Symmetry.pdf, Lecture notes of Calculus

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Symmetry

When we graphed y = x, y = x 2 , y = x , y = x 3 , y = 3 x, and y =

1 x

, we mentioned some

of the features of these members of the “Library of Functions”, the “building blocks” for much of the study of algebraic functions. Now we go deeper into the study of symmetry, exploring three main categories or types of symmetry.

TI-83 note: We will be using a variety of window settings, but unless otherwise indicated, the scales for both the horizontal and vertical axes will be 1.

Symmetry with respect to the y-axis :

The first function we’ll consider is the squaring function y = x 2 whose graph is the parabola shown below. We mentioned before that this is a simple quadratic function , and its main feature is its vertex, the point given by (0, 0). This graph clearly has symmetry about the y-axis (the line x = 0).

Notice the table values in pairs, like (-3, 9) and (3, 9), (-2, 4) and (2, 4), etc., and then read carefully the following generalization:

Notice in the table that f(-3) = f(3) = 9 and that f(-2) = f(2) = 4. When f(-x) = f(x) for all real values of x in the domain of a function, the function is called an even function , and the graph has symmetry with respect to the y-axis.

Another function in the “library” that has symmetry with respect to the y-axis is the absolute value function, (^) y = x , shown in the

figure to the right.

This, too, is an even function, with points such as (-3, 3) and (3, 3), (-2, 2) and (2, 2), (-1, 1) and (1, 1). The origin (0, 0) is its minimum point.

A graph is symmetric with respect to the y-axis if, for every point (x, y) on the

graph, the point (-x, y) is also on the graph.

Symmetry with respect to the x-axis :

A function cannot have symmetry with respect to the x-axis, so now we turn to a few relations to illustrate this type of symmetry.

The graph of y 2 = x is shown below. Compare the table for y = x 2 with the table below. These two relations have a special relationship; they are inverses of one another. (More on this, later!)

Another relation that has symmetry with respect to the x-axis is x = y , shown below.

This relation includes points such as (0, 0), (1, 1), (1, -1), (2, 2), and (2, -2).

Again, notice the relationship between this graph and its “cousin”, the absolute value function, y = x.

In our two examples, notice that a substitution of –y for y yields essentially the same equation:

(^ − y)

2 = y 2 = x and x = −y = y. Whenever this is the case, the relation has symmetry with

respect to the x-axis.

Our third and final type of symmetry to explore is the first that is not primarily reflectional, but rotational. Here we’ll begin with the algebraic generalization and then move into examples.

Symmetry with respect to the origin :

Here are a few examples.

x y 9 - 3 4 - 2 1 - 1 0 0 1 1 4 2 9 3

A graph is symmetric with respect to the x-axis if, for every point (x, y) on the

graph, the point (x, -y) is also on the graph.

A graph is symmetric with respect to the origin if, for every point (x, y) on the

graph, the point (-x, -y) is also on the graph.

Examples:

(1) Test each function for symmetry with respect to the y-axis, the x-axis, and the origin. Graph each function to verify your conclusions.

(a) y = 1 x^2 Solution: Since replacing x by -x yields an equivalent function y = 1

(^ − x)

2 =^

1 x^2

the graph is symmetric with respect to the y-axis. Substituting –y for y yields −y =

1 x^2

or y = −

1 x^2

, so this function isn’t symmetric with

respect to the x-axis. Similarly, replacing both –y for y and –x for x yields −y =

1

(^ − x)

2 or^ y^ =^ −^

1 x 2

, so this function also isn’t symmetric with

respect to the origin.

The graph below verifies symmetry with respect to the y-axis.

(b) f(x) = (x − 1 )

3

Solution: Replacing x by –x yields y = (− x − 1 )

3 , which is not equivalent to the original function. The graph, therefore, won’t have symmetry with respect to the y-axis.

Replacing y by –y yields -y = (x − 1 )

3

or y = - x( − 1 )

3 , so the graph won’t have symmetry with respect to the x-axis.

Replacing x by –x and y by –y yields -y = (-x − 1 )

3 , which simplifies to

y = − −( x − 1 )

3

or y = (x + 1 )

3 , so the graph won’t have symmetry with respect to the origin.

By the graph, it is true that the graph has point symmetry about (1,0) which is its inflection point, but it has none of the three symmetries we’re considering.

(c) x 2 + y 2 = 4

Solution: This circle relation has symmetry with respect to the y-axis, x-axis, and the origin. It also has reflectional symmetry over any line passing through the origin and rotational symmetry through any angle with the origin as a fixed point.

The 3 equations ( )-x

2

+ y 2 = 4, x 2 + ( )-y

2

= 4, and ( )-x

2

+ ( )-y

2 = 4 are each equivalent to x 2 + y 2 = 4 , so we have algebraic verification for symmetry with respect to the y-axis, x-axis, and the origin, respectively.

(2) Determine algebraically whether the given function is odd, even, or neither. Graph each function to verify your conclusions.

(a) f(x) = 3x

Solution: f (− x) = 3(−x) = 3x

−f(x) = − 3x

Since f(x) = f(-x), the function is even, and its graph has symmetry with respect to the y- axis. Since f(-x) ≠ -f(x), the function is not odd.

(b) g(x) = 1 2

x

Solution: g(-x) = 1 2

( )-x =^ −^

1 2

( )x

-g(x) = - 1 2

( )x

Since g(x) ≠ g(-x), the function is not even. Since g(-x) = -g(x), the function is odd, and its graph has symmetry with respect to the origin.

2. Graph each function to determine whether each function may have symmetry with respect to the y-axis, the x-axis, and the origin. Then verify your conclusion algebraically.

(a) y = x 4 − 9x 2 (b) y = x 3 − 27 (c) y + 2x = 0

(d) y = x^2 − 4 x

(e) x 2 = y + 5 (f) x = y 2 + 5

3. Determine algebraically whether the given function is odd, even, or neither.

(a) f x( ) = x 2 + 3 (b) g x( ) = x + x (c) y = 3 x

(d) h x( ) = 4x 3 − 6 (e) F(x) = 5 (f) G x( ) = 3 2x 2 − 1

4. Compare the graphs of f and g.

(a) Describe the effect of the absolute value sign on the graph.

(b) What kind of symmetry do the two functions have in common?

5. Complete the graph below so that it has the indicated type of symmetry.

(a) x-axis (b) y-axis (c) origin

f x( ) = x 2 − 3 g x( )^ =^ x^2 −^3

Solutions:

1. (a) y-axis (b) x-axis (c) y-axis (d) none
2. (a) The quartic function, y = x 4 − 9x 2 , has symmetry with respect to the x-axis, but it doesn’t have either of the other two symmetries.

y = (− x)

4

− 9 (− x)

2 = x 4 − 9x 2

(b) The cubic function, y = x 3 − 27 , has none of the three symmetries. It does, however, have point symmetry with respect to its inflection point, (0, -27). Note: The scale on the y-axis in the graph is 10.

(c) The linear function, y + 2x = 0, has symmetry with respect to the origin, but it doesn’t have either of the other two symmetries.

(-y) + 2(-x) = 0 is equivalent to y + 2x = 0.

(d) The rational function, y =

x^2 − 4 x

, has symmetry with respect to the origin, and it has neither of the other two symmetries.

(^ − y) =^

(− x)

2 − 4

(^ − x)

is equivalent to y =

x^2 − 4 x

(e) The quadratic function, y = x 2 − 5 , has symmetry with respect to the y-axis and none of the other two symmetries.

y = (− x)

2 − 5 = x 2 − 5

(f) The quadratic relation, x = y 2 + 5 , has symmetry with respect to the x-axis and none of the other two symmetries.

x = (− y)

2

• 5 = y 2 + 5