Power Functions (part 2)

1. Power Functions (part 2) I. Function Symmetry. A) Functions can have symmetry that is even or odd. 1) Even symmetry means it is symmetrical to the y-axis. (The left side looks like the right side). a) It looks the same on both sides of the y-axis. 2) Odd symmetry is also called point symmetry, it is a rotational symmetry. (If you rotate the function about the origin (0,0), you will get the same shape again before one complete revolution.)

2. Power Functions (part 2) II. Finding Even/Odd symmetry mathematically. A) A function y = f(x) is even if f(-x) = f(x). 1) If you replace all the x’s with –x’s and simplify, you get the original function. Example: f(x) = 3x2 – 5  3(-x)2 – 5  3x2 – 5 which is f(x). B) A function y = f(x) is odd if f(-x) = –f(x). 1) If you replace all the x’s with –x’s and simplify, you get the original function with all the signs changed. If you then factor out a –1, you get the original function. In other words –1f(x) or –f(x). Example: f(x) = 4x3 – 7x  4(-x)3 – 7(-x)  – 4x3 + 7x Now factor out a –1 which gives us –1(4x3 – 7x) which is –f(x).

3. Power Functions (part 2) III. Non-mathematical way to determine even/odd. A) This is NOT an approved method for determining even & odd symmetry. This is only for checking your answers. 1) Even: All the exponents are even (or have no variable). Example: y = -5x6 + 9x4 – 8 is even. proof: -5(-x)6 + 9(-x)4 – 8  -5x6 + 9x4 – 8 which is the same as the original function – so it is even. 2) Odd: All the exponents are odd (or have no exponent). Example: y = -5x7 + 9x3 – 8x is odd. proof: -5(-x)7 + 9(-x)3 – 8(-x)  5x7 – 9x3 + 8x Note: all the signs are switched compared to the original f(x). Factor out a -1 and all the signs are the same, -1(-5x7 + 9x3 – 8x) so f(-x) = -1f(x) which is the definition of odd symmetry.

4. Power Functions (part 2) IV. Direct, Proportional and Inverse Variation. A) Direct (or proportional) variation is when you have an equation in the form of y = kx (where k = a constant #) 1) As the value of x gets bigger, the value of y increases. 2) As the value of x gets smaller, the y decreases also. Example: The wingspan of a plane varies directly to its weight and length. wingspan = (weight)(length) B) Inverse variation is when you have the equation y = k/x 1) As the value of x increases, the value of y decreases. 2) As the value of x decreases, the y will increase. Example: The volume of a gas is inversely proportional to its temperature and pressure. Volume = temperature/pressure

5. Power Functions (part 2) V. Beginning behavior of Power Functions A) Remember that the sign of the coefficient “A” in y = Axn (where n > 0), determines if the f(x) goes up or down as x approaches infinity. B) The exponent, in conjunction with the end behavior, will determine how the graph begins, but the exponent must be a whole number (no fractions or decimals). 1) If the exponent is even, then the graph begins the same direction as it ended. (both up or both down). 2) If the exponent is odd, then the graph begins in the opposite direction as it ended. (one up, the other down). a) Think: Exponent is odd = opposite & even = same.