Trigonometric Graphs (all six) - Pre-Calculus

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

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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)

Which of the following functions is represented by this graph?

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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 $\pi$ .

Which of the trigonometric functions is represented by this graph?

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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 $2\pi$ .

Which of the following functions is represented by this graph?

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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 $\pi$ .

Which of the following functions is represented by this graph?

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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 $2\pi.$

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

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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 .

Which of the following functions is represented by this graph?

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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 $2\pi$ .

Which of the following functions is represented by this graph?

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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 $2\pi$ .

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

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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.

Where does the tangent function intercept the x-axis?

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Because the tangent function is periodic, it intercepts the x-axis in infinitely many places. We can see several of these in the graph below:

In the photo, we can see that the function is intercepting the x-axis at $x=-2\pi, -\pi, 0, \pi.$ Generalizing this, we can say that the tangent function intercepts the x-axis for $x=n\pi$ for all values of n such that n is an integer.

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

Tap to see back →

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)

Which of the following functions is represented by this graph?

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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 $\pi$ .

Which of the trigonometric functions is represented by this graph?

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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 $2\pi$ .

Which of the following functions is represented by this graph?

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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 $\pi$ .

Which of the following functions is represented by this graph?

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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 $2\pi.$

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

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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 .

Which of the following functions is represented by this graph?

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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 $2\pi$ .

Which of the following functions is represented by this graph?

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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 $2\pi$ .

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

Tap to see back →

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.

Where does the tangent function intercept the x-axis?

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Because the tangent function is periodic, it intercepts the x-axis in infinitely many places. We can see several of these in the graph below:

In the photo, we can see that the function is intercepting the x-axis at $x=-2\pi, -\pi, 0, \pi.$ Generalizing this, we can say that the tangent function intercepts the x-axis for $x=n\pi$ for all values of n such that n is an integer.