Forgot to bring your calculator? Don't worry, use ON LINE CALCULATOR. or try Maple Equation Solver or try the Interactive Maple. (First click on "software" and click on "generic Maple Forum Interface". Note that you need to know Maple Syntax to use this.

Steps of using Maple Forum Interface.

  • step 1: go to the web site according to the instructions given above.
  • step 2: type in Maple syntax within Maple input area.
  • step 3: click on "interpret Maple input", and wait for the answer. You need to scroll down the curser to read your answer.

    NOTE Youd are not required to know Maple syntax, usually, I will give you the sample syntax, what you do is to copy (Ctl+C) and paste (Ctl + V) into the "input" of the Maple Forum Interface.

    Example try " plot({x^3, (x-2)^3}, x=-5..5); " (copy ony things within " ").

    Shifting and Reflection Techiques:

  • Horizontal Shiftings: y = f(x + a) is a horizontal shifting of y = f(x). If a > 0, then the graph will be shifted to the left; if a < 0, then the graph will be shifted to the right.
  • Vertical Shiftings: y = f(x) + a is a vertical shifting of y = f(x). If a > 0, then the graph will be shifted up; if a < 0, then the graph will be shifted down.
  • y = - f(x) is a reflection of y = f(x). (Compare y = -x^2 with y = x^2).

    Standard Equation of a circle:

  • If A = (1,2) and B = (3,4). Can you find the standard equation of the circle which has AB as its diameter?
  • Write the equation of top half of the circle, which has center (1,1) and radius 3. Answer.
  • If "sqrt" stands for square root, can you graph y = sqrt(16-x^2) + 3?
  • Practice completing squares technique so you can convert a nonstandard equation to a standard equation.
  • Example: Describe the figure: 2x2 + 2y2 + 3x - 5y + 1=0. Answer.

    Exercises: Graph the following graphs together.

  • x2 + y2 = 4. (black)
  • (x + 1)2 + (y - 1)2 = 1/4. (blue)
  • (x - 1)2 + (y - 1)2 = 1/4. (red)
  • y + 1/2 + sqrt(1/4 - x2) =0. (green)

    Click here for answer.

    Line Equations

  • Know formulae for slope, point-slope, and slope-intercept.
  • Parallel lines (m1 = m2). Perpendicular lines (m1 = -1/m2).
  • Equation of a vertical line is x = a (note that the slope of a vertical line is undefined, so you can't use "point-slope" or "slope-intercept". Eaquation of a horizontal line is y = b. (note the slope of a horizontal line is 0).
  • Tips on Quiz #2

  • Example Given a point P =(-1, 2) and a line L1: 3x - y + 5 = 0.

    (a) Can you find the equation of the line L2, so that L2 passes through P and L2 is parallel to L1

    (b) Find the equatoinf the line L3 so that L3 passes through P and L3 is perpendicular to L1. Answer.

  • More Exercises on Semicircles and Shiftings (9/25).

  • (9/27) Exercise Given two points, A = (-1, 5) and B = (3, 3).

    (a) Find the line equation AB. Answer .

    (b) Find the standard equation of the circle, which has AB as its diameter.

    (c) Find the line equation for l1 so that l1 is perpendicular to line AB and passes through point A.

    (d) Find the line equation for l2 so that l2 is parallel to l1 and passes through the point B.

    Functions

  • Recall vertical line test, when will the graph represent a function?
  • Domain and Range:

    (a) When you know the graph of a function, you can tell what the domain (inputs or x) and range (outputs or y) are.

    Example : If f(x) = -2 - sqrt(1 - x^2), (the graph can be found in the previous exercies about semicircle) then domain of f = [-1, 1] and the range of f = [-3,-2].

    (b) For some functons, you will be only asked to find domain.

    Example: If f(x) = sqrt[ (x - 1)/(3*x - 1) ] then finding the domain of f amounts to solve the inequality (x - 1)/(3*x - 1) > = 0. Rewriting the inequality in the multiplication form and use the short cut, we get the domain of f = (-infinity, 1/3) union [1, infinity).

  • For further understandings of function, click here.

  • (10/1) Even functions: (a) f(x) = f(-x) (b) the graph should be symmetric to y - axis.

  • Odd functions: (a) f(x) = - f(-x) (b) the graph should be symmetric to the origin (0,0).

    Example 1 If f(x) = -2 - (1-x2)1/2 . Then f is an even function. (hint: the graph of this function is a semicircle which is symmetric to y - axis.

    Example 2 If f(x) = 10 x 3 - 8 x. then f is an odd function. (hint: check if f(x) = - f(-x).)

  • Tips on Test 2. There are 15 subproblems. (1) You need to know the shifting technique, after you get the graph, you will answer domain, range, and the interval(s) where f is increasing or decreasing. (2) You need to how to graph semicircles. From the graph, you need to answer questions listed in (1). (3) Given a function, you will be asked only to find the domain by solving an inequality. (4) Given two points, you will be asked to find the line equation and etc. There is such a problem in this web page. (5) Given a function, you need to know how to determine if it is an even or odd function without knowing its graph.