Graphing Functions - Pre-Calculus

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Question

What is the amplitude of the following function?

$y = -24\sin(x-\pi)+14$

Answer

When you think of a trigonometric function of the form y=Asin(Bx+C)+D, the amplitude is represented by A, or the coefficient in front of the sine function. While this number is -24, we always represent amplitude as a positive number, by taking the absolute value of it. Therefore, the amplitude of this function is 24.

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Question

Select the answer choice that correctly matches each function to its period.

Answer

The following matches the correct period with its corresponding trig function:

$\sin x \rightarrow 2\pi$

$\cos x \rightarrow 2\pi$

$\tan x \rightarrow \pi$

$\cot x \rightarrow \pi$

$\sec x \rightarrow 2\pi$

$\csc x \rightarrow 2\pi$

In other words, sin x, cos x, sec x, and csc x all repeat themselves every $2\pi$ units. However, tan x and cot x repeat themselves more frequently, every $\pi$ units.

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Question

What is the period of this sine graph?

Answer

The graph has 3 waves between 0 and $2\pi$ , meaning that the length of each of the waves is $2\pi$ divided by 3, or $\frac{2}{3}\pi$ .

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Question

Write the equation for a cosine graph with a maximum at $\left( \frac{ \pi }{3} , 3 \right)$ and a minimum at $\left(\frac{ 4 \pi }{3} , -3 \right)$ .

Answer

In order to write this equation, it is helpful to sketch a graph:

The dotted line is at $x = \frac{ \pi }{3}$ , where the maximum occurs and therefore where the graph starts. This means that the graph is shifted to the right $\frac{ \pi }{3}$ .

The distance from the maximum to the minimum is half the entire wavelength. Here it is $\frac{ 4 \pi }{3} - \frac{ \pi }{3} = \frac{ 3 \pi }{3} = \pi$ .

Since half the wavelength is $\pi$ , that means the full wavelength is $2 \pi$ so the frequency is just 1.

The amplitude is 3 because the graph goes symmetrically from -3 to 3.

The equation will be in the form $y = A \cdot \cos (f (x - h)) + k$ where A is the amplitude, f is the frequency, h is the horizontal shift, and k is the vertical shift.

This equation is

$y = 3 \cos (x - \frac{ \pi } {3 })$ .

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Question

Find the phase shift of .

Answer

In the formula,

.

$\frac{C}{B}$ represents the phase shift.

Plugging in what we know gives us:

$\frac{C}{B}=\frac{-4}{2}$ .

Simplified, the phase is then .

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Question

Which equation would produce this sine graph?

Answer

The graph has an amplitude of 2 but has been shifted down 1:

In terms of the equation, this puts a 2 in front of sin, and -1 at the end.

This makes it easier to see that the graph starts \[is at 0\] where $x=\frac{\pi}{4}$ .

The phase shift is $\frac{\pi}{4}$ to the right, or $x-\frac{\pi}{4}$ .

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Question

Which of the following equations could represent a cosine function with amplitude 3, period $\frac{\pi}{2}$ , and a phase shift of $\frac{\pi}{4}$ ?

Answer

The form of the equation will be $y=A\cos(Bx+C)$

First, think about all possible values of A that could give you an amplitude of 3. Either A = -3 or A = 3 could each produce amplitude = 3. Be sure to look for answer choices that satisfy either of these.

Secondly, we know that the period is $\frac{\pi}{2}$ . Normally we know what B is and need to find the period, but this is the other way around. We can still use the same equation and solve:

$\frac{2\pi}{B}=\frac{\pi}{2}$ . You can cross multiply to solve and get B = 4.

Finally, we need to find a value of C that satisfies

$-\frac{C}{B}=\frac{\pi}{4}$ . Cross multiply to get:

$-4C=B\pi$ .

Next, plug in B= 4 to solve for C:

$-4C = 4\pi$

$C=-\pi$

Putting this all together, the equation could either be:

$y = 3\cos(4x-\pi)$ or $y = -3\cos(4x-\pi)$

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Question

State the amplitude, period, phase shift, and vertical shift of the function $y=-7\sin(6x+\pi)-4$

Answer

A common way to make sense of all of the transformations that can happen to a trigonometric function is the following. For the equations y = A sin(Bx + C) + D,

amplitude is |A|

period is 2 $\pi$ /|B|

phase shift is - C/B

vertical shift is D

In our equation, A=-7, B=6, C= $\pi$ , and D=-4. Next, apply the above numbers to find amplitude, period, phase shift, and vertical shift.

To find amplitude, look at the coefficient in front of the sine function. A=-7, so our amplitude is equal to 7.

The period is 2 $\pi$ /B, and in this case B=6. Therefore the period of this function is equal to 2 $\pi$ /6 or $\pi$ /3.

To find the phase shift, take -C/B, or - $\pi$ /6. Another way to find this same value is to set the inside of the parenthesis equal to 0, then solve for x.
6x+ $\pi$ =0
6x=- $\pi$
x=- $\pi$ /6
Either way, our phase shift is equal to - $\pi$ /6.

The vertical shift is equal to D, which is -4.

y=-7\sin(6x+\pi)-4

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Question

State the amplitude, period, phase shift, and vertical shift of the function $y=-\cos(x-\pi)+3$

Answer

A common way to make sense of all of the transformations that can happen to a trigonometric function is the following. For the equations y = A sin(Bx + C) + D,

amplitude is |A|

period is 2 $\pi$ /|B|

phase shift is - C/B

vertical shift is D

In our equation, A=-1, B=1, C=- $\pi$ , and D=3. Next, apply the above numbers to find amplitude, period, phase shift, and vertical shift.

To find amplitude, look at the coefficient in front of the sine function. A=-1, so our amplitude is equal to 1.

The period is 2 $\pi$ /B, and in this case B=1. Therefore the period of this function is equal to 2 $\pi$ .

To find the phase shift, take -C/B, or $\pi$ . Another way to find this same value is to set the inside of the parenthesis equal to 0, then solve for x.
x- $\pi$ =0
x= $\pi$
Either way, our phase shift is equal to $\pi$ .

The vertical shift is equal to D, which is 3.

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Question

State the amplitude, period, phase shift, and vertical shift of the function $y=\sin(2x-3)+2$

Answer

A common way to make sense of all of the transformations that can happen to a trigonometric function is the following. For the equations y = A sin(Bx + C) + D,

amplitude is |A|

period is 2 $\pi$ /|B|

phase shift is - C/B

vertical shift is D

In our equation, A=1, B=2, C=-3, and D=2. Next, apply the above numbers to find amplitude, period, phase shift, and vertical shift.

To find amplitude, look at the coefficient in front of the sine function. A=1, so our amplitude is equal to 1.

The period is 2 $\pi$ /B, and in this case B=2. Therefore the period of this function is equal to $\pi$ .

To find the phase shift, take -C/B, or 3/2. Another way to find this same value is to set the inside of the parenthesis equal to 0, then solve for x.
2x-3=0
2x=3
x=3/2
Either way, our phase shift is equal to 3/2.

The vertical shift is equal to D, which is 2.

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Question

Information about the Hardy Weinberg equation: http://www.nature.com/scitable/definition/hardy-weinberg-equation-299

The Hardy Weinberg equation is a very important concept in population genetics. Suppose we have two "alleles" for a specific trait (like eye color, gender, etc..) The proportion for which allele is present is given by and

Then by Hardy-Weinberg:

, where both and are non-negative.

Which statement discribes the graph that would appropriately represents the above relation?

Answer

The equation can be reduced by taking the square root of both sides.

As a simple test, all other values when substituted into the original equation fail. However, works. Therefore is our answer.

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Question

Which of the trigonometric functions is represented by this graph?

Answer

This graph is the graph of y = csc x. The domain of this function is all real numbers except $n\cdot \pi$ where n is any integer. In other words, there are vertical asymptotes at all multiples of $\pi$ . The range of this function is $y\leq -1, y\geq 1$ . The period of this function is $2\pi$ .

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Question

Which of the following functions is represented by this graph?

Answer

This graph is the graph of y = cot x. The domain of this function is all real numbers except $n\cdot \pi$ where n is any integer. In other words, there are vertical asymptotes at all multiples of $\pi$ . The range of this function is $\left ( -\infty, \infty \right )$ . The period of this function is $\pi$ .

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Question

Which of the following functions is represented by this graph?

Answer

This graph is the graph of y = sec x. The domain of this function is all real numbers except $\frac{\pi}{2} + n\pi$ where n is any integer. In other words, there are vertical asymptotes at $\frac{\pi}{2}$ , $\frac{3\pi}2{}$ , $\frac{5\pi}{2}$ , and so on. The range of this function is $y\leq -1, y\geq 1$ . The period of this function is $2\pi.$

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Question

Which of the following functions is represented by this graph?

Answer

This graph is the graph of y = tan x. The domain of this function is all real numbers except $\frac{\pi}{2}+n\pi$ where n is any integer. In other words, there are vertical asymptotes at $\frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}$ , and so on. The range of this function is $\left ( -\infty, \infty \right )$ . The period of this function is $\pi$ .

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Question

True or false: If you translate a sine curve 90o to the left along the x-axis, you will have a cosine curve.

Answer

This is true! Notice the similarity of the shape between the graphs, but that they intercept the x-axis at different spots, and their peaks and valleys are at different spots.

y=sin(x), passes through the point (0,0)

y=cos(x), passes through the point (0,1)

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Question

Which of the following functions has a y-intercept of ?

Answer

The y-intercept of a function is found by substituting . When we do this to each, we can determine the y-intercept. Don't forget your unit circle!

Thus, the function with a y-intercept of is .

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Question

Which of the following functions is represented by this graph?

Answer

This graph is the graph of y = cos x. The domain of this function is all real numbers. The range of this function is $-1\leq y\leq 1$ . The period of this function is $2\pi$ .

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Question

Which of the following functions is represented by this graph?

Answer

This graph is the graph of y = sin x. The domain of this function is all real numbers. The range of this function is $-1\leq y\leq 1$ . The period of this function is $2\pi$ .

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Question

True or false: If you translate a secant function $\pi$ units to the left along the x-axis, you will have a cosecant curve.

Answer

This is false. While the graphs of secant and cosecant functions are related, in order to turn a secant function into a cosecant function, you'd need to translate the original graph $\frac{\pi}{2}$ units to the right to obtain a cosecant graph.

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