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Algebra 2

Symmetry about the vertical axis and the origin

Fold the page along the vertical axis, or turn it upside down. When does a graph land exactly on itself, and what does that tell you about it?

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Why this exists

Pania weaves borders on a loom, and she draws each design as a curve on squared paper first. She never charts the whole curve. Some borders are mirror images left to right, so she charts the right half and the left half follows. Others look the same when the cloth is turned upside down, and again half the chart is enough.

Many graphs are made of two matching halves in just these ways. Seeing it halves the work, and it is a check: a graph that should match itself and does not has a mistake in it.

Definition

A graph is symmetric about the vertical axis when folding the page along the y-axis lays the graph exactly onto itself. Point by point: whenever (a, b) is on the graph, so is (-a, b), at the same height on the other side of the axis.

A graph is symmetric about the origin when turning the page half a turn (180^\circ) about the origin lays the graph exactly onto itself. Point by point: whenever (a, b) is on the graph, so is (-a, -b), as far away on the far side of the origin.

\begin{aligned} \text{vertical axis:}&\quad (a, b) \mapsto (-a, b) \\ \text{origin:}&\quad (a, b) \mapsto (-a, -b) \end{aligned}

The first image is the reflection of a point over the y-axis, the second the rotation of a point by 180 degrees about the origin. Symmetry is a claim about every point: each one needs its partner.

Intuition

In Figure 1, fold along the y-axis and the right half of the W lands on the left half. Each dashed guide joins a point to its partner: it is level, and the axis cuts it in half.

x y −2 −1 1 2 4 8
Figure 1Example 1. The graph of y = x^4 - 2x^2. Each dashed guide joins a point to its partner, and the vertical axis cuts every guide in half.

In Figure 2, put a pin through the origin and spin the page half a turn: the peak on the left swings onto the dip on the right. Each guide runs through the origin, its midpoint.

x y −2 −1 1 2 −4 −2 2 4
Figure 2Example 2. The graph of y = x^3 - 3x. Each dashed guide passes through the origin, which sits at its midpoint.

A half turn is two folds in a row, first along the vertical axis, then along the horizontal one:

(a, b) \mapsto (-a, b) \mapsto (-a, -b)

To test a picture, pick a point, find where its partner should be, and look. Matching pairs make the symmetry believable; one missing partner rules it out.

Results

Symmetry about the vertical axis forces:

  • x-intercepts in pairs a and -a (the origin can be one on its own);
  • turning points in pairs (a, b) and (-a, b) of the same kind, peak with peak, dip with dip;
  • rising on a stretch to the right means falling on the mirror stretch to the left;
  • both ends heading the same way.

Symmetry about the origin forces:

  • x-intercepts in pairs a and -a, and if a function’s graph has a point above x = 0, that point is the origin;
  • a relative maximum at (a, b) paired with a relative minimum at (-a, -b);
  • rising on a stretch to the right means also rising on the matching stretch to the left;
  • the two ends heading opposite ways.

The parent graphs, each with a partner pair as evidence:

Parent graphSymmetric aboutA pair of points
y = x^2the vertical axis(3, 9) and (-3, 9)
y = \lvert x \rvertthe vertical axis(5, 5) and (-5, 5)
y = xthe origin(4, 4) and (-4, -4)
y = x^3the origin(2, 8) and (-2, -8)
\displaystyle y = \frac{1}{x}the origin(2, \tfrac{1}{2}) and (-2, -\tfrac{1}{2})
\displaystyle y = \sqrt{x}neither(4, 2) has no partner: nothing lies left of the axis

The graphs themselves are in the squaring and cubing functions, the absolute value graph and the square root and reciprocal functions.

Why it is true

The point rules. A fold along the y-axis carries a point a units right of the axis to a units left, at the same height: (a, b) goes to (-a, b). A half turn about the origin carries (a, b) through the origin to the point as far out on the other side, (-a, -b); the midpoint of the two is \displaystyle \left(\frac{a - a}{2}, \frac{b - b}{2}\right) = (0, 0).

Through the origin. If (0, c) is on a graph symmetric about the origin, so is its partner (0, -c). A function has one output at each input, so c = -c, and c = 0.

Peaks and dips. A fold keeps every height, so a point higher than its neighbors stays so: a peak’s partner is a peak. A half turn changes each height b to -b, so “higher than its neighbors” becomes “lower”: a peak’s partner is a dip.

Rising and falling. A fold reverses left and right but keeps heights, so the mirror stretch meets the same heights in reverse order: rising becomes falling. A half turn reverses the order and the heights; two reversals cancel, so rising stays rising. Ends follow suit: a fold keeps an end’s direction, a half turn reverses it.

Worked examples

Example 1

Folding a W

Figure 1 shows the graph of y = x^4 - 2x^2. Is it symmetric about the vertical axis, the origin, both or neither? Use the answer to list its x-intercepts and turning points.

Solution

(1, -1) and (-1, -1) are both on the graph, and so are (2, 8) and (-2, 8); the rule agrees, since x = 2 and x = -2 both give 16 - 8 = 8. Every point on the right has its mirror partner, so the graph is symmetric about the vertical axis.

It is not symmetric about the origin. The half-turn partner of (2, 8) would be (-2, -8), but the graph above x = -2 is at height 8.

The graph touches the x-axis at the origin and crosses it between 1 and 2, at \displaystyle x = \sqrt{2}, since x^2 = 2 makes x^4 - 2x^2 = 4 - 4 = 0. That crossing’s partner is a crossing too, so the x-intercepts are \displaystyle x = -\sqrt{2}, x = 0 and \displaystyle x = \sqrt{2}.

The dips at (1, -1) and (-1, -1) are a mirror pair of relative minimums, the same kind, as a fold requires. The peak at (0, 0) is on the axis and is its own partner.

Example 2

A half turn

Figure 2 shows the graph of y = x^3 - 3x. Decide which symmetry it has, and describe its turning points and ends.

Solution

(1, -2) and (-1, 2) are both on the graph, and so are (2, 2) and (-2, -2); each pair has the origin as its midpoint, and the origin itself is on the graph. The graph is symmetric about the origin.

It is not symmetric about the vertical axis: the mirror partner of (1, -2) would be (-1, -2), but the height at x = -1 is 2.

The turning points are a peak at (-1, 2) and a dip at (1, -2), one of each, because a half turn swaps peaks and dips. The right end rises and the left end falls. The x-intercepts pair up too: 0, \displaystyle \sqrt{3} and \displaystyle -\sqrt{3}, since x^2 = 3 makes x(x^2 - 3) = 0.

Example 3

Half a table is enough

Roksana is charting a bell-shaped border, y = \dfrac{6}{x^2 + 2} in Figure 3. For x = 0, 1, 2, 4 she gets the outputs 3, 2, 1, \tfrac{1}{3}. The graph is symmetric about the vertical axis. Fill in the outputs at x = -1, -2, -4 without computing, and find the highest point.

x y −4 −2 2 4 1 2 3
Figure 3Example 3. The graph of y = \dfrac{6}{x^2 + 2}, the same on both sides of the vertical axis.
Solution

The partner of (a, b) is (-a, b), at the same height, so the outputs at x = -1, -2, -4 are 2, 1 and \tfrac{1}{3}. A check: at x = -2 the rule gives \displaystyle \frac{6}{4 + 2} = 1.

The highest point is (0, 3): moving away from 0 either way makes x^2 + 2 larger and the output smaller. Symmetry says a single highest point had to be on the axis anyway. If it were at (a, b) with a \neq 0, its partner (-a, b) would be just as high, making two.

The output is never 0, since 6 over a positive number is positive, so both ends creep toward the x-axis from above: the same way, as a fold requires.

Example 4

Completing a half-drawn graph

Sakura has charted the right half of a design: a curve for x from 1 to 3 through (1, 3), (2, 0) and (3, -4). Complete it (a) so the whole graph is symmetric about the vertical axis, and (b) so it is symmetric about the origin.

x y −2 2 −4 4 A x y −2 2 −4 4 B
Figure 4Example 4. One right half, completed two ways. A is symmetric about the vertical axis, B about the origin.
Solution

(a) Each (a, b) gets the partner (-a, b), so the left piece runs through (-1, 3), (-2, 0) and (-3, -4). The right piece falls from left to right, and its mirror image rises, from -4 at x = -3 to 3 at x = -1. That is graph A in Figure 4.

(b) Each (a, b) gets the partner (-a, -b), so the left piece runs through (-1, -3), (-2, 0) and (-3, 4). It falls, from 4 to -3, just as the right piece falls. That is graph B.

(-2, 0) is in both answers: an x-intercept’s partner is an x-intercept either way, because -0 = 0. Both completions are functions, since every new point has x between -3 and -1, where nothing was drawn before.

Example 5

A center that moved

Olamide notices that y = x^3 - 3x + 2 is the curve of Example 2 lifted two units. Its table:

x-2-1012
y04204

Is it still symmetric about the origin?

Solution

No. (0, 2) is on the graph, and a function graph symmetric about the origin must pass through the origin. Nor is it symmetric about the vertical axis: the mirror partner of (1, 0) would be (-1, 0), but the height at x = -1 is 4.

The shape has kept its half turn, though. Pair (1, 0) with (-1, 4), and (2, 4) with (-2, 0): each pair has midpoint (0, 2), the point the origin was lifted to, and the curve turns onto itself about it. “Symmetric about the origin” names one particular center, and sliding a graph off it loses that symmetry though the shape is unchanged.

Example 6

Features from half a graph

Olamide’s next border curve is symmetric about the origin and has a point above every x. To the right of the axis he knows: the only x-intercept is at x = 4, there is a relative maximum at (2, 5), and the curve rises from x = 0 to x = 2 and falls for every x beyond 2. Find every x-intercept, the other turning point, where the curve rises, and how its ends behave.

Solution

The curve has a point above x = 0, so that point is (0, 0). With the partner of 4, the x-intercepts are x = -4, x = 0 and x = 4.

The relative maximum (2, 5) is partnered by a relative minimum at (-2, -5).

Under a half turn rising stays rising, so rising from 0 to 2 means rising from -2 to 0: the curve rises on the whole stretch from x = -2 to x = 2, and falls for x < -2 as it does for x > 2.

The right end goes down, and a half turn sends the ends opposite ways, so the left end goes up. It all fits: from high on the left, the curve falls to (-2, -5), rises through the origin to (2, 5), and falls through (4, 0) and on down.

Boundaries

Symmetry is about the whole graph. A picture shows only a window, and a graph can match itself inside the window and fail outside it, so matching pairs only make the symmetry believable. Proving it takes the rule. Ruling it out takes one point with no partner.

The inputs must be symmetric too. A point above x = a needs one above x = -a. A graph on one side only, like \displaystyle y = \sqrt{x}, has neither symmetry.

These are two particular symmetries. The mirror is the y-axis and the center is the origin. y = (x - 2)^2 is a mirror image about the line x = 2, and Example 5’s curve turns about (0, 2); neither has the symmetry of this lesson. Mirror lines in general are a wider idea.

A function’s graph cannot fold onto itself along the horizontal axis, unless it lies wholly on that axis, as y = 0 does: (a, b) and (a, -b) with b \neq 0 are two outputs for one input.

Symmetry is not a flip. Reflecting a graph makes a new graph; a symmetry is a move that leaves the graph as it was.

Where this goes next

The next step, even and odd functions, reads these two symmetries straight from the rule by replacing x with -x, which proves them for every point at once. The half turn is a shape’s rotational symmetry of order two, applied to a graph; the cube root graph joins the origin side of the table. Later, the cosine wave turns out to be symmetric about the vertical axis and the sine wave about the origin.

Exercises

Partners

  1. A graph is symmetric about the vertical axis and passes through (3, -7). Name another point it must pass through.
  2. A graph is symmetric about the origin and passes through (-5, 2). Name another point it must pass through.
  3. Is the set of points (1, 2), (-1, 2), (3, 0), (-3, 0), (0, 5) symmetric about the vertical axis, the origin, both or neither?
  4. Is the set of points (1, 4), (-1, -4), (2, -3), (-2, 3), (0, 0) symmetric about the vertical axis, the origin, both or neither?
  5. Is the set of points (1, 1), (-1, 1), (2, 3), (-2, -3) symmetric about the vertical axis, the origin, both or neither?

Spotting it on a graph

For Exercises 6 to 9, use Figure 5. Say whether the graph is symmetric about the vertical axis, about the origin, or neither, and give a pair of points as evidence.

x y −2 2 −1 1 A x y −2 2 −2 2 B x y −2 2 −2 2 C x y −2 2 −2 2 D
Figure 5Exercises 6 to 9. Which symmetry, if any?
  1. Graph A.
  2. Graph B.
  3. Graph C.
  4. Graph D.

Completing a graph

  1. The right half of a graph runs from (0, 1) through (2, 3) to (4, 0). The whole graph is symmetric about the vertical axis. Name the points on the left half that match (2, 3) and (4, 0).
  2. Can the right half in Exercise 10 be completed so that the whole graph is a function symmetric about the origin? Explain.
  3. A graph has a piece for x from 1 to 5 through (1, -2), (3, 1) and (5, 4). Complete it so it is symmetric about the origin: name the three matching points.
  4. A function’s graph is symmetric about the origin, has a point above every x, and passes through (1, 4), (2, 2) and (3, -1). Find its outputs at x = 0, x = -1, x = -2 and x = -3.

Parent graphs and their relatives

For each graph, say which symmetry it has and give a pair of points as evidence.

  1. y = \dfrac{1}{x^2}
  2. \displaystyle y = \sqrt[3]{x}
  3. y = 4
  4. y = -x^3

Features from symmetry

  1. A graph is symmetric about the vertical axis and has a relative minimum at (0, -2). To the right of the axis it has a relative maximum at (3, 7), and x-intercepts at x = 1 and x = 5 only. Give the matching features on the left, and the total number of x-intercepts.
  2. A graph is symmetric about the origin. It has a relative minimum at (1, -6) and rises from x = 1 to x = 4. Where is its matching turning point, of which kind, and what does the graph do from x = -4 to x = -1?
  3. A function’s graph is symmetric about the origin and its right end rises. What does its left end do?

Reasoning

  1. A graph is symmetric about the vertical axis and has exactly five x-intercepts. Explain why one of them must be x = 0.
  2. Explain why a function’s graph symmetric about both the vertical axis and the origin must lie on the x-axis.
  3. Pania says the graph of y = (x + 1)^2 has no symmetry at all. Is she right? Explain.
Answers
  1. (-3, -7), the same height on the other side.
  2. (5, -2): both coordinates change sign.
  3. The vertical axis only. (\pm 1, 2) and (\pm 3, 0) pair up, and (0, 5) is its own mirror partner. Not the origin: (0, 5) would need (0, -5).
  4. The origin only. (1, 4) pairs with (-1, -4), (2, -3) with (-2, 3), and (0, 0) with itself. Not the vertical axis: (1, 4) would need (-1, 4).
  5. Neither. The mirror partner of (2, 3), which is (-2, 3), is missing, and so is the half-turn partner of (1, 1), which is (-1, -1).
  6. The origin. (1, 1), (-1, -1) and the origin are on it. Not the vertical axis: the height at x = -1 is -1, not 1.
  7. Neither. (0, -1) is on it, so not the origin; the mirror partner of (-1, 2) would be (1, 2), but the height at x = 1 is -2. It is a mirror image about the line x = 1 instead.
  8. The vertical axis. (1, 2) and (-1, 2) pair, as do (2, -1) and (-2, -1). Not the origin: (0, 3) would need (0, -3).
  9. Neither. (0, 1) is on it, so not the origin; the mirror partner of (1, 2) would be (-1, 2), but the height at x = -1 is 0. It turns onto itself about (0, 1) instead.
  10. (-2, 3) and (-4, 0). The point (0, 1) is its own partner.
  11. No. (0, 1) would need the partner (0, -1), two outputs at x = 0, so it would not be a function.
  12. (-1, 2), (-3, -1) and (-5, -4), each (-a, -b) for a given (a, b).
  13. Output 0 at x = 0, then -4, -2 and 1. The graph passes through the origin, and each output on the left is the opposite of its partner’s.
  14. The vertical axis. (1, 1) and (-1, 1) are on it, as are (2, \tfrac{1}{4}) and (-2, \tfrac{1}{4}). Not the origin: the graph is above the x-axis everywhere.
  15. The origin. (8, 2) and (-8, -2) are on it, as are (1, 1) and (-1, -1). Not the vertical axis: the height at x = -8 is -2.
  16. The vertical axis: each (a, 4) has partner (-a, 4) on the line. Not the origin: (0, 4) would need (0, -4).
  17. The origin. (1, -1) and (-1, 1) are on it, as are (2, -8) and (-2, 8). Not the vertical axis: the height at x = -1 is 1.
  18. A relative maximum at (-3, 7) and x-intercepts at x = -1 and x = -5; the minimum is on the axis, its own partner. Four x-intercepts in all: -5, -1, 1 and 5.
  19. A relative maximum at (-1, 6), since a half turn swaps dips and peaks. From x = -4 to x = -1 the graph rises, because a half turn keeps rising as rising.
  20. It falls: heading left, it goes down, because a half turn sends the ends opposite ways.
  21. Each intercept a \neq 0 comes with a separate partner -a, so the nonzero intercepts come in pairs, an even number. Five is odd, so one intercept is its own partner, and only 0 is.
  22. If (a, b) is on the graph, so is its mirror partner (-a, b), and that point’s half-turn partner (a, -b). One output at x = a means b = -b, so b = 0 for every point.
  23. Partly. It has neither symmetry of this lesson: (0, 1) is on it but (0, -1) is not, and the mirror partner of (1, 4) would be (-1, 4), while the height at x = -1 is 0. But it is a mirror image about the line x = -1: (0, 1) and (-2, 1) are both on it.
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