Algebra and Functions

Graphs and Transformations

Pearson Edexcel International Advanced Level Mathematics

Polynomials

Method: factorise a cubic fully

Take out any common factor, such as x.
Factorise the quadratic that remains.
Write the cubic as a product of linear factors.

Cubic and quartic graphs

Each factor (x − a) gives a root, where the curve meets the x-axis at (a, 0).
A repeated root, from a squared factor, is where the curve touches the x-axis.
Positive x³ coefficient: the curve rises to the right. Negative: it falls to the right.
Positive x⁴ coefficient: both ends point up. Negative: both ends point down.

Method: sketch from factorised form

Put y = 0: each factor gives a root on the x-axis.
Put x = 0 to find where the curve meets the y-axis.
Decide the end behaviour from the sign of the highest power of x.
Draw the curve through the roots, touching the axis at a repeated root.
Label every intercept as coordinates on the sketch.

Worked example: sketch a cubic with a repeated root

Given that k is a constant with 0 < k < 3, sketch the curve with equation y = (3 − x)(x − k)², showing the coordinates of the points where the curve meets the coordinate axes.

y = 0: x = 3 and x = k (repeated, so the curve touches the axis)

x = 0: y = 3 × (−k)² = 3k²

The x³ coefficient is negative, so the curve falls to the right.

(k, 0), (3, 0) and (0, 3k²)

Reciprocal graphs

y = kx with k > 0: two branches, in the first and third quadrants. With k < 0: second and fourth.
y = kx² with k > 0: two branches, both above the x-axis.
Both have asymptotes x = 0 and y = 0.
y = kx − a + b has asymptotes x = a and y = b.
State each asymptote as an equation. Write y = 0. Never write the asymptote is the x-axis.

Method: sketch y = kx − a + b

Draw the asymptotes x = a and y = b and label each with its equation.
Draw the two branches in the quadrants given by the sign of k, each approaching both asymptotes.
Put x = 0 for the y-intercept and y = 0 for the x-intercept, and label them as coordinates.

Trap: branches that cross an asymptote

Each branch gets closer to its asymptotes and never crosses them. A branch drawn bending back past a horizontal asymptote, so that the two branches overlap vertically, is wrong, and so is a curve that levels off towards an extra horizontal line. A curve y = kx − a has the single horizontal asymptote y = 0, because kx − a gets closer to zero as x grows in either direction.

Intersections and roots

The number of real solutions of f(x) = g(x) is the number of points where y = f(x) and y = g(x) meet.
Give the number with its reason: one root because the two graphs intersect each other once.
To find the points, set the two expressions equal and solve.

Worked example: sketch a reciprocal curve, then count roots

The curve C has equation y = 4x − 2 + 1, x ≠ 2.

(a) Sketch C, stating the equations of the asymptotes and the coordinates of the points where C crosses the coordinate axes.

(b) On the same axes, sketch the curve with equation y = x². Hence state the number of real solutions of the equation 4x − 2 + 1 = x², giving a reason.

(a) Asymptotes x = 2 and y = 1

x = 0: y = 4−2 + 1 = −1, giving (0, −1)

y = 0: 4x − 2 = −1, so x − 2 = −4, giving (−2, 0)

(b) One real solution because the two graphs intersect each other once

Transformations

Curve	Transformation	Point (p, q) moves to
y = f(x + a)	translation by −a0	(p − a, q)
y = f(x) + a	translation by 0a	(p, q + a)
y = af(x)	stretch parallel to the y-axis, scale factor a	(p, aq)
y = f(ax)	stretch parallel to the x-axis, scale factor 1a	(pa, q)

Asymptotes move in the same way: a horizontal asymptote moves with q, a vertical one with p.
Describe a transformation by its type, direction and size: translate a units to the right, stretch parallel to the x-axis, scale factor 1a.

Method: transform a sketch

Move each marked point using the table.
Move each asymptote and write its new equation.
Draw the new curve with the same shape through the new points.
Label the new points as coordinates and the asymptote as an equation.

Worked example: sketch a transformed curve

The figure shows the curve y = f(x). It passes through the origin, has a maximum point at (2, 6), crosses the x-axis at (5, 0) and has the asymptote y = −3.

(a) Sketch the curve with equation y = f(x + 3), stating the coordinates of the maximum point, the points where the curve crosses the x-axis and the equation of the asymptote.

(b) State the coordinates of the maximum point and the equation of the asymptote of the curve y = 2f(x).

(a) Translation by −30: every x-coordinate decreases by 3.

Maximum (−1, 6), crosses at (−3, 0) and (2, 0), asymptote y = −3

(b) Stretch parallel to the y-axis, scale factor 2: every y-coordinate doubles.

Maximum (2, 12), asymptote y = −6

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