- Difficulty
- Introductory
- Suggested time
- 2 min
- Marks
- 1
Power functions and first sketches practice question 5
The graph of one of these power functions is shown, with the points and marked. Which function is it?
Compare each power with itself for : which powers make a small number bigger, and which make it smaller? Then check what happens for negative .
At , but , so which powers climb steeply out of the origin? Then test the choices at against the marked point .
Theory UBC Math 100 §1.2Key results
Worked solution
Strategy
Use two features of the graph: how it leaves the origin compared with far away, and what it does for negative x.
Working
For 0 < x < 1 a power above 1 shrinks x and a power below 1 enlarges it. The graph leaves the origin steeply, so the power is below 1.
The graph is below the axis for negative x, which rules out the two-thirds power.
Far from the origin the cube root grows slowly, which is why the graph flattens out.
Answer
The answer is y equals x to the power one third, the cube root of x.
Check
Swapping x and y in the picture gives a curve through the same marked points, flat at the origin and steep far out: y = x³, the inverse of x1/3.
Where this goes wrong
Choosing x³ because it also passes through (−1, −1), (0, 0) and (1, 1). The shape tells them apart: x³ is flat near the origin and steep far away, and the cube root is steep near the origin and flat far away.
Try a similar question
Let . Close to , which term dominates, and what does the graph look like there?Saved in this browser only.