The Hyperbola & Exponential Graph (Grade 10)

Sketching y = a/x and y = abᵙ, finding asymptotes and intercepts, and reading the effect of a, b and a vertical shift q — the two curved graphs of Grade 10.

Grade 10 adds two curves to the straight line and the parabola. Both are sketched the same way: find the asymptotes, find any intercepts, plot one or two points, then draw the shape.

1The hyperbola: y = a/x

y = 6/x. Both branches get closer and closer to the axes but never touch them.

y = ax + q
vertical asymptote x = 0 · horizontal asymptote y = q

An asymptote is a line the curve approaches but never reaches.
If a > 0 the branches sit in quadrants 1 and 3; if a < 0 they sit in quadrants 2 and 4.
There is no y-intercept — x can never be 0, because you cannot divide by zero.

Worked ExampleSketch y = 6x

1
Asymptotes: x = 0 and y = 0.
The standard hyperbola has the axes as its asymptotes.
2
a = 6 > 0, so the branches are in quadrants 1 and 3.
3
Plot a few points: (1;6), (2;3), (6;1) — and (−1;−6), (−2;−3).
Pick x-values that divide into 6 neatly.
4
Draw two smooth branches approaching the axes.
Never let the curve touch an asymptote.

Worked ExampleFind the asymptotes of y = 4x − 3

1
Vertical asymptote: x = 0.
Unchanged — the denominator is still x.
2
Horizontal asymptote: y = −3.
The whole graph has shifted down 3, so q = −3.

⚠️

A hyperbola never has a y-intercept, and y = ax + q never reaches y = q. If your sketch touches an asymptote, it is wrong.

2The exponential graph: y = abᵙ

y = 2ˣ. It rises steeply to the right and flattens towards y = 0 on the left, never touching it.

y = a·b^x + q
horizontal asymptote y = q · y-intercept at x = 0

If b > 1 the graph increases (growth); if 0 < b < 1 it decreases (decay).
There is one horizontal asymptote, y = q, and normally no x-intercept.
Every exponential graph of the form y = b^x passes through (0 ; 1), because b⁰ = 1.

Worked ExampleSketch y = 2^x

1
Asymptote: y = 0.
q = 0, so the x-axis.
2
y-intercept: 2⁰ = 1, giving (0 ; 1).
Every such graph passes through (0;1).
3
Points: (1;2), (2;4), (3;8); and (−1;½), (−2;¼).
Negative x gives small positive values — never negative.
4
Draw a smooth curve rising to the right, flattening left towards y = 0.

Worked ExampleFind the y-intercept and asymptote of y = 3·2^x − 4

1
Asymptote: y = −4.
q = −4.
2
y-intercept: let x = 0 → y = 3(1) − 4 = −1.
b⁰ = 1, so the a value shows through.
3
y-intercept is (0 ; −1).

💡

For any graph: the y-intercept comes from letting x = 0, and the x-intercept from letting y = 0. That one rule covers all four Grade 10 functions.

Practice exercises

Work each one out, then click to reveal the answer.

📄 Worksheet ✅ Memo 📝 Exam paper 🗝️ Marking guide

1
What are the asymptotes of y = 6x?
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x = 0 and y = 0
2
What are the asymptotes of y = 4x − 3?
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x = 0 and y = −3
3
In which quadrants does y = 6x lie?
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1 and 3 (since a > 0)
4
In which quadrants does y = −5x lie?
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2 and 4 (since a < 0)
5
Why does a hyperbola have no y-intercept?
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x cannot be 0 — you cannot divide by zero.
6
Find y when x = 3 for y = 6x.
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y = 2
7
What is the y-intercept of y = 2^x?
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(0 ; 1)
8
What is the asymptote of y = 2^x + 5?
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y = 5
9
Find the y-intercept of y = 3·2^x − 4.
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3(1) − 4 = (0 ; −1)
10
Does y = (½)^x increase or decrease?
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Decreases — the base is between 0 and 1.