- Difficulty
- Introductory
- Suggested time
- 3 min
- Marks
- 2
Power functions and first sketches practice question 6
The graph of is shown, where and are nonzero constants. What are the signs of and ?
Near the lower power dominates; far from the higher power does. Match each term to the part of the graph it controls.
Near the origin , and far from it . Read off the shape of the graph in each place, then ask which sign of the coefficient gives that shape.
Theory UBC Math 100 §1.2Key results
Worked solution
Strategy
Near the origin x⁴ is tiny next to x², so the shape there is decided by a. Far away x⁴ swamps x², so the shape there is decided by b.
Working
The graph is an upward bowl at the origin, so a > 0.
The graph falls steeply far from the origin, so b < 0.
A check: the zeros at x = ±2 need −a/b > 0, so a and b have opposite signs. Here a = 4 and b = −1.
Answer
The answer is a is positive and b is negative.
Check
With a = 4 and b = −1, y = 4x² − x⁴ = 4 − (x² − 2)² has zeros at 0 and ±2 and peaks of height 4 at x = ±√2, as drawn.
Where this goes wrong
Letting the x⁴ term shape the graph near the origin. Near 0 the higher power is the smaller one, so the bowl at the origin belongs to ax² and the steep fall far away belongs to bx⁴.
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