Exponential and degree-based trigonometric graphs · Higher
| English | 中文 | Pinyin |
|---|---|---|
| period/ˈpɪərɪəd/ | 周期 | zhōu qī |
Which graph repeats and which keeps growing?
- A tide repeats, while a doubling population keeps growing. Which graph feature distinguishes these patterns?
- This lesson studies period 周期: The input interval after which a repeating graph has the same values again.
Choose the mathematical structure
- For y=k^x with k>0,k≠1, the y-intercept is 1; k>1 gives growth and 0<k<1 gives decay. In degrees, sine and cosine repeat every 360° with range [-1,1]; tangent repeats every 180° and is undefined at 90°+180°n. Label axes in degrees.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines period?
The input interval after which a repeating graph has the same values again.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
For y=2^x, x=-1,0,1,3 give 1/2,1,2,8. Sine has values 0,1,0,-1,0 at 0°,90°,180°,270°,360°. Cosine starts at 1; cos180°=-1. Tangent has tan45°=1 but no finite value at 90°. For k=1/2, increasing x produces decay, never a negative output.
Exponential and degree-based trigonometric graphs
For y=k^x with k>0,k≠1, the y-intercept is 1; k>1 gives growth and 0<k<1 gives decay
Compare the model with the worked case and explain one change.
Find 2³.
Three factors of 2 give 8.
Test a tempting shortcut
- Degrees and radians are different units. A steep tangent branch is not a finite point at its asymptote. Exponential growth is not repeated addition of a constant amount.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
The sine graph and tangent graph both have period 360° as their smallest positive period. This claim is false. Explain which definition or assumption it violates.
Find sin270°.
The unit-circle ordinate at 270° is -1.
The sine graph and tangent graph both have period 360° as their smallest positive period.
Degrees and radians are different units. A steep tangent branch is not a finite point at its asymptote. Exponential growth is not repeated addition of a constant amount.
Interpret a new situation
- AQA A12 Higher requires exponential functions with positive bases and sine/cosine/tangent for angles of any size in degrees. No differentiation or radian sector formula is introduced here.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find tan45°.
The right triangle has equal opposite and adjacent lengths.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 8300 · Higher · 3.2. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
The input interval after which a repeating graph has the same values again. Choose the relationship, show the method, check its assumptions and interpret the result.