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 §1.1Key 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.
The idea behind it
The power functions and first sketches lesson, a figure to move, in a few steps
Try a similar question
Let . Close to , which term dominates, and what does the graph look like there?Saved in this browser only.