Trigonometric Graphs Test

1.

What is the shape of a cosine graph like?

A.

Banana
B.

Dracula Teeth
C.

Mountain
D.

Orange

Correct Answer
B. Dracula Teeth

Explanation
The shape of a cosine graph is like Dracula Teeth. This is because a cosine graph oscillates between a maximum and minimum value, creating a pattern that resembles the shape of Dracula's teeth. The graph starts at the maximum value, then decreases to the minimum value, before increasing again to the maximum value. This pattern repeats itself in a periodic manner, just like the shape of Dracula's teeth.

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2.

What is the amplitude of the graph y = 2sin(x+pi/4) ?

Correct Answer
2

Explanation
The amplitude of a sine function is the distance from the midline to the maximum or minimum point on the graph. In the given equation, the coefficient of the sine function is 2, which represents the amplitude. Therefore, the amplitude of the graph y = 2sin(x+pi/4) is 2.

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3.

What is the period of the graph y = tan 5x. (Answer in degrees. But, you do not have to write the word "degrees" in the answer.)

Correct Answer
180

Explanation
The period of the graph y = tan 5x is 180 degrees. This means that the graph repeats itself every 180 degrees. The tangent function has a period of π radians or 180 degrees, so when the coefficient of x is 5, it means that the graph will complete one full cycle every 180 degrees.

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4.

What is the amplitude of the graph y = tan 2x?

Correct Answer
Infinite

Explanation
The graph of y = tan 2x is a periodic function that oscillates between positive and negative infinity as x approaches certain values. This occurs because the tangent function has vertical asymptotes at these values, causing the amplitude to be infinite. As x increases or decreases, the graph of y = tan 2x repeats this pattern indefinitely, resulting in an infinite amplitude.

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5.

What is the period of the graph y = tan 3x? (Answer in degrees. But, you do not have to write the word "degrees" in the answer)

Correct Answer
60

Explanation
The period of the graph y = tan 3x is 60 degrees. This means that the graph repeats itself every 60 degrees. In other words, after every 60 degrees, the graph will have the same shape and pattern.

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6.

Sketch the graph of y = 2 sinx -1 and state its rangeLeave your answer as "a

Correct Answer
-2

Explanation
The graph of y = 2 sinx -1 is a sinusoidal function with an amplitude of 2 and a vertical shift downward by 1 unit. The range of this function is all real numbers between -2 and 0, inclusive. This is because the maximum value of the function is 1 and the minimum value is -3, so the range is [-2, 0].

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7.

Sketch the graph of y = 3 - cos2x and state its range. Leave your answer as "a

Correct Answer
2

8.

Sketch the graphs y=4sin 2x and y = 2cosx -1 for 0<360 degrees on the same diagram. How many solutions are there in this interval for 4 sin2x + 1 = 2 cosx?

Correct Answer
4

Explanation
The equation given is 4 sin2x + 1 = 2 cosx. To find the number of solutions in the interval 0

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9.

Sketch the graphs of y = 2|cos2x| and y = 1+sinx for 0<360 degrees.What is the amplitude of the graph y =1+sinx?

Correct Answer
2

Explanation
The amplitude of a sine function is the distance between the maximum and minimum values of the graph. In the equation y = 1 + sinx, the coefficient of sinx is 1, which means the amplitude is 1. Therefore, the amplitude of the graph y = 1 + sinx is 1.

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10.

From the previous question, find the number of distinct values of x in this interval for which: 2cos2x = sinx+1

Correct Answer
7

Explanation
The equation 2cos2x = sinx+1 can be rewritten as 2(1-2sin^2x) = sinx+1. Simplifying further, we get 4sin^2x + sinx - 1 = 0. This is a quadratic equation in sinx. Solving for sinx using the quadratic formula, we find two distinct roots. Since sinx is periodic with a period of 2π, we can add 2π to each root to find additional solutions. Therefore, there are a total of 4 distinct values of sinx in the interval [0, 2π]. However, we need to find the number of distinct values of x. Since sinx is a periodic function, each distinct value of sinx corresponds to two distinct values of x in the interval [0, 2π]. Therefore, there are a total of 4 * 2 = 8 distinct values of x. However, one of the roots, sinx = -1/2, does not lie in the interval [0, 2π]. Therefore, the correct answer is 7.

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