Cubic, Reciprocal and Exponential Graphs
Foundation & Higher — exponential graphs in section 5 are Higher.
Recognise and plot cubic and reciprocal graphs, then use their features to read values and solve equations. Higher students extend the same ideas to exponential growth and decay.
Before you start: Linear Graphs, Quadratic Graphs and Graphical Solutions and Powers and Roots.
1. Recognise graph families
Start with the rule, the intercepts and the behaviour on both sides of zero. A few familiar shapes help you recognise a family, but a supplied rule may have extra terms that change the detail.
| Rule | Characteristic feature |
|---|---|
| y = x³ | Passes through the origin; negative inputs give negative outputs. |
| y = 1/x | Two disconnected branches; x = 0 is undefined. |
| Higher: y = 2ˣ | Positive for every real x; grows by a fixed factor for equal steps in x. |
For an accurate plot, calculate a table, choose even axis scales, plot ordered pairs and draw a smooth curve. Check values around zero and near any bend. Do not join across an input where the rule is undefined.
2. Cubic graphs — Foundation & Higher
Worked example — plot y = x³
Cube each input: (−2)³ = (−2) × (−2) × (−2) = −8. A negative input cubed stays negative. This is different from squaring it, and different from calculating 3x.
| x | x³ |
|---|---|
| -2 | -8 |
| -1 | -1 |
| 0 | 0 |
| 1 | 1 |
| 2 | 8 |
Both intercepts are (0, 0). The graph has 180° rotational symmetry about the origin. Although it flattens there, it keeps increasing: the origin is not a maximum or minimum.
Worked example — a cubic with extra terms
For y = x³ − 2x, calculate afresh. At x = −1, y = −1 + 2 = 1. At x = 0.5, y = 0.125 − 1 = −0.875.
| x | x³ − 2x |
|---|---|
| -2 | -4 |
| -1.5 | -0.375 |
| -1 | 1 |
| -0.5 | 0.875 |
| 0 | 0 |
| 0.5 | -0.875 |
| 1 | -1 |
| 1.5 | 0.375 |
| 2 | 4 |
Read the x-axis crossings as x ≈ −1.4, 0, 1.4. The middle root is exactly zero by substitution. Unlike x³, this cubic has two turning points. Never assume that every cubic is increasing throughout or has only one root.
3. Reciprocal graphs — Foundation & Higher
Worked example — plot y = 1/x
| x | 1/x |
|---|---|
| −2 | −½ |
| −1 | −1 |
| −½ | −2 |
| ½ | 2 |
| 1 | 1 |
| 2 | ½ |
There is no value at x = 0 because division by zero is undefined. Leave a gap in a table and keep the branches separate. Positive x gives positive y; negative x gives negative y, so both branches matter.
An asymptote is a line a graph approaches. Here the axes x = 0 and y = 0 are asymptotes. As positive x approaches zero, y grows without bound; from the negative side y becomes very negative. As |x| grows, y approaches zero. This graph never touches either axis and has no intercepts.
For y = −1/x the signs reverse: the branches lie in the upper-left and lower-right quadrants. A reciprocal, 1/x, means “one divided by x”; it is not the same idea as an inverse function.
Worked example — read an equation graphically
To solve 1/x = 2, draw y = 2 and read its crossing with the positive branch: x = 0.5. To solve 1/x = −2, use y = −2 and the negative branch: x = −0.5. The axes themselves are not branches of the curve.
For positive quantities related by inverse proportion, see Inverse Proportion. Those contextual domains may use only one branch; the full algebraic graph uses both.
4. Plot an unfamiliar supplied rule
Worked example — y = x + 1/x
Do both operations. At x = 2, y = 2 + ½ = 2.5; at x = −2, y = −2 − ½ = −2.5. The rule is still undefined at zero.
| x | x + 1/x |
|---|---|
| −2 | −2.5 |
| −1 | −2 |
| −½ | −2.5 |
| ½ | 2.5 |
| 1 | 2 |
| 2 | 2.5 |
The positive branch bends at (1, 2); the negative branch bends at (−1, −2). Use extra points if you need a more accurate plot. No roots appear: for positive x both terms are positive, and for negative x both are negative. This sign argument confirms the reading without needing a solving formula.
A table or diagram for a new rule deserves its own checks. Do not borrow all the features of a familiar graph merely because one term looks similar.
5. Higher — exponential growth and decay
In y = kˣ, x is the exponent and k is a positive constant. For k > 1 the graph increases; for 0 < k < 1 it decreases; for k = 1 it is the constant line y = 1. In all three cases y = 1 when x = 0. These are not the power graphs y = xᵏ.
Worked example — compare 2ˣ and (½)ˣ
| x | 2ˣ | (½)ˣ |
|---|---|---|
| −2 | ¼ | 4 |
| −1 | ½ | 2 |
| 0 | 1 | 1 |
| 1 | 2 | ½ |
| 2 | 4 | ¼ |
For a positive base different from 1, the output stays positive. One end approaches the horizontal asymptote y = 0; there are no x-intercepts. The graphs above meet at (0, 1). Equal steps in x multiply y by the same factor, rather than adding a fixed amount.
Worked example — interpret a threshold
A supplied model is B(n) = 80 × 1.25ⁿ. Here B is a modelled quantity observed after n whole days. At n = 0, 1, 2 the values are 80, 100, 125. The graph starts at (0, 80). Its height is multiplied by 1.25 for each extra day.
Draw B = 150 and read the continuous crossing at n ≈ 2.8. But this question records whole days. B(2) = 125 and B(3) = 156.25, so the first recorded value above 150 is day 3. State the real domain: n = 0, 1, 2, …, not negative or fractional observations.
A model a × kˣ has y-intercept a because k⁰ = 1. If time is genuinely continuous, a fractional-time crossing may be meaningful. Let the context decide. Numerical repeated-change calculations belong in Compound Interest and Growth; powers and negative indices are in Powers and Roots.
Think Like a Mathematician
A decreasing graph is not automatically inverse proportion. For y = 1/x, the product xy is always 1. Higher y = (½)ˣ is also decreasing, but equal steps multiply the output by ½. The equation and the numerical pattern distinguish the families.
Common Mistakes
x³ is not 3x. Cube the whole input. Undefined is not zero. Do not plot 1/0. Separate branches stay separate. Do not draw a line across x = 0. Higher: kˣ and xᵏ are different rules, and a continuous graph extension does not automatically allow fractional days.
Practice — Try It Yourself
Work on squared paper where a graph is requested. Try each question before opening its answer. Graph readings are estimates: give only the precision your scale supports.
1. Match each rule to a shape and justify it: y = x³, y = 1/x, and Higher y = 2ˣ. Shapes: two branches; increasing S-shaped curve through the origin; always-positive increasing curve.
x³: the S-shaped curve, with (0, 0) and negative outputs for negative x. 1/x: two branches, with x = 0 excluded. Higher 2ˣ: always positive, increasing, with y-intercept 1. The intercepts and domain help confirm a visual match.
2. Plot y = x³ for x = −2, −1, 0, 1, 2. A student writes (−2, 8). Correct the point and explain the curve near the origin.
The values are −8, −1, 0, 1, 8. The correct point is (−2, −8), because three negative factors give a negative product. Compare with the worked cubic graph: smooth, increasing and flat near (0, 0), without turning back.
3. For y = 1/x, find y when x = −0.25 and x when y = 4. Explain why (0, 0) is invalid.
At x = −0.25, y = −4. If y = 4, x = 0.25. At x = 0 the division is undefined; it does not give zero. Neither axis is part of this graph.
4. On the supplied graph of y = x³ − 2x, estimate all roots and the value when x = 1.5.
Roots are about −1.4, 0, 1.4. At x = 1.5, read y ≈ 0.4; substitution gives 3.375 − 3 = 0.375, explaining the rounded reading.
5. Calculate y = x + 1/x at x = 4 and x = −4. Could this graph cross the x-axis? Give a reason.
Outputs are 4.25 and −4.25. It cannot cross: for nonzero x, x and 1/x have the same sign, so their sum cannot be zero. The rule is undefined at zero.
6. Sketch y = −1/x using x = −2, −1, −½, ½, 1, 2. State the quadrants and asymptotes.
Corresponding y-values: ½, 1, 2, −2, −1, −½. Draw two smooth branches in the upper-left and lower-right quadrants. Both axes remain asymptotes; leave x = 0 undefined and never join across it.
7. Higher: classify y = 3ˣ, y = 0.8ˣ and y = 1ˣ as growth, decay or constant. Give their common y-intercept.
3ˣ grows; 0.8ˣ decays; 1ˣ is the constant y = 1. All pass through (0, 1) because a positive number to the power zero is 1.
8. Higher: use the paired exponential graph to estimate the solution of 2ˣ = 3. Explain why x = 3 is not the answer.
Cross y = 2ˣ with y = 3: x ≈ 1.6 (about ±0.1 from a suitable grid). The 3 is the height, not the input. At x = 3 the height is 8.
9. Higher: for the whole-day model B(n) = 80 × 1.25ⁿ, estimate when the continuous curve reaches 150 and give the first whole day above 150.
Continuous reading: n ≈ 2.8. The first allowed whole day is 3, since B(2) = 125 and B(3) = 156.25. Do not report 2.8 as a recorded whole-day observation.
Frequently Asked Questions
Do cubic graphs always have three real roots?
No. x³ has one distinct real root, while x³ − 2x has three. Count crossings and consider the full shape; repeated roots can touch rather than cross.
Does an asymptote mean the curve stops?
No. For 1/x, the curve continues indefinitely while approaching an axis. The boundary of the picture is only the viewing window.
Do I need logarithms for the exponential questions here?
No. Use the supplied graph or a sensible table to estimate crossings, then interpret the answer in its domain.
Continue Learning
Previous: Quadratic Graphs and Graphical Solutions
Higher next: Trigonometric Graphs
Foundation applications: Inverse Proportion and Rates of Change.