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Graphs of equations: intercepts and symmetry

Level: 101 · for anyone starting precalculus

One line: The graph of an equation is the set of every point whose coordinates make it true, and the rest of the section is that sentence applied: a point is on the graph if it passes the equation, an intercept is a point of the graph with one coordinate 0, and a symmetry is a mirror image that passes the equation too.

The idea

An equation in x and y, such as y = x² − 4, is a test that each point of the plane passes or fails. (2, 0) passes, since 0 = 4 − 4; (1, −2) fails, since −2 ≠ 1 − 4. The graph of the equation is the set of all the points that pass. That is the whole definition, and it is the same idea as the solution set of a linear equation: an equation in two unknowns has infinitely many solutions, each a pair (x, y), and plotting them all draws the graph.

The book's section 1.2 has three objectives, and all three are this definition put to work:

  1. Is a point on the graph? Substitute its coordinates. The equation holds, or it does not.
  2. Intercepts. The points where the graph meets an axis. A point on the x-axis has y = 0, so to find the x-intercepts, set y = 0 and solve for x; for the y-intercepts, set x = 0 and solve for y.
  3. Symmetry. A graph is symmetric about the y-axis if the mirror image (−x, y) of every point on it is also on it, and so on for the other mirrors. To test, replace x by −x and see if you get an equivalent equation.

This page follows section 1.2, Graphs of Equations in Two Variables; Intercepts; Symmetry, in Michael Sullivan's Precalculus.

Symbol by symbol

Piece Say it What it means
graph of an equation the set of all points (x, y) whose coordinates satisfy it. Not "the drawing": the drawing is a picture of the set.
satisfies "satisfies" makes the equation true when its coordinates are put in for x and y.
x-intercept "x-intercept" the x-coordinate of a point where the graph meets the x-axis. A number, not a point: the x-intercepts of y = x² − 4 are −2 and 2.
y-intercept "y-intercept" the y-coordinate of a point where the graph meets the y-axis.
symmetric about the x-axis whenever (x, y) is on the graph, so is (x, −y): the lower half is the upper half reflected.
symmetric about the y-axis whenever (x, y) is on it, so is (−x, y): left and right halves mirror each other.
symmetric about the origin whenever (x, y) is on it, so is (−x, −y): turning the graph half a turn about O leaves it unchanged.

A word on the intercepts: Sullivan defines them as numbers, the coordinates, and some other books mean the points. Say which you mean if it matters: "the x-intercept 3" and "the point (3, 0)" are safe either way.

The three symmetry tests

Symmetric about Replace Example that passes
the x-axis y by −y x = y², since x = (−y)² is the same equation
the y-axis x by −x y = x² − 4, since (−x)² = x²
the origin x by −x and y by −y y = x³ − x, since −y = −x³ + x is y = x³ − x times −1

"Equivalent" allows rearranging: multiplying both sides by −1, as in the origin test above, gives back the same equation, so the test passes. A test fails if a single point on the graph has a mirror image that is not, and the program finds such a witness: (2, 6) is on y = x³ − x, and (−2, 6) is not, so the graph is not symmetric about the y-axis.

Two consequences, both in the program:

  • Any two symmetries force the third. Reflecting in the x-axis and then in the y-axis is the same as half a turn about the origin, so an equation that passes two tests passes all three. The ellipse x² + 4y² = 16 does.
  • A graph of y = f(x) is almost never symmetric about the x-axis. If (x, y) and (x, −y) are both on it, one x has two y's, which a function cannot have unless y = 0.

What the program prints

The pictures mark with * every point with whole-number coordinates that satisfies the equation exactly, and with o every square of the grid the curve passes through between them. The intercepts and symmetries are computed exactly with fractions.

Verified output of graphs_intercepts_symmetry.py — regenerated by tools/run_examples.py, never hand-typed.

1. A GRAPH IS A SET OF POINTS: y = x^2 - 4
     6   .  .  .  o  .  .  |  .  .  o  .  .  .
     5   .  .  .  *  .  .  |  .  .  *  .  .  .
     4   .  .  .  o  .  .  |  .  .  o  .  .  .
     3   .  .  .  o  .  .  |  .  .  o  .  .  .
     2   .  .  .  o  o  .  |  .  o  o  .  .  .
     1   .  .  .  .  o  .  |  .  o  .  .  .  .
     0   -  -  -  -  *  -  +  -  *  -  -  -  -
    -1   .  .  .  .  o  .  |  .  o  .  .  .  .
    -2   .  .  .  .  o  o  |  o  o  .  .  .  .
    -3   .  .  .  .  .  *  |  *  .  .  .  .  .
    -4   .  .  .  .  .  o  *  o  .  .  .  .  .
    -5   .  .  .  .  .  .  |  .  .  .  .  .  .
    -6   .  .  .  .  .  .  |  .  .  .  .  .  .
        -6 -5 -4 -3 -2 -1  0  1  2  3  4  5  6   x
   * is a whole-number point (x, y) that makes the equation true exactly;
   o is a square the curve passes through between such points.

2. IS THE POINT ON THE GRAPH? SUBSTITUTE AND SEE
   ( 2,  0):  y =  0,  x^2 - 4 =  0   on the graph
   ( 1, -3):  y = -3,  x^2 - 4 = -3   on the graph
   ( 1, -2):  y = -2,  x^2 - 4 = -3   not on it
   (-3,  5):  y =  5,  x^2 - 4 =  5   on the graph
   ( 0,  4):  y =  4,  x^2 - 4 = -4   not on it
   (0, 4) fails although (0, -4) passes: a sign is a different point.

3. INTERCEPTS: SET THE OTHER COORDINATE TO 0, THEN SOLVE
   equation              x-intercepts (y = 0)    y-intercepts (x = 0)
   y = x^2 - 4                          -2, 2                      -4
   x = y^2                                  0                       0
   y = x^3 - x                       -1, 0, 1                       0
   x^2 + 4y^2 = 16                      -4, 4                   -2, 2
   y = x + 2                               -2                       2
   y = x^2 + 1                           none                       1
   An intercept is a single number, a coordinate: the x-intercepts of
   y = x^2 - 4 are -2 and 2, and the points are (-2, 0) and (2, 0).
   y = x^2 + 1 never reaches the x-axis: x^2 = -1 has no real solution.
   x = y^2 has 0 as both: the graph goes through the origin.

4. SYMMETRY: DOES THE MIRROR IMAGE OF THE EQUATION SAY THE SAME THING?
   equation              x-axis    y-axis    origin
   y = x^2 - 4               no       yes        no
   x = y^2                  yes        no        no
   y = x^3 - x               no        no       yes
   x^2 + 4y^2 = 16          yes       yes       yes
   y = x + 2                 no        no        no
   y = x^2 + 1               no       yes        no
   x-axis: replace y by -y.  y-axis: replace x by -x.  origin: replace both.
   If the new equation is equivalent to the old one, the graph has that symmetry.

5. THE TEST, BY HAND, ON y = x^3 - x
   origin: -y = (-x)^3 - (-x) = -x^3 + x; multiply by -1: y = x^3 - x.  Same: symmetric.
   y-axis:  y = (-x)^3 - (-x) = -x^3 + x.  Different equation: not symmetric.
   A witness for 'not': (2, 6) is on the graph, its y-axis mirror (-2, 6) is not: F(2, 6) = 0, F(-2, 6) = -12.

6. ANY TWO SYMMETRIES FORCE THE THIRD
   Reflecting in the x-axis and then the y-axis is a half turn about the origin:
   (x, y) -> (x, -y) -> (-x, -y). So x^2 + 4y^2 = 16, with both axis symmetries,
   gets origin symmetry for free. The ellipse:
     6   .  .  .  .  .  .  |  .  .  .  .  .  .
     5   .  .  .  .  .  .  |  .  .  .  .  .  .
     4   .  .  .  .  .  .  |  .  .  .  .  .  .
     3   .  .  .  .  .  .  |  .  .  .  .  .  .
     2   .  .  .  o  o  o  *  o  o  o  .  .  .
     1   .  .  o  o  .  .  |  .  .  o  o  .  .
     0   -  -  *  -  -  -  +  -  -  -  *  -  -
    -1   .  .  o  o  .  .  |  .  .  o  o  .  .
    -2   .  .  .  o  o  o  *  o  o  o  .  .  .
    -3   .  .  .  .  .  .  |  .  .  .  .  .  .
    -4   .  .  .  .  .  .  |  .  .  .  .  .  .
    -5   .  .  .  .  .  .  |  .  .  .  .  .  .
    -6   .  .  .  .  .  .  |  .  .  .  .  .  .
        -6 -5 -4 -3 -2 -1  0  1  2  3  4  5  6   x

7. A GRAPH OF y = f(x) WITH x-AXIS SYMMETRY IS FLAT
   If (x, y) and (x, -y) are both on it, one x has two y's unless y = 0.
   So among graphs of functions, only y = 0 is symmetric about the x-axis;
   x = y^2 is symmetric about it precisely because y is not a function of x there.

Questions

1. Is (1, −3) on the graph of y = x² − 4? Is (−3, 5)? Is (0, 4)?

Answer (1, −3): 1 − 4 = −3, yes. (−3, 5): 9 − 4 = 5, yes. (0, 4): 0 − 4 = −4, not 4, so no. The point (0, −4) is on it.

2. Find the intercepts of 4x² + 9y² = 36.

Answer y = 0: 4x² = 36, x² = 9, x-intercepts −3 and 3. x = 0: 9y² = 36, y² = 4, y-intercepts −2 and 2.

3. Find the intercepts of y = x² + 1. What does the answer say about the picture?

Answer x = 0 gives the y-intercept 1. y = 0 gives x² = −1, which has no real solution: no x-intercepts. The graph stays above the x-axis; it is a parabola with its lowest point at (0, 1).

4. Test y = x⁴ − 2x² for symmetry.

Answer y-axis: (−x)⁴ − 2(−x)² = x⁴ − 2x², the same: symmetric. x-axis: −y = x⁴ − 2x² is a different equation: not. Origin: −y = x⁴ − 2x², not. Only even powers of x means y-axis symmetry.

5. Test y = x³ for symmetry.

Answer Origin: −y = (−x)³ = −x³, so y = x³: symmetric. y-axis: y = −x³, different: not. x-axis: −y = x³, different: not.

6. A graph is symmetric about the y-axis and (3, −2) is on it. Name another point on it. What if it is symmetric about the origin instead?

Answer y-axis: (−3, −2). Origin: (−3, 2). About the x-axis it would be (3, 2).

7. A graph is symmetric about both axes. Must it be symmetric about the origin?

Answer Yes. From (x, y), x-axis symmetry gives (x, −y), and then y-axis symmetry gives (−x, −y), which is origin symmetry.

8. Symmetric about the origin: does that mean the graph goes through the origin?

Answer No. xy = 1 is symmetric about the origin, since (−x)(−y) = xy, but (0, 0) fails: 0 ≠ 1. Symmetry says where points come in pairs, not which points are there.

9. Why are the x-intercepts found by setting y = 0, and not x = 0?

Answer The x-intercepts are where the graph meets the x-axis, and every point of the x-axis has y = 0. It sounds backwards and is the most common slip in the section.

10. Is (0, 0) an x-intercept or a y-intercept of y = x³ − x?

Answer Both: the point is on both axes, so 0 is an x-intercept and also the y-intercept. The x-intercepts are −1, 0 and 1.

11. Can the graph of an equation of the form y = (an expression in x) be symmetric about the x-axis?

Answer Only if it is y = 0. Otherwise some x would need two y's, y and −y. That is why x = y², solved for x, is the book's x-axis example.

12. The program tested symmetry by comparing F(−x, y) with F(x, y) at a grid of fractions. Why is that trustworthy for these equations?

Answer F(−x, y) − F(x, y) is a polynomial of degree at most 3, and a nonzero polynomial of low degree cannot vanish on that many points of a grid, so agreeing everywhere on the grid means agreeing everywhere. For an equation with square roots or fractions in x, the sample test would need more care.

How to practise

  1. Two lines per intercept problem: "y = 0: …" and "x = 0: …". Writing the substitution first prevents the swap in question 9.
  2. Symmetry: substitute, simplify, compare. Write the new equation in full, then decide whether it is the old one, maybe times −1.
  3. Guess before testing. Look at the powers: only even powers of x suggests y-axis symmetry, only even powers of y x-axis symmetry, and all odd powers in both origin symmetry. Then test, because mixed cases fool the guess.
  4. The cards, especially trap.

Flashcards

The page as a deck of Anki cards: graphs_intercepts_symmetry.txt. Import with File → Import. Tags: definition, intercepts, symmetry, problems, trap.

Where this goes next

Two families of graphs get sections of their own: straight lines, whose equations are first degree, in lines and slope, and circles, which are the distance formula held fixed, in circles. Symmetry comes back in chapter 3 of the book as even and odd functions: y-axis symmetry is even, origin symmetry is odd.

Po polsku, w skrócie

Wykres równania to zbiór wszystkich punktów (x, y), których współrzędne spełniają równanie. Cała reszta wynika z tego zdania. Czy punkt leży na wykresie: podstaw i sprawdź. Miejsca przecięcia z osiami: punkt na osi x ma y = 0, więc żeby znaleźć przecięcia z osią x, podstawiamy y = 0 i rozwiązujemy; dla osi y podstawiamy x = 0. Pułapka: brzmi to na odwrót i łatwo się pomylić.

Symetria: wykres jest symetryczny względem osi y, jeśli razem z punktem (x, y) zawiera (−x, y); test to zamiana x na −x i sprawdzenie, czy wychodzi to samo równanie. Względem osi x zamieniamy y na −y, względem początku układu obie zmienne. Dwie symetrie wymuszają trzecią. Symetria względem początku nie znaczy, że wykres przez niego przechodzi (xy = 1). Program rysuje wykresy na siatce tekstowej i sprawdza wszystko dokładnie.

Auf Deutsch: Stichwörter

Der Graph einer Gleichung ist die Menge ihrer Lösungen; Achsenschnittpunkte und Symmetrie liest man an der Gleichung ab.

Stichwörter: Graph einer Gleichung, Lösungsmenge, Nullstelle (x-Achsenabschnitt), y-Achsenabschnitt, Achsensymmetrie, Punktsymmetrie zum Ursprung, Probe durch Einsetzen.

See also