A graph can mirror across the vertical axis or match itself after a half-turn around the origin. These are different symmetries. Even functions use equal outputs at opposite inputs; odd functions use opposite outputs. Before checking either pattern, confirm that the function actually allows both x and −x.
Predict
Predict whether the formula x² guarantees an even function on every domain. Restrict it to [0,3] and most opposite inputs disappear. Predict whether a graph symmetric at five sampled points must be symmetric everywhere. Without a specified connecting rule, an asymmetric bend between those points can remain unseen.
Experiment
Graphs P and Q. P is the navy peaked graph; Q is the red increasing graph. Both are complete line-segment models on the symmetric domain [−3,3].
Supplied Graph P and Graph Q are complete piecewise-linear functions on [−3,3]. Their vertices are connected by the shown straight segments.
x
P(x)
Q(x)
−3
−1
−2
−1
1
−1
0
2
0
1
1
1
3
−1
2
For P, compare points reflected across the y-axis. For Q, rotate each point by 180 degrees around the origin. Then explain why the connecting segments inherit the same symmetry. The interval domain is explicitly symmetric about zero.
Understand
A domain D is symmetric about zero when x in D implies −x in D. On such a domain, f is even if f(−x)=f(x) for every allowed x, and odd if f(−x)=−f(x). If the domain is not symmetric, the standard even/odd classification does not apply as either symmetry; we call the function neither. It is not enough that an algebraic formula would have the identity on a larger domain.
For an even function, the point (x,y) is paired with (−x,y), a reflection across the vertical axis. For an odd function it is paired with (−x,−y), a half-turn around the origin. Oddness does not mean every output is negative. If zero belongs to an odd function’s domain, its output there must be zero because f(0)=−f(0). An odd function whose domain excludes zero need not have an origin point at all.
To prove parity from a formula, replace x by −x and simplify while keeping the domain visible. Even powers remain unchanged and odd powers change sign, but mixtures can have neither symmetry. One counterexample pair is enough to disprove a claimed identity. To prove it for a whole formula, a few matching numerical cases are not enough.
For a complete piecewise-linear graph, it is sufficient to verify that every segment and endpoint maps to another segment or endpoint under the proposed symmetry. For an unspecified sampled curve, checking vertices only certifies symmetry of those observations, not of the unknown values between them. This distinction explains why the supplied graph’s joining rule matters.
Apply
Construct an even extension of a graph given only for nonnegative inputs by reflecting across the vertical axis. Construct an odd extension by a half-turn, checking that any assigned value at zero is compatible. Then restrict the resulting domain asymmetrically and explain why the classification changes even though the plotted formula has not changed.
Continue
Symmetry belongs to a function together with its domain. Use graph structure and algebraic evidence appropriately, and distinguish proof from a finite sample. Next, sequences introduce domains that are subsets of the integers, making the difference between plotted points and a continuous curve especially important.
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An odd function is defined at both 4 and −4, and f(4)=−7. Find f(−4).
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
An even function has h(5/2)=6. Find h(−5/2).
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Classify supplied Graph P and Graph Q using their complete piecewise-linear definitions, not only a few points. Explain the symmetry of each segment.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Classify f(x)=x² with domain [−1,3] as even, odd or neither under the standard symmetry definition.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
On the real line, classify x⁴−2x²+5, x³−4x and x²+x. Justify algebraically.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Can a function be both even and odd? Also explain whether a symmetric finite sample proves parity of an otherwise unspecified continuous function.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
