The Hyperbola & Exponential Graph — Grade 10 Maths (CAPS)

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

Also known as: reciprocal graphs · asymptotes · exponential growth

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 = a/x + q
vertical asymptote x = 0 · horizontal asymptote y = q

An asymptote is a line the curve gets closer and closer to as it runs out to the side or up and down. The hyperbola and the exponential graph in this lesson never cross theirs.
For the PARENT graph y = a/x (that is, q = 0): if a > 0 the branches sit in quadrants 1 and 3; if a < 0 they sit in quadrants 2 and 4. Adding q slides the whole graph up or down, so a shifted hyperbola can reach into other quadrants.
There is no y-intercept — x can never be 0, because you cannot divide by zero.

Worked ExampleSketch y = 6/x

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 = 4/x − 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 = a/x + q never reaches y = q. If your sketch touches an asymptote, it is wrong.

2The exponential graph: y = ab^x

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

With a > 0: if b > 1 the graph increases (growth); if 0 < b < 1 it decreases (decay). A negative a flips the curve over its asymptote, so each case does the opposite. The base itself is always b > 0, b ≠ 1.
There is one horizontal asymptote, y = q. Whether there is an x-intercept depends on a, b and q: solve a·b^x + q = 0. y = 3·2^x − 4 does cut the x-axis; y = 3·2^x + 4 never does, because 3·2^x is always positive.
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 = 6/x?
Show answer ▾Hide answer ▴
x = 0 and y = 0
2
What are the asymptotes of y = 4/x − 3?
Show answer ▾Hide answer ▴
x = 0 and y = −3
3
In which quadrants does y = 6/x lie?
Show answer ▾Hide answer ▴
1 and 3 (since a > 0)
4
In which quadrants does y = −5/x lie?
Show answer ▾Hide answer ▴
2 and 4 (since a < 0)
5
Why does a hyperbola have no y-intercept?
Show answer ▾Hide answer ▴
x cannot be 0 — you cannot divide by zero.
6
Find y when x = 3 for y = 6/x.
Show answer ▾Hide answer ▴
y = 2
7
What is the y-intercept of y = 2^x?
Show answer ▾Hide answer ▴
(0 ; 1)
8
What is the asymptote of y = 2^x + 5?
Show answer ▾Hide answer ▴
y = 5
9
Find the y-intercept of y = 3·2^x − 4.
Show answer ▾Hide answer ▴
y = 3(1) − 4 = −1, so the intercept is (0 ; −1)
10
Does y = (½)^x increase or decrease?
Show answer ▾Hide answer ▴
Decreases — the base is between 0 and 1.