In this discussion, we learn how to Graph the Tangent Function with Transformations.
Concepts will be discussed first, then an example.
Begin with the parent graph of y=tan(x).
The parent graph has:
an x-intercept at 0
a vertical asymptote at pi/2
a vertical asymptote at -pi/2
Draw the parent graph, beginning at the x-intercept, then up and out to the asymptote on the right.
Go back to the x-intercept and draw down and out to the asymptote on the left.
This pattern continues to the right and to the left.
Now, the trig function is of the form:
y=atan[b(x-c)]+d.
Tangent does not have an amplitude, which is what “a” represents for sine and cosine.
However, for tangent and cotangent, “a” is used to find the midpoints between the x-intercepts and asymptotes.
We won’t worry about the midpoint in this discussion.
That will be discussed in another blog.
b is the horizontal stretch/compress
c indicates how many units to shift right/left
d indicates how many units to shift up/down
To find the exact placements on the graph, compute the period.
period =pi/b
Next, compute the x-scale.
The x-scale is the increment value that is added along x-axis.
x-scale = period/2
Now begin at c (which is the x-intercept) and then add and subtract increments of of the x-scale.
Draw the new graph, just like the parent graph, but with the new transformations.
Now, an example.
Given a function, y=tan(x-pi/4).
Rewrite the function in the form
atan[b(x-c)]+d,
Thus, we have,
y=1tan[1(x-pi/4)]+0.
We see that
a=1 b=1 c=pi/4 d=0
For this example, we mirror the parent graph, but shift all x-values c units to the right (c is pi/4).
To find the exact placements on the graph, compute the period.
period =pi/b = pi/1 = pi
Next, compute the x-scale.
x-scale = period/2 = pi/2
Now begin at c (pi/4) and add and subtract increments of the x-scale (pi/2).
Here are the values:
pi/4
pi/4 + pi/2 = 3pi/4
pi/4 – pi/2= -pi/4
So now we have
an x-intercept at pi/4
a vertical asymptote at 3pi/4
a vertical asymptote at -pi/4
Draw the graph, just like the parent graph, but with the new transformations.
If you have any questions, or if this reading helps, let me know in the comments!
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