GCSE · Maths · Edexcel · Spec 1MA1 · Higher

Sketching and interpreting standard graphs

This curve spends its whole life chasing a line it can never catch. Drag the point and watch, then learn to name any graph from a few clues.

Chase the point along y = 10/x. Can you make y reach 0?

x: 10.25. y: 0.98

drag me right, then left

Watch out: Near x = 0 the curve runs off the top of this window. That is the window ending, not the curve. Keep making x smaller and y keeps growing: x = 0.1 gives y = 100 and x = 0.01 gives y = 1000. And x = 0 itself? 10 ÷ 0 is undefined, so there is no point on the y-axis at all.

Predict, then check

Picture a horizontal line at each height and count where it meets the curve.

Here is a table of values for y = x³ − 3x + 3. When x = −2, −1, 0, 1 and 2, y = 1, 5, 3, 1 and 5. The curve rises to a local maximum (a peak) at (−1, 5), falls to a local minimum (a dip) at (1, 1), then rises again. How many solutions do x³ − 3x + 3 = 6, x³ − 3x + 3 = 5, x³ − 3x + 3 = 3 and x³ − 3x + 3 = 1 have, in that order?

Your turn to supply the steps

Sketching a shifted reciprocal

Sketch y = 2.4/x − 0.6. Choose each missing step.

Match it to y = k/x + b. Here k = 2.4 and b = −0.6.
Vertical asymptote: the y-axis, because x = 0 would mean dividing by 0.
missing step

Which line is step 3?

Horizontal asymptote: the x-axis.Horizontal asymptote: the line y = −0.6.Horizontal asymptote: the line y = 0.6.

What do you think?

Does y = 2ˣ ever reach 0?

Follow the graph of y = 2ˣ to the left, where x becomes more and more negative.

Which is closest to what you think happens to y?

Eventually y gets to 0. Go far enough left and the curve lands on the x-axis.Once x is negative, y is negative too, so 2⁻² = −4.It keeps going down, crosses the x-axis and carries on below it.It keeps halving, so it gets closer and closer to 0 but never gets there.

How sure are you?

I'm sureI think soI'm guessing

Eventually y gets to 0. Go far enough left and the curve lands on the x-axis.

This is a common misconception

Try it: 2⁰ = 1, 2⁻¹ = 1/2, 2⁻² = 1/4, 2⁻³ = 1/8. Every step to the left halves y, and halving a positive number always leaves a positive number. However far left you go, y is never 0. The curve creeps ever closer to the x-axis without touching it, which is exactly what makes the x-axis an asymptote.

Once x is negative, y is negative too, so 2⁻² = −4.

This is a common misconception

A negative power does not make a negative number. 2⁻¹ = 1/2 and 2⁻² = 1/4. For y = bˣ with b greater than 1, every negative x gives a y value between 0 and 1.

It keeps going down, crosses the x-axis and carries on below it.

This is a common misconception

To get below the x-axis, y would have to pass through 0 first, and no power of 2 equals 0. The curve stays above the axis for every value of x, squeezed ever closer to it but never through it.

It keeps halving, so it gets closer and closer to 0 but never gets there.

This matches the evidence

Exactly right. Each step of 1 to the left halves y, so the x-axis is an asymptote. Two points pin the curve down: (0, 1), because b⁰ = 1, and (1, b), because b¹ = b. So y = 2ˣ passes through (0, 1) and (1, 2), and y = 3ˣ passes through (0, 1) and (1, 3).

For y = bˣ with b greater than 1, y grows quickly to the right and shrinks towards 0 to the left without ever reaching it. With b between 0 and 1, such as y = (1/2)ˣ, it runs the other way: y decreases as x increases ((1/2)⁷ is already 1/128) but still never reaches 0, and it still passes through (0, 1).

Asymptotes from the equation

Where are the asymptotes?

Pick an equation, then choose the category it belongs to.

Where is the vertical asymptote?

Still to sort

The y-axis (x = 0) (0)

x = 0 would mean dividing by 0.

Where the line is: y = k/x + b still has the y-axis: adding b moves the curve up or down, not sideways.

Another vertical line, x = −a (0)

The x value that makes the denominator 0.

Where the line is: For y = 1/(x − 4) the asymptote is x = 4, not x = −4. Solve x − 4 = 0.

No vertical asymptote (0)

An exponential graph has one asymptote only, and it is horizontal.

8 of 8 still to sort.

Sort each equation by its vertical asymptote. Then switch the rule and sort the same equations by their horizontal asymptote.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Quadratics, cubics, reciprocals and exponentials, plus the lines some curves chase for ever and never reach.

What you need to know

An asymptote is a line a curve approaches but never touches. Reciprocal graphs have two, and exponential graphs have one.
y = k/x has both axes as asymptotes: x = 0 would mean dividing by 0, and no x makes k/x equal 0.
y = k/x lies in the first and third quadrants when k is positive, and in the second and fourth when k is negative.
y = k/x + b has asymptotes x = 0 and y = b. y = k/(x + a) has asymptotes x = −a and y = 0.
Every y = bˣ passes through (0, 1) and (1, b). y = bˣ + c has the horizontal asymptote y = c and y-intercept (0, 1 + c).
A cubic has one, two or three roots. A squared bracket gives a repeated root, where the curve touches the x-axis at a turning point.
The solutions of f(x) = c are the x values where y = f(x) meets the line y = c.

The big picture

Standard graphs can be recognised and sketched from a few key features: shape, intercepts, turning points and asymptotes. Quadratics and cubics have turning points, and a squared bracket in a factorised cubic gives a repeated root where the curve touches the x-axis. Reciprocal and exponential graphs have asymptotes, lines the curve approaches but never touches, and adding a constant or changing the denominator moves them. Solutions of f(x) = c are read from where the curve meets the line y = c.

Key points

1A sketch needs axes, the right shape and every key feature labelled, but no scale.

2Factorise a cubic to read its roots: each bracket equals 0 at one root.

3A cubic's turning points are usually local: highest or lowest only in their own neighbourhood.

4Find a vertical asymptote by asking which x makes the denominator 0.

5Find a horizontal asymptote by asking which y value the curve can never equal.

6Once an asymptote has moved off an axis, the curve can cross that axis, so test y = 0 and x = 0.

7y = −bˣ is y = bˣ reflected in the x-axis, through (0, −1) and (1, −b).

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Worked example

Problem

Sketch y = x³ − 6x² + 9x, marking its intercepts and turning points.

⚠ Watch out

Thinking every reciprocal graph has both axes as asymptotes. Only y = k/x does. For y = k/x + b the horizontal asymptote is y = b, and for y = k/(x + a) the vertical asymptote is x = −a, where the denominator is 0.

🧠

Memory hook

Asymptotes come from the impossible. Ask: which x can never happen? That is the vertical one (dividing by 0). Which y can never happen? That is the horizontal one (a fraction or power that can never be 0).

✓

Check yourself

Without drawing anything: name both asymptotes of y = 3/x − 2, say whether the graph has a y-intercept, and find where it crosses the x-axis.

Flashcards

(14)

What is an asymptote?

A line that a curve gets closer and closer to but never touches.

Why does y = k/x never meet the y-axis?

On the y-axis x = 0, and dividing by 0 is undefined, so there is no point there.

Why does y = k/x never meet the x-axis?

On the x-axis y = 0, and no value of x makes k/x equal 0.

Which quadrants does y = k/x lie in?

k positive: first and third (top right, bottom left). k negative: second and fourth (top left, bottom right).

Asymptotes of y = k/x + b?

The y-axis (x = 0) and the line y = b.

Vertical asymptote of y = k/(x + a)?

x = −a, the value of x that makes the denominator 0. The x-axis is the horizontal asymptote.

Which two points does every graph of y = bˣ pass through?

(0, 1), because b⁰ = 1, and (1, b), because b¹ = b.

What does y = bˣ look like when b is between 0 and 1?

y decreases as x increases, still passes through (0, 1), and still has the x-axis as its asymptote.

Asymptote and y-intercept of y = bˣ + c?

Horizontal asymptote y = c; y-intercept (0, 1 + c). No vertical asymptote.

How is y = −bˣ related to y = bˣ?

It is the reflection in the x-axis: still the x-axis as asymptote, now through (0, −1) and (1, −b).

How do you read the solutions of f(x) = c from a graph?

Draw the horizontal line y = c and read the x values where it meets the curve y = f(x).

How many roots can a cubic have?

One, two or three. If it has exactly two, one of them is a repeated root, where the curve touches the x-axis at a turning point.

What is a local maximum?

A turning point where y changes from increasing to decreasing. It is the highest point only in its own neighbourhood, not necessarily on the whole graph.

What must a sketch of a graph show?

The axes and the correct shape, with the key features (intercepts, turning points, asymptotes) marked and labelled. No scale is needed.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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Sketching and interpreting standard graphs

Chase the point along y = 10/x. Can you make y reach 0?

What can each graph do?

Sketching a shifted reciprocal

Does y = 2ˣ ever reach 0?

Where are the asymptotes?

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