Let's recall the graph of the function shown below.
We can stretch the graph of the sine function in the -direction by multiplying the right-hand-side of the function by a number. This type of transformation is a vertical stretch.
For instance, the graph stretches the sine curve by a stretch factor of
Notice that the minimum and maximum values have changed to and respectively.
Alternatively, multiplying by a positive number less than has the effect of shrinking the graph. For instance, the graph shrinks the original sine curve by a stretch factor of
Again, the minimum and maximum values have changed to and respectively.
Now recall the graph of the function shown below.
We can apply a vertical stretch to the cosine function in the same way. For example, the graph stretches the cosine curve by a stretch factor of
Again, the minimum and maximum values have changed to and respectively.
What is the equation of the graph shown above?
Let's add the graph of to the plot.
To plot the given curve, we follow these steps:
Start with the graph of
Then, stretch it parallel to the -axis by a stretch factor of
Therefore, the given curve is the graph of
What is the equation of the graph shown above?
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a
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$y=\dfrac 15\cos x$ |
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b
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$y=6\cos x$ |
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c
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$y=\dfrac 16\cos x$ |
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d
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$y=5\cos x$ |
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e
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$y=4\cos x$ |
What is the equation of the graph shown above?
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a
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$y=\dfrac{3}{2}\cos{x}$ |
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b
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$y=2\sin{x}$ |
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c
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$y=\pi\sin(x)$ |
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d
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$y=\dfrac{3}{2}\sin{x}$ |
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e
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$y=2\cos{x}$ |
What is the minimum value of
Method 1
To plot the given curve, we follow these steps:
Start with the graph of
Then, stretch it parallel to the -axis by a stretch factor of
The resulting graph is shown below.
From the graph, we see that the minimum value is
Method 2
We know that that has a minimum value of
Multiplying both sides of the above inequality by gives:
Therefore, the minimum value of is
What is the maximum value of $y=\dfrac{3}{4}\sin{x}?$
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a
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$\dfrac{3}{2}$ |
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b
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$\dfrac{1}{2}$ |
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c
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$\dfrac{3}{5}$ |
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d
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$\dfrac{3}{4}$ |
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e
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$\dfrac{2}{3}$ |
What is the minimum value of $y=2\cos{x}?$
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a
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$-1$ |
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b
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$0$ |
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c
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$1$ |
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d
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$-2$ |
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e
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$-\dfrac{1}{2}$ |
The amplitude of a sine function or cosine function represents half the distance between the maximum and minimum values of the function.
We can calculate the amplitude using the minimum and maximum values and
For example, the amplitude of the function shown above, where and , is
What is the amplitude of
To plot the given curve, we follow these steps:
Start with the graph of
Then, stretch it parallel to the -axis by a stretch factor of
The resulting graph is shown below.
From the graph, we see that the maximum value is and the minimum value is Therefore, the amplitude is
What is the amplitude of $f(x)=\dfrac{3}{2}\sin{x}?$
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a
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$\dfrac{3}{4}$ |
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b
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$\dfrac{1}{2}$ |
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c
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$\dfrac{3}{2}$ |
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d
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$3$ |
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e
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$5$ |
What is the amplitude of $f(x)=4\cos{x}?$
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a
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$8$ |
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b
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$6$ |
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c
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$5$ |
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d
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$2$ |
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e
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$4$ |
Now, let's recall the graph of shown below.
We can also apply vertical stretches to the graph of in the same way. The function is the result of stretching the tangent by a stretch factor of
Likewise, the function is the result of stretching the tangent by a stretch factor of
Note that we can't define an amplitude for the tangent function, as it has no minimum or maximum value.
Given that which of the following shows the curve
To plot the given curve, we follow these steps:
Start with the graph of
Then, stretch it parallel to the -axis by a stretch factor of
The resulting graph is shown below.
Therefore, the correct answer is I.
Given that $f(x) = \dfrac{1}{3}\tan x,$ which of the following shows the curve $y=f(x)?$
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a
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b
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c
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d
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e
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Given that $f(x) = 3\tan x,$ which of the following shows the curve $y=f(x)?$
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a
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b
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c
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d
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e
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