Exponential and Trigonometric Graphs
Beyond straight lines and parabolas lies a fascinating world of graphs that model rapid growth and repeating cycles. This guide will help you master the key features of exponential and trigonometric graphs for your higher-tier GCSE Maths exam.
Key Graphs to Know for Higher GCSE
Exponential Growth ($y = k^x$, $k > 1$)
Shows rapid growth. Always passes through $(0,1)$ and never touches the x-axis (its asymptote).
Exponential Decay ($y = k^x$, $0 < k < 1$)
Shows rapid decay. Also passes through $(0,1)$ and never touches the x-axis.
Sine Graph ($y = \sin x$)
A wave that starts at the origin $(0°, 0)$. Repeats every $360°$. Values range from −1 to 1.
Cosine Graph ($y = \cos x$)
A wave that starts at its maximum $(0°, 1)$. Repeats every $360°$. Values range from −1 to 1.
Tangent Graph ($y = \tan x$)
A series of repeating curves separated by vertical asymptotes (e.g., at $x = 90°$). Repeats every $180°$.
Worked Examples
Plotting an Exponential Graph
Complete the table for $y = 3^x$ and plot the graph.
| $x$ | −2 | −1 | 0 | 1 | 2 |
|---|---|---|---|---|---|
| $y$ | $\frac{1}{9}$ | $\frac{1}{3}$ | 1 | 3 | 9 |
Plot these points to create a smooth curve that passes through $(0,1)$ and gets closer to the x-axis for negative $x$ values.
Sketching a Trigonometric Graph
Sketch $y = \cos x$ for $0° \le x \le 360°$.
To sketch this, remember the key landmark points:
- Starts at maximum $(0°, 1)$.
- Crosses the x-axis at $90°$.
- Reaches minimum $(180°, -1)$.
- Crosses the x-axis again at $270°$.
- Finishes the cycle at $(360°, 1)$.
Connect these points with a smooth wave.
Tutor Insights
🤔 Common Misunderstandings
- Exponential Graphs: Confusing $y = x^2$ (a parabola) with $y = 2^x$ (exponential growth), or drawing the graph touching the x-axis.
- Trig Graphs: Mixing up the sine and cosine graphs. Remember: “Sine starts at the origin, Cosine starts at the crest”.
📝 Common Exam Mistakes
- Missing labels on axes or key points.
- Incorrect key points, especially for trig graphs.
- Forgetting asymptotes for the tangent graph.
- Drawing jagged lines instead of smooth curves.
Practice Questions
- Sketch the graph of $y = \sin x$ for $0° \le x \le 360°$. Label the coordinates of the maximum and minimum points.
- Which equation could represent a graph showing exponential decay?
a) $y = x^2$, b) $y = 0.5^x$, c) $y = 2^x$. - State two differences between the graph of $y = \sin x$ and the graph of $y = \cos x$.
Show Answers
- Your sketch should be a smooth wave starting at $(0°, 0)$, with a maximum at $(90°, 1)$ and a minimum at $(270°, -1)$.
- Answer: b) $y = 0.5^x$. The base ($k$) is between 0 and 1.
- Two differences: 1. Sine starts at $(0°, 0)$ while Cosine starts at $(0°, 1)$. 2. Sine’s first peak is at $90°$ while Cosine’s first peak is at $0°$.
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