A function is even when f(−x) = f(x): symmetric about the y-axis. Odd when f(−x) = −f(x): rotational symmetry about the origin. Most are neither. The leading term alone decides a polynomial’s end behavior.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Try it first
f(x) = x⁴ − 3x². Work out f(1), f(−1), f(2) and f(−2). What do the four values show?
Stuck is fine. Let’s Learn shows the way next. Got it first time? You can jump ahead to Watch an Example.
Substitute −x
x⁴ − 2x²: f(−x) = x⁴ − 2x² = f(x), even. x² + x: f(−x) = x² − x, neither f(x) nor −f(x). Substitute rather than trusting the picture; a small window can hide the asymmetry.
A shortcut for polynomials
All even powers: even. All odd powers: odd. A mix: neither. Cosine is even and sine is odd, the same way.
End behavior
For a polynomial, the leading term settles it alone. At x = 100, x³ is a million and 4x is four hundred; the rest is swamped. Even degree: both ends the same way. Odd: opposite ways. A negative coefficient flips both.
An example
−2x⁴ + x³ + 7: even degree, negative coefficient. Both ends fall. The middle terms shape the bumps, but the ends are fixed by −2x⁴.
Continuity
A function is continuous where its graph has no break. Breaks appear at holes, at jumps and at vertical asymptotes. Polynomials never break; rational functions break where the denominator is zero.
Read before you compute
Symmetry halves the work: an even function needs plotting on x ≥ 0 only. End behavior says which way the ends go. A break says where to be careful. Three facts, read before any table is built.
Step 2: Try It Yourself
Tap and try it out.
Analyze g(x) = −2x⁴ + x² and h(x) = (x + 1)/(x − 3): symmetry, end behavior, breaks
- 1g(−x) = −2x⁴ + x² = g(x): evenall even powers
Step 0 of 4
Step 3: In Real Life
Reading a stock chart
Before computing anything, an analyst reads a chart’s shape: where it heads at the ends, whether it breaks, whether it mirrors. End behavior and symmetry tell the story before the numbers do.
Step 4: Watch an Example
One step at a time.
Watch Noor Read f(x) = 2x⁵ − 8x³ + x
A fresh polynomial, read for all three facts — symmetry, end behavior and continuity — and checked with numbers at the end.
- Step 1
Substituting −x is the whole symmetry test, and it is worth doing term by term. (−x)⁵ is five minus signs multiplied together, and they cancel in pairs until one is left over: −x⁵. (−x)³ leaves one over too: −x³. So f(−x) = 2(−x⁵) − 8(−x³) + (−x) = −2x⁵ + 8x³ − x.
Watch Ana Read g(x) = −3x⁴ + 5x² − 1
The same three readings on a quartic. Ana does the substitution; you name the symmetry, the ends and the check.
- Step 1
Substituting −x leaves this rule exactly as it was. (−x)⁴ is four minus signs, which cancel in pairs, and (−x)² is two, which also cancel: g(−x) = −3x⁴ + 5x² − 1, the very expression Ana started with, the constant included.
Watch Sam Find the Breaks in h(x) = (2x + 1)/(x² − 4x + 3)
A quotient rather than a polynomial, so continuity is suddenly a real question. Sam factors the bottom; you count the breaks.
- Step 1
A polynomial never breaks, but this is one polynomial divided by another, and division by zero is the one piece of arithmetic with no answer at all. So the breaks, if there are any, sit exactly where the denominator is zero, and nowhere else.
Step 5: Your Turn
Practice makes it stick.
The Garage
Problem 1 of 2
A downtown garage charges $5 for any time up to and including 1 hour, $10 for any time up to 2 hours, and $15 for any time up to 3 hours. A graph of cost in dollars against time parked runs from the moment a car arrives to 3 hours. How many times does that graph jump?
The Model
Problem 2 of 2
A print shop’s monthly profit, in dollars, is modeled by P(x) = −2x⁴ + 40x² − 50, where x is the number of thousands of flyers printed. The model was fitted to figures for x between 0 and 4. Someone asks what it says for a very large x, far past the data. Which is it?
Read the Shape
Three readings, one rule at a time: replace x by −x for the symmetry, find the highest power for the ends, and look underneath for the breaks.
1 of 5
f(x) = x⁶ − 4x². Even, odd or neither?
2 of 5
f(x) = 5x⁵ − 2x³. Even, odd or neither?
3 of 5
f(x) = −4x³ + x². As x runs far to the left, does the graph rise or fall?
4 of 5
f(x) = 3x⁶ − 20x⁴. Which describes the two far ends?
5 of 5
h(x) = (x + 5)/(2x − 7). At which value of x is h discontinuous?
Whose Step?
One line on each page is where the slip starts. Find that line, then put it right.
1 of 6
Priya decides whether f(x) = x³ + 5x is even, odd or neither. One line on her page is where the slip starts. Which?
Priya tests f(x) = x³ + 5x
- 1f(−x) = (−x)³ + 5(−x)
- 2= x³ − 5x
- 3x³ − 5x is not f(x), and it is not −f(x)
- Answerf is neither even nor odd
2 of 6
Priya’s line 2, put right: type f(−x) for f(x) = x³ + 5x.
3 of 6
Dev reads the far ends of f(x) = 6x³ − 2x⁴ off the page below. One line is where the slip starts. Which?
Dev reads the ends of f(x) = 6x³ − 2x⁴
- 1f(x) = 6x³ − 2x⁴
- 2leading term 6x³
- 3odd degree, positive coefficient
- Answerfalls to the far left, rises to the far right
4 of 6
Dev’s page, put right. Which describes the two far ends of f(x) = 6x³ − 2x⁴?
5 of 6
Maya decides where h(x) = (x² − 9)/(x − 3) breaks. One line is where the slip starts. Which?
Maya finds the breaks in h(x) = (x² − 9)/(x − 3)
- 1h(x) = (x² − 9)/(x − 3)
- 2x² − 9 = (x − 3)(x + 3)
- 3h(x) = x + 3
- Answera polynomial, so h never breaks
6 of 6
At which value of x does h(x) = (x² − 9)/(x − 3) break?
Pick the Move
Decide which move answers the question asked. Nothing here needs working all the way through.
1 of 2
f(x) = x⁴ − 5x² has to be sketched between x = −4 and x = 4. Which move halves the plotting, and is allowed here?
2 of 2
g(x) = (x + 6)/(x² − 5x). Which move finds every x where g breaks?
Same Behavior?
Judge these without plotting anything. Select every rule the question describes.
1 of 2
Select every rule whose graph is its own mirror image in the y-axis.
2 of 2
Select every rule whose two far ends go the same way as each other.
Step 6: Quick Check
Show what you know.
Question 1 of 2
f(x) = x⁴ − x². Even, odd or neither?
Question 2 of 2
What decides the end behavior of a polynomial?
Ready for more?Optional. Past the lesson, for anyone who wants it.
Stretch 1 of 2
f is an even function and g is an odd function, and neither of them is the rule that gives 0 at every input. What kind of function is their product p(x) = f(x)g(x)?
Stretch 2 of 2
A function f is even and odd at the same time, and it is defined for every real number. What is f(2)?
What You Learned
- Even means f(−x) = f(x); odd means f(−x) = −f(x); most functions are neither.
- Substitute −x rather than trusting the picture.
- The leading term alone decides a polynomial’s end behavior.
- Discontinuities appear at holes, jumps and vertical asymptotes.
For the grown-up
The misconception this lesson targets is that symmetry and end behavior can be read off a picture. A curve drawn between −2 and 2 can look symmetric when it is not, and a curve that has not turned yet can look as though it never will; both readings are settled by substituting −x and by finding the term of highest degree, and neither needs a plot at all. Two cousins of it show up in the practice: the leading term is the highest power and not the first term written, so 6x³ − 2x⁴ is governed by its −2x⁴; and cancelling a factor out of a fraction does not heal a break, because the input that made the bottom zero is still an input the original rule has no value at. When a student is sure from the picture, the fastest question to ask is "what is f(−x)?" — one line of algebra outranks any window.