2 Power Function Definition Where k and p are constantsPower functions are seen when dealing with areas and volumesPower functions also show up in gravitation (falling bodies)
3 This is a power function Direct ProportionsThe variable y is directly proportional to x when: y = k * x(k is some constant value)AlternativelyAs x gets larger, y must also get largerkeeps the resulting k the sameThis is a power function
4 Direct Proportions Example: The harder you hit the baseballThe farther it travelsDistance hit is directly proportional to the force of the hit
5 Direct Proportion Suppose the constant of proportionality is 4 Then y = 4 * xWhat does the graph of this function look like?
6 Again, this is a power function Inverse ProportionThe variable y is inversely proportional to x whenAlternatively y = k * x -1As x gets larger, y must get smaller to keep the resulting k the sameAgain, this is a power function
7 Inverse ProportionExample: If you bake cookies at a higher temperature, they take less timeTime is inversely proportional to temperature
8 Inverse Proportion Consider what the graph looks like Let the constant or proportionality k = 4Then
9 Power Function Looking at the definition Recall from the chapter on shifting and stretching, what effect the k will have?Vertical stretch or compressionfor k < 1
10 Special Power Functions Parabola y = x2Cubic function y = x3Hyperbola y = x-1
12 Special Power Functions Most power functions are similar to one of these sixxp with even powers of p are similar to x2xp with negative odd powers of p are similar to x -1xp with negative even powers of p are similar to x -2Which of the functions have symmetry?What kind of symmetry?
13 Variations for Different Powers of p For large x, large powers of x dominatex5x4x3x2x
14 Variations for Different Powers of p For 0 < x < 1, small powers of x dominatexx4x5x2x3
15 Variations for Different Powers of p Note asymptotic behavior of y = x -3 is more extreme0.520100.5y = x -3 approaches x-axis more rapidlyy = x -3 climbs faster near the y-axis
16 Think About It… Given y = x –p for p a positive integer What is the domain/range of the function?Does it make a difference if p is odd or even?What symmetries are exhibited?What happens when x approaches 0What happens for large positive/negative values of x?
17 Formulas for Power Functions Say that we are told that f(1) = 7 and f(3)=56We can find f(x) when linear y = mx + bWe can find f(x) when it is y = a(b)tNow we consider finding f(x) = k xpWrite two equations we knowDetermine kSolve for p
18 Finding Values(8,t)Find the values of m, t, and k