CHAPTER 1, SECTION 2 Functions and Graphs

Increasing and Decreasing Functions increasing functions rise from left to right  any two points within this interval must have a positive slope decreasing functions fall from left to right  any two points within this interval must have a negative slope constant functions follow a horizontal path from left to right  any two points within this interval must have a zero slope one function can follow increasing, decreasing, and constant patterns

Identify intervals on which the following functions are increasing, decreasing, and constant.

Boundedness bounded below a function is bounded below if there is some number b that is less than or equal to every number in the range of the function  this number is called a lower bound  same idea as a minimum bounded above a function is bounded above if there is some number b that is greater than or equal to every number in the range of the function  this number is called an upper bound  same idea as a maximum bounded – a function is bounded if it is bounded above and below

Determine whether the function is bounded above, bounded below, or bounded.

Local and Absolute Extrema local/relative maxima – a function has a local/relative maxima if there is some value f(x) that is greater than or equal to all range values in some interval of the function  if the point is greater than or equal to all range values, it is called the absolute maxima local/relative minima – a function has a local/relative minima if there is some value f(x) that is less than or equal to all range values in some interval of the function  if the point is less than or equal to all range values, it is called the absolute minima these points are also a part of what is called critical points of functions

Symmetry basic definition: the graph looks the same when folded according to the line of symmetry Symmetric with respect to the y-axis  Graphs that are symmetric with the y-axis are considered even functions, f(-x)=f(x) Symmetric with respect to the x-axis  Graphs that are symmetric with the x-axis are not functions, f(x)=-f(x) Symmetric with respect to the origin  Graphs that are symmetric with the origin are considered odd functions, f(-x)=-f(x)

Determine whether the functions are even, odd, or neither and describe their symmetry.

Horizontal Asymptotes Horizontal asymptotes  A line y=b that the graph approaches but never crosses  For rational functions:  If degree of num.>degree of den., then there is NO horizontal asymptote.  If degree of num.<degree of den., then y=0 is the horizontal asymptote  If degrees are equal, then the ratio of leading coefficients is the horizontal asymptote  Can be found by examining the graph and looking for the y- value the graph is approaching from both directions, but isn’t crossing.

Vertical Asymptotes Vertical asymptotes  A line x=a that the graph approaches but never crosses  Recall to find: find the value that makes the denominator=0, look for error on the table, look for an infinite discontinuity on the graph. *Functions can have both vertical and horizontal asymptotes *Some functions DO cross their asymptotes *Some functions have more than one horizontal asymptote – they cannot have more than 2 WHY? – extra credit

Find all horizontal and vertical asymptotes of the functions below.

What can we determine from a graph? Domain Range X-intercepts Y-intercepts Increasing, decreasing, constant intervals Values for x Values for y or f(x) Max/min Symmetry Discontinuities, asymptotes boundedness

Use the graph to determine each of the following: 1. Domain of f 2. Range of f 3. X-intercepts 4. Y-intercepts 5. Increasing intervals 6. Decreasing intervals 7. Values of x for which f(x)≤0 8. The relative max of f 9. F(-2) 10. Values of x for which f(x)=0 11. Is f even, odd, or neither? 12. Is f bounded?

In conclusion Exit Slip: With a partner, complete problem 80 on page 100 of your textbook Homework: