- Difficulty
- Standard
- Suggested time
- 4 min
- Marks
- 2
Power functions and first sketches practice question 8
Which description matches the graph of ?
Three questions settle the sketch: which term wins near 0, which wins far away, and where is y = 0?
Factor: . Then check what happens when is replaced by .
Theory UBC Math 100 §1.2Key results
Worked solution
Strategy
Sketch a two-term polynomial by finding the dominant term near zero, the dominant term far away, and the zeros in between. Symmetry is a bonus check.
Working
The lower power wins for small x, so the graph passes through the origin shaped like a cubic.
The higher power wins far away, and its sign is negative, so the graph falls for large positive x.
Three x-intercepts.
Every term is an odd power, so the function is odd: symmetric about the origin, not the y-axis.
Answer
The answer is Like x cubed near the origin, zeros at −1, 0 and 1, falling like negative x to the fifth for large x, symmetric about the origin..
Check
f(0.1) = 0.00099, close to 0.1³, while f(2) = −24 and f(−2) = 24: falling on the right and symmetric about the origin.
Where this goes wrong
Choosing the description in which the higher power rules near the origin. The higher power rules far away; near zero it is the lower power, which is why the graph leaves the origin looking like x³.
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