The graphs of rational functions m and n have the same vertical asymptotes, and both have a single x-intercept at x = 5. Which equation could represent function n?

A.
n(x) = 5m(x)
B.
n(x) = m(x) + 5
C.
n(x) = m(x − 5)
D.
n(x) = m(x + 5)

A

Step-by-step explanation:

We are given two rational functions m(x) and n(x) that have the same vertical asymptotes both with a single x-intercept at x = 5.

The correct choice will be A.

Recall the transformations of functions.

B represents m(x) being shifted up 5 units. If the function is shifted up, the vertical asymptotes will be the same, but the x-intercept will change.

C represents m(x) being shifted 5 units to the right. This changes both the x-intercept and the vertical asymptotes.

Likewise, D represents m(x) being shifted 5 units to the left. Again, this will change both the x-intercept and the vertical asymptotes.

Therefore, the only choice left is A. It represents a vertical stretch by a factor of 5. This preserves the x-intercepts and the vertical asymptotes. Consider the function:

If n(x)=5m(x), we can see that:

So, the x-interceps and vertical asymptotes are preserved.

δabc ≅ δdcb by sas because

db ≅ ac . . given (side)

bc ≅ bc . . reflexive property (side)

∠dbc ≅ ∠ acb . . alternate interior angles where transversal bc crosses parallel lines bd and ac (included angle)

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