How does the graph change if the rational function f(x)= 2x – 3 3x + 1 is changed to f(x)= 4x – 3 3x + 1 ?

A. The vertical and horizontal asymptotes stay the same.
Incorrect. The vertical asymptote stays the same but the horizontal asymptote changes when the variable “a” is changed.

B. Only the vertical asymptote changes.
Incorrect. The vertical asymptote does not change when the variable “a” is changed.

C. Only the horizontal asymptote changes.
Correct! The vertical asymptote stays the same and the horizontal asymptote changes when the variable “a” is changed.

D. The x-intercept stays the same.
Incorrect. The numerator controls the x-intercept, and the numerator of the function changes.

A. x ≠ -4
Incorrect. A rational function cannot exist when the denominator equals 0. (x + 4) is not in the denominator.

B. x ≠ 6
Incorrect. A rational function cannot exist where the denominator equals zero so 6 is one answer but there is also another answer.

C. x ≠ -1
Incorrect. A rational function cannot exist where the denominator equals zero so -1 is one answer but there is another answer also.

D. x ≠ -1 or 6
Correct! A rational function cannot exist when x equals -1 or 6.

How does the graph change if the rational function f(x) = (x – 3)(x + 6) (x + 1)(x – 5) is changed to f(x) = (x – 6)(x + 6) (x + 1)(x – 5)?

A. The asymptotes stay the same but the graph reverses direction where x ≥ -1.
Correct! The vertical asymptotes stay the same and the graph changes shape where x ≥ -1.

B. The asymptotes stay the same but the graph reverses direction where x ≤ -1.
Incorrect. The vertical asymptotes stay the same but the location where the graph changes shape is incorrect.

C. The vertical asymptote changes from 3 to 5 and the graph stays the same.
Incorrect. The vertical asymptotes do not change and the graph does not stay the same.

D. The vertical asymptote changes from 3 to 5 and the graph changes shape where x ≥ -1.
Incorrect. The vertical asymptotes do not change but the graph does reverse direction where x ≥ -1.

If the graph of a rational function, g(x), has a vertical asymptote at x = -3, a hole in the graph at x = 1 2 and a horizontal asymptote at y = 1, which of the following binomials did not influence the asymptotic behavior of this rational function?

A. 2x − 1
Incorrect. This binomial would be the common factor in both the numerator and denominator of the function that cause the hole in the graph.

B. x + 3
Incorrect. This binomial represents the factor that caused the vertical asymptote.

C. x − 3
Correct! Although this factor could be part of the rational function, it did not influence either the vertical or horizontal asymptote, or cause a hole in the graph.

D. x - 1 2
Incorrect. This is binomial is an equivalent expression of the factor that would be common in both the numerator and denominator of the function that caused a hole in the graph.

Which statement or statements below best describes the domain of the graph of the function
f(x) = x(x - 3)(x  + 2)x² - 4??

I.   The range does not include {2} because there is a horizontal asymptote at y = 2.
II.  The domain does not include {2} because there is a vertical asymptote at x = 2.
III. The domain does not include {-2} because there is a hole in the graph at x = -2.
IV. The domain includes {0 and 3} because there are x-intercepts at (0, 0) and (3, 0).

A. I and II only
Incorrect. Graph the function to look for asymptotes.

B. I, II, and IV only
Incorrect. If you substitute the value x = -2 into the function, what number do you have in the denominator?

C. II and III only
Incorrect. Although both statements are accurate, look for possible horizontal asymptotes and the x-intercepts.

D. II, III, and IV only
Correct!