Nonlinear Functions and their Graphs Lesson 4.1

Polynomials General formula a0, a1, … ,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 Consider what happens when x gets very large negative or positive Called “end behavior” Also “long-run” behavior Basically the leading term anxn takes over Compare f(x) = x3 with g(x) = x3 + x2 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 ) A function, f is an increasing function if the values of f increase as t increases The average rate of change > 0 A function, f is an decreasing function if the values of f decrease as t increases The average rate of change < 0

Extrema of Nonlinear Functions Given the function for the Y= screen y1(x) = 0.1(x3 – 9x2) 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 • Local maximum f(c) ≥ f(x) when x is near c Local minimum f(n) ≤ f(x) when x is near n c n

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

Extrema of Nonlinear Functions The calculator can find maximums and minimums When viewing the graph, use the F5 key pulldown menu Choose Maximum or Minimum Specify the upper and lower bound for x (the "near") Note results

Assignment Lesson 4.1A Page 232 Exercises 1 – 45 odd

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

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

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

Assignment Lesson 4.1B Page 234 Exercises 45 – 69 odd