D12-D14 Graphs of rational functions Quiz: AQA Further Maths, Unit 2

The 20 questions

1. What is the horizontal asymptote of y = (2x + 1)/(x - 3)?

   • y = 3
   • y = 2
   • y = 1
   • y = 2x

2. What is the vertical asymptote of y = (2x + 1)/(x - 3)?

   • x = -1/2
   • x = 3
   • x = 2
   • x = -3

3. Which condition gives an oblique asymptote for a rational function?

   • The numerator's degree is exactly one more than the denominator's
   • The function has no real zeros
   • The denominator has higher degree than the numerator
   • The numerator and denominator have the same degree

4. Where does y = (x^2 - 4)/(x + 1) meet the x-axis?

   • x = 2 and x = -2
   • x = 2 only
   • x = 4 and x = -4
   • x = -1 only

5. What is the y-intercept of y = (x + 2)/(x - 1)?

   • -2
   • 2
   • 1/2
   • -1/2

6. What is the oblique asymptote of y = (x^2 + 1)/(x - 1)?

   • y = x + 1
   • y = x
   • y = 2x + 1
   • y = x - 1

7. What is the maximum value of y = 3/(x^2 + 1), and where does it occur?

   • 3, at x = 1
   • There is no maximum
   • 1/3, at x = 0
   • 3, at x = 0

8. What is the range of y = x/(x^2 + 1)?

   • -1/2 <= y <= 1/2
   • 0 <= y <= 1, since the function is non-negative for all real x
   • -1 <= y <= 1, since the maximum of x over x^2 + 1 is attained at x = 1
   • All real y, since the rational function takes every real value

9. For y = (x + 1)/(x^2 + 2x + 5), what is the range?

   • -1/4 <= y <= 1/2
   • -1/2 <= y <= 1/2
   • 0 <= y <= 1/4
   • -1/4 <= y <= 1/4

10. Which are the stationary points of y = x + 1/x?

    • (1, 2) and (-1, 2)
    • (2, 2.5) and (-2, -2.5)
    • (0, 0) only
    • (1, 2) and (-1, -2)

11. What is the range of y = 2x/(x^2 + 4)?

    • -1 <= y <= 1
    • -1/2 <= y <= 1/2
    • -1/4 <= y <= 1/4
    • 0 <= y <= 1/2

12. What is the oblique asymptote of y = (x^2 - x + 1)/(x - 1)?

    • y = x + 1
    • y = x
    • y = x - 1
    • y = 2x

13. Which statement about y = (x - 2)/(x^2 - x - 2) is correct?

    • Vertical asymptotes at x = -1 and x = 2
    • A horizontal asymptote at y = 1 and no hole
    • A vertical asymptote at x = -1 and a hole at x = 2
    • A vertical asymptote at x = 2 only

14. Find k so that y = (kx + 1)/(2x - 3) has horizontal asymptote y = 4.

    • k = 2
    • k = 1/4
    • k = 4
    • k = 8

15. What is the horizontal asymptote of y = (3x^2 + 1)/(x^2 - 4)?

    • y = 3/4
    • y = 3
    • y = 1
    • y = -4

16. Which value of x makes y = (x^2 + 2)/(x + 2) undefined?

    • x = -1
    • x = 0
    • x = 2
    • x = -2

17. What is the x-intercept of y = (x - 3)/(x + 1)?

    • x = -3
    • x = -1
    • x = 1
    • x = 3

18. What is the vertical asymptote of y = 5/(x - 4) + 2?

    • x = 2
    • x = 4
    • y = 2
    • x = -4

19. For y = (x^2 - 1)/(x - 1), which statement is correct?

    • It has a vertical asymptote at x = 1 and y = x - 1
    • It has a horizontal asymptote y = 1
    • It simplifies to y = x + 1 with a hole at x = 1
    • It has no hole and no asymptote

20. What is the minimum value of y = (x^2 + 4)/x for x > 0?

    • 4, at x = 1
    • 2, at x = 2
    • 4, at x = 2
    • There is no minimum

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