1.03 Even and Odd Symmetry (filled in) PDF

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MHF4U

1.3 EVEN AND ODD SYMMETRY

KEY FEATURES OF GRAPHS OF POLYNOMIAL FUNCTIONS WITH ODD DEGREE Positive Leading Coefficient





the graph extends from quadrant _____ to

Negative Leading Coefficient



the graph extends from quadrant _____ to

quadrant _________

quadrant _________

similar to the graph

similar to the graph

_________________

_________________

The domain is __________________ and the range is _____________________

KEY FEATURES OF GRAPHS OF POLYNOMIAL FUNCTIONS WITH EVEN DEGREE Positive Leading Coefficient

Negative Leading Coefficient

the graph extends from quadrant _____ to

the graph extends from quadrant _____ to

quadrant _________

quadrant _________

similar to the graph

similar to the graph

_________________

_________________



The domain is ______________________________ ; The range is _________________________ Even/Odd Degree vs. Even/Odd Symmetry A polynomial function can be described as even or odd depending on its degree. 3

2

eg. y  x + x  1 __ ____________________

4

y  3x + x  7 _________________________

Additionally, a function can be described as having even or odd symmetry based on symmetrical properties of the graph. Even Symmetry means that the function is symmetric, specifically, over the y-axis, as opposed to line symmetry where the function is symmetric over any line. Odd Symmetry means that the function is symmetric, specifically, about the origin, as opposed to point symmetry where the function is symmetric over any point.

Beside each of the functions shown below, write the type of symmetry demonstrated (i.e. even, odd, or neither).

1. f ( x)  2 x

3

4. f ( x)  2 x3 + 3x 2 + 5 x

5

3

7

3

2. f ( x)   x + 3x + 2 x

3. f ( x)  4 x + 3x + 2 x

5. f ( x)  2 x 5 + 3x 4 + 2 x  5

**6. f ( x)  2 x 7 + 3x 5  4 x + 5

Can odd degree functions ever have even symmetry? Explain.

Observe the exponents on each term in the examples in the table above. What conclusion can we draw about odd degree functions and their symmetry?

MHF4U

1.3 EVEN AND ODD SYMMETRY

Beside each of the functions shown below, write the type of symmetry demonstrated (i.e. even, odd, or neither).

1. f ( x)  x 4 + x3  2 x 2  3x + 1

2. f ( x)  x 4  2 x 2 + 1

3. f ( x)  x4 + 2 x3

**4. f ( x)  x 4  2 x 3 + x

5. f ( x)  2 x 4 + x 2

6. f ( x)  2 x4  x 2 + 4 x

Can even degree functions ever have odd symmetry? Explain.

Observe the exponents on each term in the examples in the table above. What conclusion can we draw about even degree functions and their symmetry?

Algebraically Verifying Even and Odd Symmetry Functions can be identified as having even or odd symmetry without graphing as they will follow certain algebraic patterns based on the properties of reflection. Functions with even symmetry will satisfy the equation: f ( x )  f ( x ) Functions with odd symmetry will satisfy the equation: f ( x )   f ( x ) Recall: -if an even function has symmetry, the symmetry can only be even (or line). -if an odd function has symmetry, the symmetry can only be odd (or point). Examples: Without graphing, determine if each polynomial function’s symmetry is even, odd, or neither. a)

y  5 x7  4 x4 + x

c)

y  x ( x + 2) ( x  2 )

2

2

b)

y  x6  5 x4 + 1

d)

y  x (x + 1)(x + 7)(x + 9)...