Cogito
Precalculus · Chapter 1 · Lesson 3
Symmetry, End Behaviour, and Continuity
Read a graph before you compute anything.
12 problems · about 22 minutes · TEKS P.2.A, TEKS P.2.I
What this lesson teaches
The student analyses functions for even and odd symmetry, end behaviour, and points of discontinuity.
- Even means f(−x) = f(x); odd means f(−x) = −f(x); most functions are neither.
- The leading term alone decides a polynomial’s end behaviour.
- Discontinuities appear at holes, jumps and vertical asymptotes.
Warm Up
Straightforward practice. Get the method working first.
5 problemsf(x) = x⁴ − x². Even, odd or neither? 1, 2 or 3.
Answer 1
Why Even.
What decides the end behaviour of a polynomial?
Answer The leading term: its degree and its sign.
Why The leading term.
f(x) = x². Even, odd or neither? 1, 2 or 3.
Answer 1
Why Symmetric about the y-axis.
f(x) = x³. Even, odd or neither? 1, 2 or 3.
Answer 2
Why Rotational symmetry about the origin.
f(x) = x² + x. Even, odd or neither? 1, 2 or 3.
Answer 3
Why Mixing even and odd powers destroys both symmetries.
Build It Up
The same ideas with more to keep track of.
3 problemsf(x) = 3x⁵. How many ends go up?
Answer 1
Why Odd degree sends the ends opposite ways.
f(x) = x⁶. How many ends go up?
Answer 2
Why Even degree, positive coefficient.
f(x) = 1 ÷ (x − 3). At which x is it discontinuous?
Answer 3
Why Where the denominator is zero.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsSort each function by its symmetry.
Answer Even: x⁴ − 3, cos x · Odd: x⁵ + x, sin x
Why Cosine keeps its value at a negative angle; sine flips.
f(x) = −x³. As x runs far right, rise or fall? 1 for rise, 2 for fall.
Answer 2
Why The negative flips the usual cubic.
The Test: f(x) = x⁴ + 2. Is it even, odd or neither? Enter 1 for even, 2 for odd, 3 for neither.
Answer 1
Why Even.
The Ends: f(x) = −2x⁴. As x runs far right, does the graph rise or fall? Enter 1 for rise, 2 for fall.
Answer 2
Why It falls.