(0, - 17)$

Step 5

Select a few $x$ values from the domain. It would be more useful to select the values so that they are next to the $x$ value of the radical expression end point.

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Step 5.1

Substitute the $x$ value $1$ into $f (x) = - 29 \sqrt{x} - 17$ . In this case, the point is $(1, - 46)$ .

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Step 5.1.1

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

$f (1) = - 29 \sqrt{1} - 17$

Step 5.1.2

Simplify the result.

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Step 5.1.2.1

Remove parentheses.

$f (1) = - 29 \sqrt{1} - 17$

Step 5.1.2.2

Simplify each term.

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Step 5.1.2.2.1

Any root of $1$ is $1$ .

$f (1) = - 29 \cdot 1 - 17$

Step 5.1.2.2.2

Multiply $- 29$ by $1$ .

$f (1) = - 29 - 17$

Step 5.1.2.3

Subtract $17$ from $- 29$ .

$f (1) = - 46$

Step 5.1.2.4

The final answer is $- 46$ .

$y = - 46$

Step 5.2

Substitute the $x$ value $2$ into $f (x) = - 29 \sqrt{x} - 17$ . In this case, the point is $(2, - 29 \sqrt{2} - 17)$ .

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Step 5.2.1

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

$f (2) = - 29 \sqrt{2} - 17$

Step 5.2.2

Simplify the result.

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Step 5.2.2.1

Remove parentheses.

$f (2) = - 29 \sqrt{2} - 17$

Step 5.2.2.2

The final answer is $- 29 \sqrt{2} - 17$ .

$y = - 29 \sqrt{2} - 17$

Step 5.3

The square root can be graphed using the points around the vertex $(0, - 17), (1, - 46), (2, - 58.01)$

$\begin{array}{c} x & y \\ 0 & - 17 \\ 1 & - 46 \\ 2 & - 58.01 \end{array}$

Step 6