OCR GCSE Mathematics: Exponential and trigonometric graphs, and transformations of curves (J560) – Revision Notes

These notes condense OCR GCSE (9-1) Mathematics J560 statements 7.01b, 7.01d, 7.01e and 7.03a (specification Version 2.1, August 2026). Everything here is Higher tier only; Foundation tier candidates are not assessed on it. For full explanations and worked examples, read the study guide. Then try the practice questions, see the course hub and tick items off the checklist. If you are unsure where to start, take a free 10-minute diagnostic.

1. Exponential graphs (7.01b, 7.01d) – Higher tier only

Key facts for y = kˣ with k > 0

Fact	Detail
Fixed point	(0, 1), since k⁰ = 1
Position	Always above the x-axis; y > 0
Asymptote	y = 0 (the curve approaches but never touches)
k > 1	Growth: rises left to right, steeper for larger k
0 < k < 1	Decay: falls left to right
k = 1	Horizontal line y = 1
Link	0.5ˣ = 2⁻ˣ, the mirror image of 2ˣ in the y-axis

Method: plotting from a table of values

1. Choose x-values, including 0 and some negative ones.
2. Work each y out separately. Negative power means reciprocal: 3⁻² = 1/9.
3. For y = a × kˣ, work out kˣ first and then multiply by a.
4. Plot every point, then draw one smooth curve. Do not use a ruler.
5. Check the y-axis scale: exponential values grow fast.

Quick reminders

• y = 3 × 1.1ˣ has y-intercept 3. At x = 5 it is 3 × 1.61051 = 4.83 (3 s.f.).
• To find k from a point (p, q) on y = kˣ: solve kᵖ = q. For (3, 64), k³ = 64, so k = 4.
• For a point with x = −1, say (−1, 1/5): k⁻¹ = 1/5, so k = 5.
• In a real-life model such as P = 200 × 0.8ˣ the 200 is the starting value and 0.8 is the multiplier each step. Because 0.8 < 1, P decays.

2. Trigonometric graphs (7.01e) – Higher tier only

x is an angle in degrees.

Graph	Period	Range	Start (x = 0)	Key points
y = sin x	360°	−1 to 1	(0, 0)	max at 90°, zero at 180°, min at 270°
y = cos x	360°	−1 to 1	(0, 1)	zero at 90°, min at 180°, zero at 270°, max at 360°
y = tan x	180°	all values	(0, 0)	zeros at 0°, 180°, 360°; asymptotes at 90° and 270°

Must-know distinctions

• Sine starts at 0 and goes up. Cosine starts at 1 and goes down. Cosine is sine shifted 90° to the left.
• Sine and cosine have a maximum and minimum. Tangent has neither: it has asymptotes.
• Sine and cosine repeat every 360°. Tangent repeats every 180°.
• tan 45° = 1 and tan 225° = 1. The curve is below the axis between 90° and 180°.

Method: sketch y = sin x or y = cos x

1. Draw axes labelled in 90° steps from 0° to 360° (or longer if asked).
2. Mark the y-values 1 and −1.
3. Plot the five key points for one wave.
4. Join with a smooth wave, not straight segments.

Method: how many solutions of sin x = c?

Draw the line y = c on your sketch and count the crossings. For 0 < c < 1, there are two per 360°. So sin x = 0.4 has four solutions between 0° and 720°.

3. Translations and reflections (7.03a) – Higher tier only

Function notation is not required, so just work with the equation.

Change	Equation	Effect
Move up a	y = (original) + a	Translate a up
Move down a	y = (original) − a	Translate a down
Move left a	replace x by (x + a)	Translate a left
Move right a	replace x by (x − a)	Translate a right
Reflect in x-axis	y = −(original)	Change sign of every y
Reflect in y-axis	replace x by −x	Change sign of every x

Warning: the horizontal change goes the opposite way to the sign. (x + 2) moves the graph 2 left. To check, find the x-value that makes the bracket zero: x + 2 = 0 gives x = −2.

Method: sketch a transformed graph

1. Sketch the original and mark its key points (turning point, intercepts, asymptote).
2. Apply the change to each key point: translate by adding, reflect by changing a sign.
3. Move or reflect any asymptote too.
4. Redraw the same shape through the new points, then label coordinates.

Small worked reminders

• y = (x + 2)² − 1: turning point (−2, −1); y-intercept (0, 3); roots x = −3 and x = −1.
• y = −x²: reflection of y = x² in the x-axis; maximum at (0, 0).
• y = sin x + 2: range 1 to 3; passes through (0, 2).
• y = cos x − 1: range −2 to 0; passes through (0, 0).
• y = 2ˣ − 3: asymptote y = −3; y-intercept (0, −2).
• y = −2ˣ: below the x-axis; passes through (0, −1).
• y = 2⁻ˣ: reflection of y = 2ˣ in the y-axis; passes through (0, 1).
• y = x² − 6x reflected in the x-axis: y = −x² + 6x, turning point (3, 9) rather than (3, −9).

Combining changes: worked reminder

Starting from y = x², the curve y = −(x − 1)² + 4 is built in order: move 1 right (turning point (1, 0)), reflect in the x-axis (an upside-down U), then move 4 up. The maximum is at (1, 4). Put x = 0 to find the y-intercept: −(0 − 1)² + 4 = 3. Roots: (x − 1)² = 4, so x − 1 = 2 or −2, giving x = 3 and x = −1.

Describing a transformation in words

Say what happens and give the size and direction.

• “Translation 3 units to the right” (not “moves 3”).
• “Translation 2 units down”.
• “Reflection in the x-axis” or “reflection in the y-axis”. Always name the axis.
• If a question asks for a single transformation, check that you have not described two. A move and a reflection together are two transformations.

Reading a given graph

When a graph is supplied without its equation, use its key points. Read off the turning point, the intercepts and any asymptote, then apply the same change to each one. Plot the new key points first and only then draw the curve.

Symmetry shortcuts

• y = x² reflected in the y-axis is unchanged.
• y = x³ reflected in the y-axis gives y = −x³, the same as reflecting in the x-axis.
• y = sin x reflected in the y-axis gives y = −sin x.

Exam habits for sketches

• Label the axes and put numbers on the points the question asks for.
• Use a pencil sketch first if the question asks for a neat final graph.
• For an exponential, write the asymptote next to the dashed line.
• For a trigonometric graph, mark 90°, 180°, 270° and 360° on the x-axis and 1 and −1 on the y-axis.
• Compare your sketch with a value: y = 3ˣ at x = 2 must give 9, so the curve has to pass well above 3 there.

Quick self-test

1. What are the coordinates of the point where every graph y = kˣ crosses the y-axis?
2. State the asymptote of y = 5ˣ.
3. Does y = 0.7ˣ increase or decrease as x increases?
4. A curve y = kˣ passes through (2, 2.25). Find k, given k > 0.
5. Work out the value of 3 × 1.1⁵ to 3 significant figures.
6. State the period of y = tan x.
7. Give the equations of the two vertical asymptotes of y = tan x between 0° and 360°.
8. State the range of y = cos x − 1.
9. What is the turning point of y = (x − 4)² + 3?
10. Write the equation of y = x² − 6x after reflection in the x-axis.
11. State the asymptote of y = 2ˣ − 3.
12. How many solutions has sin x = 0.4 between 0° and 720°?

Answers

1. (0, 1).
2. y = 0.
3. Decreases, because 0.7 < 1.
4. k² = 2.25, so k = 1.5.
5. 3 × 1.61051 = 4.83.
6. 180°.
7. x = 90° and x = 270°.
8. −2 to 0 (that is, −2 ≤ y ≤ 0).
9. (4, 3).
10. y = −x² + 6x.
11. y = −3.
12. Four.

Where marks are usually lost

• Drawing straight lines between plotted points on a curve, or sketching a curve that has kinks.
• Letting an exponential curve cross the x-axis, or forgetting that it passes through (0, 1).
• Moving (x + 2) to the right instead of the left.
• Giving the turning point of (x + 2)² − 1 as (2, −1).
• Reflecting in the wrong axis, or changing the sign of only part of the equation. The minus applies to the whole right-hand side.
• Not moving the asymptote when the graph is translated up or down.
• Sketching sin x starting at 1, or cos x starting at 0.
• Missing the asymptotes of tan x, or giving tan x a maximum.
• Leaving key coordinates off the sketch when the question asks you to label them.

Official syllabus

OCR GCSE (9-1) Mathematics J560, specification Version 2.1 (August 2026), first assessment 2017, published by OCR: statements 7.01b, 7.01d, 7.01e and 7.03a.