Nonlinear Functions and their Graphs Lesson 4 1

Nonlinear Functions and their Graphs Lesson 4. 1

Polynomials l General formula l l a 0, a 1, … , an are constant coefficients n is the degree of the polynomial Standard form is for descending powers of x anxn is said to be the “leading term”

Polynomial Properties l Consider what happens when x gets very large negative or positive l l Called “end behavior” Also “long-run” behavior Basically the leading term anxn takes over Compare f(x) = x 3 with g(x) = x 3 + x 2 l l Look at tables Use standard zoom, then zoom out

Increasing, Decreasing Functions A decreasing function An increasing function

Increasing, Decreasing Functions Given Q = f ( t ) l A function, f is an increasing function if the values of f increase as t increases l l The average rate of change > 0 A function, f is an decreasing function if the values of f decrease as t increases l The average rate of change < 0

Extrema of Nonlinear Functions l Given the function for the Y= screen y 1(x) = 0. 1(x 3 – 9 x 2) l l l Use window -10 < x < 10 and -20 < y < 20 Note the "top of the hill" and the "bottom of the valley" These are local extrema • •

Extrema of Nonlinear Functions l Local maximum l l f(c) ≥ f(x) when x is near c Local minimum l f(n) ≤ f(x) when x is near n • c n •

Extrema of Nonlinear Functions l Absolute minimum l f(c) ≤ f(x) for all x in the domain of f • l Absolute maximum l f(c) ≥ f(x) for all x in the domain of f l Draw a function with an absolute maximum

Extrema of Nonlinear Functions l The calculator can find maximums and minimums l l l When viewing the graph, use the F 5 key pulldown menu Choose Maximum or Minimum Specify the upper and lower bound for x (the "near") Note results

Assignment l l l Lesson 4. 1 A Page 232 Exercises 1 – 45 odd

Even and Odd Functions l l If f(x) = f(-x) the graph is symmetric across the y-axis It is also an even function

Even and Odd Functions l l If f(x) = -f(x) the graph is symmetric across the x -axis But. . . is it a function ? ?

Even and Odd Functions l A graph can be symmetric about a point l l l Called point symmetry If f(-x) = -f(x) it is symmetric about the origin Also an odd function

Assignment l l l Lesson 4. 1 B Page 234 Exercises 45 – 69 odd