Video Transcript
Consider the graph below. Which function does the plot in the
graph represent? 𝑦 equals two sin of 𝑥. 𝑦 equals two cos of 𝑥. 𝑦 equals one-half sin of 𝑥. 𝑦 equals one-half cos of 𝑥. Or 𝑦 equals one-half tan of
𝑥.
In this example, we have been given
a graph and need to decide which of the functions represents the graph. Because the plot is a periodic
curve, we are looking at either a sine or cosine graph. Although tangent is also a periodic
function, it does not form a single curve across the set of real numbers as we see
here. The remaining options include
various sine and cosine functions with a constant multiple in front of each. So we should begin by reviewing the
properties of both the 𝑦 equals sin of 𝑥 and the 𝑦 equals cos of 𝑥
functions.
First, we recall the 𝑦-intercepts
of sine and cosine. This will be the value of the
function where 𝑥 equals zero. We know that sin of zero equals
zero and cos of zero equals one. This means the sine graph has a
single 𝑦-intercept at zero, zero and the cosine graph has a single 𝑦-intercept at
zero, one. Both sine and cosine have a maximum
𝑦-value of one and a minimum 𝑦-value of negative one. Comparing this to the given graph,
we see that the maximum 𝑦-value is 0.5 and minimum 𝑦-value is at negative 0.5. So it seems the 𝑦-values have been
divided by two. We also notice that the given plot
has a 𝑦-intercept of zero, 0.5.
Next, we can determine which of the
given functions have this specific 𝑦-intercept by substituting 𝑥 equals zero. We’ll begin this process with
option (A). We know that sin of zero is zero,
so two times sin of zero also equals zero. The 𝑦-intercept of our graph is
not at zero, so we eliminate option (A).
Next, we’ll calculate the
𝑦-intercept of 𝑦 equals two cos of 𝑥. We know that the cos of zero is
one, so two times cos of zero equals two. So this function has a 𝑦-intercept
of two. So option (B) is not the correct
answer. We are looking for the function
with a 𝑦-intercept of 0.5. At this point, we are leaning
towards option (D), since the plot seems to match the cosine function if all the
𝑦-values were multiplied by one-half, whereas if we took half of the 𝑦-intercept
of sine, it would still be at zero.
To confirm we have the right match,
we will substitute zero for 𝑥 in both remaining functions. As expected, the 𝑦 equals one-half
cos of 𝑥 function has a 𝑦-intercept matching the graph, whereas the 𝑦 equals
one-half sin of 𝑥 function does not.
Therefore, by examining the maximum
and minimum values and calculating the 𝑦-intercepts, we were able to eliminate
options (A), (B), (C), and (E). And we were able to match the key
features of the given graph to the function 𝑦 equals one-half cos of 𝑥.
