Practice Set 5
10 Rational Challenge Questions
This challenge set focuses on asymptote reasoning, restrictions, extraneous solutions, and real-world rational models.
Practice Questions
EOC Challenge Set: Rational Functions
Use this set when you want a stronger mix of graph reasoning, modeling, and equation solving that feels closer to tougher EOC-style rational work.
Question 1
For f(x) = (ax + 2) / (x - 1), if the horizontal asymptote is y = 3, what is a?
Desmos Move: When the degrees match, the horizontal asymptote is the ratio of leading coefficients, so compare that ratio to 3.
Question 2
Which expression has a hole at x = 4 and no vertical asymptote?
Desmos Move: A hole with no vertical asymptote happens when the entire denominator factor cancels and nothing is left in the denominator.
Question 3
Solve 5 / (x + 2) = x / (x + 2) for all valid x-values.
Desmos Move: Since the denominators are the same, compare the numerators for allowed x-values and remember x ≠ -2.
Question 4
Which statement best explains why x = 2 is an extraneous solution to (x² - 4) / (x - 2) = 0?
Desmos Move: Factor the numerator and look at the original denominator before deciding whether the zero is allowed.
Question 5
Which function has vertical asymptote x = -3 and horizontal asymptote y = -2?
Desmos Move: Use the denominator to place the vertical asymptote and the outside constant to place the horizontal asymptote.
Question 6
A car trip costs $150 total, split equally among x riders, plus a fixed $8 parking fee per rider. Which expression models the cost per rider?
Desmos Move: One part is shared and changes with x. The other part is fixed for each rider and stays outside the fraction.
Question 7
For f(x) = (x + 1) / (x - 2), what x-value makes f(x) = -2?
Desmos Move: Set the expression equal to -2, solve, and check that the result does not make the denominator 0.
Question 8
A rational function approaches the same y-value on both the far left and far right. Which feature are you identifying?
Desmos Move: Watch what happens to the graph as x becomes very large positive and very large negative.
Question 9
Why can a simplified rational expression and the original expression have the same graph shape but not the exact same graph?
Desmos Move: Compare an original expression and its simplified rule on the same screen. A canceled factor often leaves one missing point.
Question 10
For y = (x - 4) / (x + 1), what is the x-value when y = 2?
Desmos Move: Set the rational function equal to 2 and solve. Then check that the result does not make the denominator 0.