11.7.A.1Multiple Choice
For r(x) = (x² + 4)/(x − 2), the value r(6) = 10. Which statement best interprets r(6) = 10?
Select A, B, C, or D to check your answer.
Directions: Work each problem on your own paper first. For multiple choice, select A, B, C, or D. You will see right away whether you are correct. If you miss, you will see the most likely reason, and you can try again. For short response, type your answer, then compare it with the model solution. Questions 1–2 cover 1.7.A.1; 3–5 cover 1.7.A.2; 6–10 cover 1.7.A.3; 11–14 cover 1.7.A.4; 15–17 cover 1.7.A.5; 18–19 cover 1.7.A.6; question 20 covers them all.
For r(x) = (x² + 4)/(x − 2), the value r(6) = 10. Which statement best interprets r(6) = 10?
Select A, B, C, or D to check your answer.
For r(x) = (2x + 3)/(x² + 1), the value r(100) ≈ 0.0203. Which statement best interprets this value?
Select A, B, C, or D to check your answer.
For r(x) = (5x⁴ − x)/(2x⁶ + 3), which polynomial dominates for inputs of large magnitude, and what is the quotient of the leading terms?
Select A, B, C, or D to check your answer.
Let r(x) = (6x³ + 5)/(2x³ − x). (a) Find r(10), r(100), and r(1,000). (b) Write the quotient of the leading terms. (c) Divide r(100) by your answer to (b) and explain what this tells you about the quotient of the leading terms.
What is the quotient of the leading terms of r(x) = 2(x + 3)²(x − 1) / [(x − 4)(x + 2)]?
Select A, B, C, or D to check your answer.
Which statement describes the end behavior of r(x) = (3x³ − x)/(x² + 4)?
Select A, B, C, or D to check your answer.
Which rational function has a slant asymptote parallel to the line y = 3x?
Select A, B, C, or D to check your answer.
Let r(x) = (−2x³ + x)/(x² + 3). (a) Write the quotient of the leading terms. (b) Write lim x→∞ r(x) and lim x→−∞ r(x). (c) Does the graph have a slant asymptote? If so, parallel to which line? (d) Check your answer to (b) by finding r(10) and r(100) and r(−10) and r(−100).
Which of the following graphs could be the graph of r(x) = (x² + 1)/(x − 1)?
Select A, B, C, or D to check your answer.
Which statement about the ends of r(x) = (x⁴ + 1)/(x² − 3) is true?
Select A, B, C, or D to check your answer.
What is the horizontal asymptote of the graph of r(x) = (9x² − 4)/(3x² + x + 5)?
Select A, B, C, or D to check your answer.
What is the horizontal asymptote of the graph of r(x) = (5 − 4x³)/(2x³ + x)?
Select A, B, C, or D to check your answer.
Let r(x) = (7x² − 3x)/(2x² + 5). (a) Find the horizontal asymptote and write both limit statements. (b) Find r(10), r(100), r(1,000), and r(−100). (c) The graph of r crosses the line y = 7/2 exactly once. Find the x-value where it happens, and explain why this does not contradict the horizontal asymptote.
The graph of a rational function r is shown, with its horizontal asymptote as a dashed line. Which of the following could be r(x)?
Select A, B, C, or D to check your answer.
Which of the following has a horizontal asymptote at y = 0?
Select A, B, C, or D to check your answer.
Which statement is true about r(x) = (2x + 1)/(x² + 3)?
Select A, B, C, or D to check your answer.
Let r(x) = (5x + 2)/(x² + 6). (a) Which polynomial dominates, and what is the quotient of the leading terms? (b) Write the horizontal asymptote and both limits. (c) Find r(10), r(100), and r(1,000). (d) The numerator 5x + 2 equals 0 at x = −0.4, so r(−0.4) = 0. Does that contradict the horizontal asymptote y = 0? Explain.
For a rational function r, lim x→∞ r(x) = 4. Which statement is the best interpretation?
Select A, B, C, or D to check your answer.
The table gives values of r(x) = (10x − 3)/(4x + 1). The horizontal asymptote is y = 2.5. What is the first input in the table at which r(x) is within 0.01 of 2.5?
| x | 10 | 100 | 1,000 | 10,000 |
| r(x) | 2.3659 | 2.4863 | 2.4986 | 2.49986 |
Select A, B, C, or D to check your answer.
Consider g(x) = (12x² − 5x)/(3x² + 7), h(x) = (x² + 3)/(x⁴ + 1), and k(x) = (5x³ + 2)/(x² + 4). (a) For each function, say which polynomial dominates (or that neither does) and write the quotient of the leading terms. (b) Give the end behavior of each using limit notation, and say whether it has a horizontal or a slant asymptote. (c) For g, find g(100) and g(−100) and use them to support your answer.