Midterm II Review for Math 121, Summer 2012

Topics we covered.

  1. circles
    1. using the standard form $(x-h)^2+(y-k)^2=r^2$
      1. read off the center
      2. read off the radius
      3. graph
      4. find $x$- and $y$-intercepts
    2. getting to the standard form from the general form $x^2+y^2+ax+by+c=0$.
      1. complete the square twice
    3. building an equation for a circle from given geometric information
      1. e.g.,when told the coordinates of the ends of a diameter, find the midpoint to get the center of the circle, find the distance between these endpoints to get twice the radius of the circle...
  2. variation
    1. terms
      1. direct
      2. inverse
      3. joint
      4. constant of proportionality
    2. plugging in given numerical information to get the formula, using it to get numerical information with new values of the independent variable(s)
  3. basics of functions
    1. a relation between two sets
    2. when a relation is a function
    3. the domain and range of a function
    4. plugging values into functions -- uses parentheses
    5. sum, difference, product, and quotient of two functions, and their domains
    6. graphing a (real-valued) function (of a real variable)
    7. the Vertical Line Test
    8. reading function values off a graph
    9. plotting a function by putting in lots of $x$ values
    10. even and odd functions
    11. increasing and decreasing functions
    12. local and absolute maxima and minima of a function
    13. average rate of change of a function over an interval in its domain, which is the slope of the secant line
  4. library of functions
    1. $x$
    2. $x^2$
    3. $x^3$
    4. $1/x$
    5. $\sqrt{x}$
    6. $\sqrt[3]{x}$
    7. constant functions
    8. $|x|$
    9. step function, greatest integer function $\operatorname{int}$
  5. piecewise defined function
    1. graphing
    2. writing the expression, given a graph
  6. graphing functions by transformations:
    1. the graph of $f(x)+c$ in terms of the graph of $f(x)$
    2. the graph of $f(x+c)$ in terms of the graph of $f(x)$
    3. the graph of $cf(x)$ in terms of the graph of $f(x)$, where $c>1$
    4. the graph of $cf(x)$ in terms of the graph of $f(x)$, where $0
    5. the graph of $-f(x)$ in terms of the graph of $f(x)$
    6. the graph of $f(cx)$ in terms of the graph of $f(x)$, where $c>1$
    7. the graph of $f(cx)$ in terms of the graph of $f(x)$, where $0
    8. the graph of $f(-x)$ in terms of the graph of $f(x)$
    9. combining transformations