Question 1Graphs of functions
The curve y = x2 − 4x + 7 and the line y = kx + 1 meet at exactly one point. What are the possible values of k?
TMUA Graphs of Functions — Practice Questions by Topic
These are SummitPapers original questions, not official past paper questions. Official TMUA questions sorted by topic are on TMUA past papers by topic.
- MM8Graphs of functions
- 72 questions46 on Paper 1 · 26 on Paper 2
- 9 free solutionsThe rest show the correct letter only
Covers: transformations of a graph, asymptotes and points of intersection, how many regions a graph divides the plane into, reading a graph of f or of its derivative.
What this topic tests
Graphs of functions here means transformations, asymptotes and intersections, how many regions a graph divides the plane into, and reading a graph of f or of its derivative.
How it is assessed
This is Section 1, on both papers in this set. Each question has five options. A paper is 20 questions in 75 minutes, with no calculator.
This note is written for SummitPapers. The official list of what can be examined is the specification, together with the Notes on Mathematics and, for Paper 2, the Notes on Logic and Proof.
Key methods
A transformation
y = f(x − a) shifts the graph of y = f(x) to the right by a. y = f(x) + a shifts it up by a. y = f(ax) stretches horizontally by factor 1/a. The order of two transformations is part of the answer.
The derivative’s graph
Where f is increasing, f' is positive. A stationary point of f is a root of f'. The gradient of f at a point is the height of the graph of f', not the height of f.
Common mistakes
Shifting the wrong way
f(x + 2) moves the graph left, not right. The sign inside the brackets is opposite to the direction of the shift.
Reading an asymptote as a root
A vertical asymptote is where the function is undefined. It is not a point where the graph crosses the axis.
Worked example
This question is also in the list below, with the solution folded. It is opened here so the method is on the page before the other questions.
The curve y = x2 − 4x + 7 and the line y = kx + 1 meet at exactly one point. What are the possible values of k?
Answer: C. k = −4 ± 2√6
Worked solution. Set x2 − 4x + 7 = kx + 1, so x2 − (4 + k)x + 6 = 0. Exactly one intersection means the discriminant is zero: (4 + k)2 − 24 = 0, so 4 + k = ±√24 = ±2√6. Therefore k = −4 ± 2√6.
Why the other options look right. A keeps only the positive square root of 24. B keeps only the negative square root. D replaces √24 by √6 and drops the factor 2. E solves 4 + k = ±2√6 as k = 4 ± 2√6.
Paper 1 questions
Paper 1 is Applications of Mathematical Knowledge. Calculators are not allowed.
Q1. The curve y = x2 − 4x + 7 and the line y = kx + 1 meet at exactly one point. What are the possible values of k?
Free · worked solution included
Answer and worked solution
Answer: C. k = −4 ± 2√6
Worked solution. Set x2 − 4x + 7 = kx + 1, so x2 − (4 + k)x + 6 = 0. Exactly one intersection means the discriminant is zero: (4 + k)2 − 24 = 0, so 4 + k = ±√24 = ±2√6. Therefore k = −4 ± 2√6.
Why the other options look right. A keeps only the positive square root of 24. B keeps only the negative square root. D replaces √24 by √6 and drops the factor 2. E solves 4 + k = ±2√6 as k = 4 ± 2√6.
Q2. The function f is defined for all real x. Symmetry about a point (p, 0) means that the graph has 180° rotational symmetry about that point. Which statements are true? I: The graph of y = f(x − 2) is symmetric about (2, 0) if and only if f is odd. II: The graph of y = f(x − 2) is symmetric about the line x = 2 if and only if f is even. III: The graph of y = f(x − 2) is symmetric about the y-axis if and only if f is even.
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Answer and worked solution
Answer: D. I and II only
Worked solution. Let g(x) = f(x − 2). Rotational symmetry of g about (2, 0) means g(2 + t) + g(2 − t) = 0, so f(t) + f(−t) = 0, and conversely. Thus I is true. Symmetry of g about x = 2 means g(2 + t) = g(2 − t), so f(t) = f(−t), and conversely. Thus II is true. For III, f(x) = x2 is even, but f(x − 2) = (x − 2)2 gives 1 at x = 1 and 9 at x = −1, so it is not symmetric about the y-axis. Thus III is false. Only I and II are true.
Why the other options look right. A thinks that translating the graph 2 units to the right destroys the symmetry of f, so it rejects all three statements. B drops the reflection in the line x = 2, which is exactly evenness of f. C drops the half-turn about (2, 0), which is exactly oddness of f. E includes III, but f(x) = x2 is a counterexample.
Q3. The graph of y = log base 10 of x is translated by 3 units in the positive y-direction. This translation is equivalent to a stretch of factor k parallel to the x-axis. What is the value of k?
Free · worked solution included
Answer and worked solution
Answer: A. 0.001
Worked solution. A translation by 3 in the positive y-direction gives y = log base 10 of x, plus 3, which is log base 10 of (1000x). A stretch of factor k parallel to the x-axis replaces x by x/k, giving y = log base 10 of (x/k). These agree when 1/k = 1000, so k = 0.001, which is option A.
Why the other options look right. E is 1000, from writing the stretch as log base 10 of (kx) instead of log base 10 of (x/k), so that kx = 1000x. B is −3, from solving log base 10 of x − log base 10 of k = log base 10 of x + 3 to get log base 10 of k = −3 and reporting log k instead of k. C is 1/3, from writing log base 10 of x + 3 as log base 10 of (3x), so x/k = 3x. D is the translation distance 3 itself, mistaken for the stretch factor.
Q4. What is the complete range of k for which the curves y = x³ − 3x and y = k − (x + 1)² meet at three distinct points, of which exactly two have positive x-coordinates?
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Answer
Answer: E. 22/27 < k < 1
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Q5. The curve y = 2x2 is translated by the vector (2, −3), then reflected in the x-axis, then stretched parallel to the x-axis with scale factor 2. What is the equation of the resulting curve?
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Answer
Answer: B. y = −(1/2)x2 + 4x − 5
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Q6. How many real solutions does the equation 4 cos x = √x have, where x is in radians?
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Answer
Answer: D. 5
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Q7. Find the complete set of values of m such that the line y = m(x + 1) and the curve y = 2√x have two distinct points of intersection with positive x-coordinate.
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Answer
Answer: C. 0 < m < 1
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Q8. What is the area of the region of points (x, y) with |x| + |y| ≤ 3 and |x| ≤ 2?
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Answer
Answer: C. 16
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Q9. Let floor(x) denote the greatest integer less than or equal to x. How many real solutions does the equation x − floor(x) = (x + 2)/6 have?
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Answer
Answer: C. 5
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Q10. A translation is applied to y = x3. Which of these graphs could be the result? I: y = x3 − 6x2 + 12x + 1. II: y = x3 − 6x2 + 9x + 2. III: y = x3 + 12x2 + 48x − 7.
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Answer
Answer: E. I and III only
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Q11. Let f(x) = (x − 1)2 (x − 3) and g(x) = −p(x − q)2, where p and q are real and p > 0. What is the complete set of values of q for which f(x) = g(x) has three distinct real solutions for every p > 0?
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Answer
Answer: A. 1 < q ≤ 3
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Q12. Let n be the number of points where the graphs of y = |x² − 1| and y = m(x + 1) meet, where m is a real constant. What is the smallest positive integer that n cannot equal?
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Answer
Answer: D. 4
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Q13. The graphs of y = x² + 4x + 5 and y = mx + 1 do not meet. What is the complete range of possible values of m?
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Answer
Answer: A. 0 < m < 8
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Q14. Let f(x) = x² − 10x. The curve y = f(x) is stretched parallel to the x-axis by scale factor s, where s > 0, and then translated c units in the positive y-direction. The minimum point of the new curve is (15, −20). Find s + c.
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Answer
Answer: B. 8
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Q15. The function f takes every real value from −1 to 4, and no other values. Find the difference between the maximum and minimum values of (f(x))² − 2f(x).
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Answer
Answer: D. 9
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Q16. The curve y = sin x is reflected in the line y = 2, and the image is then translated by π/6 in the positive x-direction. What is the equation of the final curve?
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Answer
Answer: D. y = 4 − sin(x − π/6)
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Q17. The graph of y = f(x) is translated 2 units in the positive x-direction. What is the equation of the resulting graph?
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Answer and worked solution
Answer: B. y = f(x − 2)
Worked solution. A translation of a units in the positive x-direction replaces x by x − a. Here a = 2, so the equation is y = f(x − 2). A point that was at x = 0 on the original graph is now at x = 2.
Why the other options look right. A replaces x by x + 2, which moves the graph 2 units in the negative x-direction. C moves the graph 2 units up. D moves it 2 units down. E stretches the graph vertically by a factor of 2 and does not translate it.
Q18. Let f(x) = 9x2 − 18x + 13. What are the coordinates of the minimum point of y = √(f(x − 2))?
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Answer
Answer: A. (3, 2)
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Q19. Let f(x) = x² + 6x + 13. What is the turning point of y = 3f(x + 2) − 1?
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Answer
Answer: A. (−5, 11)
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Q20. Let f(x) = x² − 6x + 11. What are the coordinates of the turning point of y = 2f(x − 1) − 4?
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Answer
Answer: A. (4, 0)
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Q21. Let f(x) = x² − 6x + 13. What are the coordinates of the turning point of y = 2f(x + 2) − 5?
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Answer
Answer: C. (1, 3)
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Q22. Let f(x) = x² + 4x + 7. After the graph is changed to y = 3f(x − 2) − 5, the turning point is
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Answer
Answer: A. (0, 4)
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Q23. Let f(x) = x² − 6x + 14. Find the turning point of the graph of y = f(2x) + 1.
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Answer
Answer: E. (3/2, 6)
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Q24. The function f is given by f(x) = x² − 6x + 13. What are the coordinates of the turning point of y = 3f(x − 1) − 2?
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Answer
Answer: C. (4, 10)
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Q25. The graphs of y = |x − 3| and y = 9 − 2|x| enclose a finite region. What is the area of that region?
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Answer
Answer: C. 21
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Q26. A quadratic graph has the line x = 3 as its line of symmetry and passes through (1, 1) and (4, −5). Which of the following is an equation of the graph?
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Answer
Answer: A. y = 2x2 − 12x + 11
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Q27. How many distinct real roots does the equation x4 − 4x3 + 5 = 0 have?
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Answer
Answer: C. 2
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Q28. A graph is transformed by a translation of 2 units in the positive x-direction followed by a reflection in the x-axis. For which of the following functions f is the graph of y = f(x) mapped onto itself? I: f(x) = sin(πx/2) II: f(x) = cos(πx) III: f(x) = sin(πx/2) cos(πx)
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Answer
Answer: B. I and III only
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Q29. Let f(x) = 2x2 − 8x + 14. What are the coordinates of the minimum point of y = √(f(x − 3))?
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Answer
Answer: A. (5, √6)
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Q30. The graph of a quadratic y = f(x) has its vertex at (2, 6) and passes through (−1, −3). Which expression is equal to f(x)?
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Answer
Answer: D. −x2 + 4x + 2
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Q31. The function f is given by f(x) = x2 − 3x + 4. What is the sum of the x- and y-coordinates of the minimum point of y = f(x + 2)?
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Answer
Answer: A. 5/4
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Q32. Which statement describes the symmetry of the curve x3 + y3 = 3xy + 1?
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Answer
Answer: B. It has the line y = x as a line of symmetry, but neither coordinate axis.
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Q33. How many real solutions does (sin x)2 = (x/360)2 have, where x is measured in degrees?
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Answer
Answer: E. 7
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Q34. The curves y = x2, y = x2 − 4x + 5 and y = x2 + 2 are drawn in the plane. The coordinate axes are not counted as extra curves. Into how many regions do these three curves divide the plane?
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Answer
Answer: D. 6
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Q35. What is the largest x-coordinate of a point on the curve xy2 − 4y + x2 = 0?
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Answer
Answer: D. the cube root of 4
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Q36. How many real solutions does (cos x)2 = x4 have?
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Answer
Answer: E. 2
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Q37. The graph of y = 2x is transformed to the graph of y = 8 × 4x. Which of the following could be the sequence of transformations?
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Answer
Answer: E. a stretch parallel to the x-axis, followed by a stretch parallel to the y-axis
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Q38. The graph of y = f(x) meets the x-axis at exactly two distinct points, both with positive x-coordinates. How many of the graphs y = f(|x|), y = |f(x)|, y = f(x) + |f(x)|, y = f(1/x) and y = f(x − 1) + 1 necessarily meet the x-axis at exactly two distinct points?
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Answer
Answer: C. 2
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Q39. The curve y2 = x4 − 4x2 + 5 is drawn for real x and y. Which statement describes the possible values of y?
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Answer
Answer: C. y ≤ −1 or y ≥ 1
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Q40. The curve y = x3 + bx2 + cx is rotated through 180° about the origin. The image is the same as the image obtained by translating the original curve 2 units in the negative x-direction. What is the value of c?
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Answer
Answer: A. 2
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Q41. How many distinct x-intercepts does the graph of y = 5 − |3 − |x|| have?
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Answer
Answer: D. 2
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Q42. What can be said about the asymptotes of y = (x2 − 5x + 1)/(x − 2)?
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Answer
Answer: B. The graph has asymptotes x = 2 and y = x − 3.
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Q43. A cubic function f has a local maximum at (1, 4) and a local minimum at (3, 0). Which of the following graphs has a local minimum at (−1, 2)?
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Answer
Answer: B. y = 6 − f(−x)
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Q44. What is the smallest positive solution of cos(π√x) = −1/2?
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Answer
Answer: A. 4/9
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Q45. How many solutions does (x2 − 4) sin(πx) = 0 have on the interval −3 ≤ x ≤ 3?
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Answer
Answer: C. 7
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Q46. What is the least value of log base 2 of (x2 − 10x + 29)?
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Answer
Answer: D. 2
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Paper 2 questions
Paper 2 is Mathematical Reasoning. It can test this same topic. Argument, proof, and identifying errors are the topics that appear on Paper 2 only.
Q1. Let y = 3−x cos2(x). Which statement is true?
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Answer and worked solution
Answer: C. y = 0 when x = π/2.
Worked solution. At x = π/2, cos(π/2) = 0, so y = 3−π/2 × 0 = 0. The other statements fail. The factor 3−x is not periodic, so y is not periodic. Both 3−x and cos2(x) are nonnegative, so y is never negative. At x = 0, y = 1, but at x = −2π, cos(−2π) = 1 and 32π > 1, so y exceeds 1 and 1 is not the greatest value. As x → ∞, 3−x → 0, so y → 0 rather than 1.
Why the other options look right. A would hold for cos2(x) alone, but multiplication by 3−x destroys the period. B treats 3−x as negative when x is positive, confusing a negative exponent with a negative value, but 3−x > 0 for every x and cos2(x) ≥ 0. D evaluates y at x = 0 and ignores the larger values for negative x, such as y = 32π at x = −2π. E uses the limit of 3−x × 1 as if the exponential tended to 1.
Q2. The function f is differentiable for all real x. It satisfies f(1) = f(5) = 0 and f(x) > 0 for 1 < x < 5. Which statement must be true?
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Answer and worked solution
Answer: D. f(x − 2) > 0 for 3 < x < 7
Worked solution. If 3 < x < 7, then 1 < x − 2 < 5, so f(x − 2) > 0; this holds for every such f. The other statements need not hold. For f(x) = (x − 1)(5 − x)(x + 1), which is positive on 1 < x < 5, f'(x) = (6 − 2x)(x + 1) + (x − 1)(5 − x), so f'(3) = 0 + 4 = 4. The integral from 1 to 5 of f'(x) dx equals f(5) − f(1) = 0, which is never positive. For f(x) = (x − 1)(5 − x), f(−x) = −(x + 1)(x + 5), which is negative for 1 < x < 5. For f(x) = −(x − 3)4 + 3(x − 3)2 + 4, write t = x − 3: f = (4 − t2)(t2 + 1) is zero at t = ±2 and positive between, and f' = 2t(3 − 2t2) vanishes at t = 0 and t = ±√(3/2), three points of (1, 5). The statement that must be true is f(x − 2) > 0 for 3 < x < 7.
Why the other options look right. A assumes the graph is symmetric about x = 3, but f(x) = (x − 1)(5 − x)(x + 1) has f'(3) = 4. B forgets that the integral of f' from 1 to 5 is f(5) − f(1) = 0, whatever the shape between the roots. C reflects the interval the wrong way: f(−x) > 0 is guaranteed only for −5 < x < −1, and for f(x) = (x − 1)(5 − x) it is negative on 1 < x < 5. E relies on Rolle's theorem, which guarantees at least one zero of f', but f(x) = −(x − 3)4 + 3(x − 3)2 + 4 has three.
Q3. The graph of a quartic with positive leading coefficient is crossed by horizontal lines y = k. As k increases through four values, the numbers of distinct intersections are recorded. Which of the following sequences is impossible?
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Answer and worked solution
Answer: A. 4, 2, 4, 3
Worked solution. A quartic has at most three turning points. If the leading coefficient is positive and the two local minima have different heights, the local maximum lies above both of them. As the horizontal level rises, the number of distinct intersections runs through 0, 1, 2, 3, 4, 3, 2, hitting the odd values only at the turning levels. After the four-root band, the count falls through 3 to the two-root region above the local maximum, and it cannot return to 4. If the minima have equal heights the counts are 0, 2, 4, 3, 2, and with only one minimum they are 0, 1, 2. In every case the levels with four intersections form a single interval, so 4 cannot reappear after a 2. The sequence 4, 2, 4, 3 is impossible.
Why the other options look right. B is the region below the graph, then the lower minimum, then the two-root band, then the four-root band. C is the two-root band, the four-root band, the local maximum, then the region above it. D is the lower minimum, the two-root band, the four-root band, then the local maximum. E skips the turning levels: below the graph, between the minima, in the four-root band, then above the local maximum.
Q4. The graph of y = ax2 + bx + c, where a, b and c are constants with a ≠ 0, has its vertex at (2, −3) and meets the y-axis below the x-axis. Which of the following must be true? (I) a > 0; (II) ab < 0; (III) c < −3.
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Answer and worked solution
Answer: B. II only
Worked solution. The vertex has x-coordinate −b/(2a) = 2, so b = −4a and ab = −4a2, which is negative for every a ≠ 0. So II must be true. The vertex value is c − b2/(4a) = c − 4a = −3, so c = 4a − 3. Meeting the y-axis below the x-axis means c < 0, that is 4a − 3 < 0, so a < 3/4 with a ≠ 0. Both signs of a are possible. a = 1/2 gives y = (1/2)x2 − 2x − 1, with vertex (2, −3) and c = −1, so III is false. a = −1 gives y = −x2 + 4x − 7, with vertex (2, −3) and c = −7, so I is false. Only II must be true, which is option B.
Why the other options look right. A overlooks that the vertex condition −b/(2a) = 2 forces b = −4a, so ab = −4a2 < 0 always. C assumes a vertex below the x-axis must be a minimum point, so a > 0; but a = −1 gives y = −x2 + 4x − 7, with vertex (2, −3) and y-intercept −7. D assumes the vertex is the highest point, so the y-intercept would lie below −3; but a = 1/2 gives y = (1/2)x2 − 2x − 1, with vertex (2, −3) and y-intercept −1. E accepts I and III as well, but they cannot both hold, since c = 4a − 3 < −3 needs a < 0; the two examples above refute I and III separately.
Q5. The function f is defined for all real numbers, and a and b are non-zero real constants. Which conditions are necessary and sufficient for the graph of y = f(x) to have rotational symmetry of order 2 about the point (a, b)? I: f(a + x) + f(a − x) = 2b for every real x. II: f(2a − x) = 2b − f(x) for every real x. III: The function g(x) = f(x + a) − b is odd.
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Answer and worked solution
Answer: E. I, II and III
Worked solution. A half-turn about (a, b) sends the point (a + t, y) to (a − t, 2b − y). The graph is unchanged exactly when every point (a + t, f(a + t)) is sent to a point of the graph, that is f(a − t) = 2b − f(a + t) for every t. This is condition I. Putting x = a + t in II gives f(a − t) = 2b − f(a + t), so II is the same condition. For III, g(−x) = −g(x) means f(a − x) − b = −(f(a + x) − b), which rearranges to f(a + x) + f(a − x) = 2b, again condition I. All three conditions are necessary and sufficient.
Why the other options look right. A accepts only the defining condition I and rejects both rewritings, although II is I with x replaced by a + x and III is I after moving the centre to the origin. B rejects III, overlooking that moving the centre (a, b) to the origin turns half-turn symmetry into the odd-function condition. C rejects II, overlooking that 2a − x and x lie the same distance either side of a. D rejects I, which is the definition of the symmetry written with the points a + x and a − x.
Q6. It is given that f(x) = x3 − 3x + q, where q is a real constant, and that f(x) = 0 has three distinct real roots. Which of the following must be true? I: The equation f(x) + 2 = 0 has three distinct real roots. II: The equation f(x − 1) = 0 has three distinct real roots. III: The equation f(x) f(−x) = 0 has six distinct real roots.
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Answer
Answer: B. II only
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Q7. The graph of y = f(x), defined for all real x, is stretched parallel to the x-axis with scale factor 2 and then translated 3 units in the positive x-direction. The result is y = g(x). The same graph is translated 3 units in the positive x-direction and then stretched parallel to the x-axis with scale factor 2. The result is y = h(x). Which condition is necessary and sufficient for g and h to be the same function?
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Answer: E. f(x) = f(x + 3/2) for all x
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Q8. Which statement about the graph of y = (x² − 1)/(x² − 4) is true?
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Answer: E. y > 1 for every x with |x| > 2.
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Q9. The real number q is chosen so that |x| × |x − 2| = q has exactly k distinct real solutions. Which list is the complete set of possible values of k?
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Answer: D. 0, 2, 3, 4
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Q10. Which statement about the interval 0 < x < π/2 is true?
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Answer: C. (sin x)tan x tends to 1 as x tends to π/2 from below
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Q11. Which statement describes the set of all points (x, y) in the plane with (x + y)3 = x3 + y3?
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Answer: D. It is the x-axis, the y-axis and the line y = −x.
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Q12. Which statement describes the graph of y = log2(x) × logx(8), drawn for every real x for which the expression is defined?
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Answer: A. It is the part of the line y = 3 with x > 0, except the point (1, 3).
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Q13. For which values of the constant k does the equation |x2 − 4x + 3| = x + k have exactly three distinct real solutions?
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Answer: D. k = −1 or k = −3/4
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Q14. The circle x2 + y2 = 4 and the lines y = 2 and x = 1 are drawn in the plane. Into how many regions do they divide it?
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Answer: C. 7
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Q15. Find the sum of all distinct real solutions of |x² − 4| = x + 2.
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Answer: E. 2
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Q16. Into how many regions do the parabola y = x² + 1 and the line y = 0 divide the plane?
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Answer: B. 3
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Q17. Into how many regions do the graphs of y = x³ − 3x² and y = −4 divide the plane?
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Answer: D. 5
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Q18. How many distinct real solutions does the equation x³ − 16x + 3 = 0 have?
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Answer: A. 3
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Q19. The graph of y = (2x + 1)/(x − 3) has rotational symmetry of order 2. What is its centre of symmetry?
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Answer: A. (3, 2)
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Q20. Into how many regions do the parabola y = 6 − x², the line y = 2 and the line x = 0 divide the plane?
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Answer: A. 8
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Q21. What is the range of f(x) = (x² + 1)/(x² + 4) for real x?
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Answer: B. 1/4 ≤ f(x) < 1
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Q22. Into how many regions do the parabola y = x² and the lines y = 4 and y = −1 divide the plane?
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Answer: B. 6
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Q23. The function f(x) = x² − 4x + 7 is restricted to x ≥ 2. Which expression and domain give f⁻¹(x)?
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Answer: A. 2 + √(x − 3), x ≥ 3
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Q24. How many real solutions does the equation 2x = x² have?
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Answer: B. 3
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Q25. How many real solutions does x4 − 4x2 = x − 1 have?
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Answer: E. exactly four
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Q26. Into how many regions do the curves y = x2, y = x + 2 and y = −x + 2 divide the plane?
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Answer: A. 9
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