Power functions and first sketches practice question 17
A student describes the graph of the function below for large negative as shown. Find the first line that is wrong.
Read each line as a claim you can test: put in a specific , large and negative or close to , and see whether the size and sign come out as the line says.
Test the sign with a number: what is ?
An even power never changes sign across zero; only the minus sign in front of it matters.
Theory UBC Math 100 §1.2Key results
Corrected working
Strategy
Far from the origin the highest power decides, near the origin the lowest does, and an even power is never negative.
Working
x6 is positive on both sides of zero, so −x6 is negative on both sides.
Near the origin the lower power wins; the graph touches the origin from above.
The zeros, for completeness.
Answer
The answer is the graph falls on both sides far out, touches the origin from above, and has zeros at 0 and plus or minus 1.
Check
y(−2) = 4 − 64 = −60, the same as y(2) because both powers are even, so the graph falls on both sides.
Where this goes wrong
Giving −x6 a sign that follows x. An even power is never negative, so −x6 is negative on both sides of zero.
The idea behind it
The power functions and first sketches lesson, a figure to move, in a few steps
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