Study Guide for Exam2, MAT170
Section 5.1 – Vertical and Horizontal shifts
Given a function f(x), know and understand how to interpret the vertical shift, written f(x)+k, and the horizontal shift, written f(x+h). Summary: given f(x), you can define a new function y=f(x+h)+k, where h is the shift left/right, and k is the shift up/down. For example, if , then is the original parabola moved left 2 and down 4. Use your graphing calculator to assist. Know how to work with functions defined by tables. Look over all the examples in this section.
Section 5.2 – Reflections and symmetry
· Study pages 189-190 for a good introduction of the effects of negating the input and/or output of a function. Examples 1 on page 190 is also good. The summary is given in the box on page 191.
· Know how to recognize symmetry, pages 191-193. An Even function is symmetric across the y-axis (and || are good examples). An Odd function is symmetric about the origin, (is a good example).
Study all examples in this section.
Section 5.3 – Vertical stretches and compression.
This section is summarized on page 200. Negative stretch factors are covered on page 199. Remember: the vertical stretch affects the y-coordinate, not the x-coordinate. Example 3 is good, combining stretches with shifts.
Section 5.4 – Horizontal stretches and compression.
If f(kx) represents a horizontal stretch of a function, then the stretch factor is 1/k. Compare example 1 and example 2. The summary is given on page 209. The horizontal stretch factor affects the x-coordinate, not the y-coordinate.
Know how to combine all the transformations in Section 5.1-5.4.
Study all examples in this section.
Section 8.1 – Composition of functions
· Know how to compose two (or more) functions given by tables or by formulas.
· Know how to decompose functions pages 333-4. Sometimes it’s useful to think of an “inside” function and an “outside” function. See example 5.
Section 8.2 – Inverse functions
· Know the definition of an inverse function, page 338.
· Know how to show whether a function has an inverse function (horizontal line test).
· Know how to find the inverse of a function (solve for the input variable).
· Know how to decipher in words the meaning of expressions such as and in context.
· Know how to graphically show inverses (reflection across 45° line) and how to use restricted domains when applicable (page 345).
· Know how to show whether two functions are inverse of one another (their composition will equal x). See page 343-4.
· Know how the domain and range of a function and the domain of range of the inverse are related, page 342-3.
Study all examples in this section.
Section 9.1 - Power Functions
· Recognize the general form of a power function, page 366. Powers can also be fractional (roots) and negative. Graphs on pages 367-9 show examples of positive integer powers, fractional and negative integer powers.
· Be able to recognize the effect of the value of the power on the graph. Page 370 has good examples.
Section 9.2 - Comparing Power, Exponential, and log functions
This section compares power functions to log and exponential functions. Graphs of these superimposed upon one another are found on pages 374-375. Log functions generally grow the slowest, while exponential functions (with a base > 1) grow faster than any power function, regardless of the power and the leading coefficient. Page 374 top-right has a good example.
Section 9.3 - Polynomial Functions
· Be able to recognize the general behavior of a polynomial function (definition and samples on page 380).
· Be able to locate roots of a polynomial by algebraic and graphical means (x-intercepts of the graph) . Recognize patterns for the various polynomials (for instance, a cubic polynomial has up to 3 real roots, an nth degree poly has up to n real roots, etc).
· Be familiar with the long-run behavior of a polynomial. When viewed on a large enough scale the graph of the polynomial looks like the graph of its leading term.
Section 9.4 – The Short-Run behavior of polynomials
· Be able to determine roots of a polynomial.
· Understand what is meant by the multiplicity of a root and how the graph behaves at multiple zeros (see box, bottom of page 388).
· Be able to determine the formula for a polynomial given its graph by looking at the long term behavior, at the zeros, at the multiplicity of the zeros and at any other point you are given on the graph. Good examples are found on pages 388-389.
Section 9.5 – Rational Functions
· Be able to recognize the general form of a rational function (with polynomials for numerator and denominator), page 393.
· Be able to discern the long term behavior of a rational function (see box, bottom of page 394). This involves comparing the degrees of the numerator and denominators. Pages 394-5 have one graph of each case (degrees same, degree top less than bottom, degree bottom less than top).
Section 9.6 – The short – run behavior of ration functions.
· Be able to determine vertical asymptotes (set bottom = 0 and solve).
· Be able to determine y-intercept (set x = 0) and x-intercept(s) (set y = 0 and solve; usually means top = 0).
· Be able to sketch an accurate graph of a rational function by means of identifying vertical and horizontal asymptotes, x and y intercepts, and common sense.