About this chapter
Graphs and transformations tests whether you can draw what the algebra says, and read the algebra back from a drawing. Sketches are marked on detail: the shape, every point where the curve meets an axis, and the equation of every asymptote. A curve that looks right but is missing a labelled intercept will not get full marks.
The chapter covers cubics with repeated roots, reciprocal graphs of the forms k/x and k/x², translations and stretches, and using intersections to count the solutions of an equation. The harder questions find a quadratic and a cubic from their features, reverse a transformation to recover a function, and decide when a translated cubic just touches the x-axis.
The ten questions
- Q1 (5 marks): factorise x³ − 6x² + 9x completely and sketch the cubic, which touches the x-axis at its repeated root.
- Q2 (5 marks): sketch y = 3/x², then the same curve moved down 1, with its asymptote and exact x-axis crossings.
- Q3 (6 marks): where a maximum and a minimum move under y = f(x+2), y = 3f(x) and y = f(2x).
- Q4 (5 marks): sketch a cubic and y = 4/x on one set of axes, then count the real solutions of x³(x−3) = 4, giving a reason.
- Q5 (5 marks): translate y = 4/x, describe a transformation fully, and find where the new curve crosses the x-axis.
- Q6 (9 marks): sketch (2x+1)²(x−3), read off where it is positive, expand it, and find where a horizontal line meets it again.
- Q7 (8 marks): find a quadratic and a cubic from their graphs, then the third point where they meet by cancelling common factors.
- Q8 (8 marks): sketch y = 3/x − 2, use a given point to find k for a line, then find the second intersection algebraically.
- Q9 (8 marks): transform a curve with two asymptotes in two ways, then find the constants a and b in its equation.
- Q10 (9 marks): counting solutions of f(x) = k and f(x+3) = 2, and the values of c that make f(x) + c touch the x-axis.
Key skills tested
Sketching cubics. Find the roots and the y intercept. A squared factor means the curve touches the x-axis, and a negative x³ term turns the shape upside down. For example, y = (x−1)²(x+2) touches at (1, 0).
Factorise, then sketch. Take out a factor of x first, then factorise the quadratic that is left. For example, x³ − 4x² + 4x = x(x−2)².
Reciprocal graphs. For positive k, y = k/x sits in the first and third quadrants, while y = k/x² has both branches above the x-axis.
Asymptotes. State each one as an equation, and move it with the graph. For example, y = 1/x + 3 has asymptotes x = 0 and y = 3.
Counting solutions. Rearrange into f(x) = g(x) for two graphs you can sketch. The number of solutions is the number of intersections, and that is the reason to give.
Intersections by algebra. Set the two equations equal, and look for common factors before expanding anything.
Translations. y = f(x+a) moves the graph a units to the left; y = f(x) + a moves it a units up.
Stretches. y = af(x) multiplies every y coordinate by a; y = f(ax) divides every x coordinate by a.
Equations from graphs. The roots give the factors, and a constant k is always needed. For example, a repeated root at 3 and a root at 0 give y = kx(x−3)², then one more point gives k.
Worked example
Question 5 from this chapter. The curve C has equation y = 4/x. (a) C is translated 3 units in the negative x direction: write down the new equation and its vertical asymptote. (b) Describe fully the single transformation that maps C onto y = 4/x − 2. (c) Find where y = 4/x − 2 crosses the x-axis.
(a) Moving 3 units left replaces x with x + 3, so the new curve is y = 4/(x+3). The vertical asymptote moves with it, from x = 0 to x = −3.
(b) Subtracting 2 from the whole function is a translation of 2 units in the negative y direction. The word translation is needed for the first mark, and the size and direction for the second.
(c) Set y = 0: 4/x = 2, so x = 2 and the crossing point is (2, 0).
Seen on real papers
Every question in our booklets is original. These are the recent papers where each question type has appeared.
- January 2023, Q7: sketching y = 6/x, describing its translation, then a line meeting the curve, as in Q5 and Q8.
- October 2024, Q4: a quadratic and a cubic with a repeated root, both found from their graphs, then where they meet, as in Q7.
- October 2024, Q6(a): sketching a translated reciprocal curve with its asymptote, as in Q8(a).
- June 2024, Q3: transformations of a sketched graph and where its turning point goes, as in Q3 and Q9.
- October 2024, Q9(a) and (d): where a cubic is positive, and where a point moves under a translation and a stretch, as in Q6(b) and Q3.
Where marks are lost
- Shift is not a transformation. In January 2023 almost everyone knew the graph moved 2 units right, but very few called it a translation, and the mark needs that word.
- Asymptote on the wrong side. In October 2024 the most common sketching error put the vertical asymptote at x = −2 instead of x = 2.
- No constant in front. Writing a cubic from its graph as x(x−4)² with no constant k was the most frequent error in October 2024, and the shared factors of the two curves were seldom cancelled.
- Wrong direction or factor. Moving a point the wrong way under f(x+3), or using a scale factor of ¼ instead of 4, were the usual slips when transforming a point in October 2024.
Common questions
Does f(x+2) move the graph left or right?
Left by 2. Changes inside the bracket act on x and go the opposite way to the sign; changes outside act on y and go the way you expect.
What must every sketch show?
The correct shape, the coordinates of every point where it meets an axis, and the equation of every asymptote, even when the question only hints at them.
How do I justify a number of solutions?
Say that it is the number of times the two graphs intersect, and make sure your sketch actually shows that many crossings.
Where this chapter leads
- Pure 1 Chapter 6: Trigonometric Ratios: translating and stretching sine, cosine and tangent graphs.
- Pure 1 Chapter 7: Radians: the same transformations with angles in radians.
- Pure 1 Chapter 8: Differentiation: a tangent that meets a cubic again.
- Pure 2 Chapter 1: Algebraic Methods: the factor theorem, used to factorise cubics before sketching them.