Study Guide for Exam 1
Study Guide for Exam 1 Math 11010-001
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This 7 page Study Guide was uploaded by Sarah Cross on Monday February 8, 2016. The Study Guide belongs to Math 11010-001 at a university taught by Tracy A. Laux in Spring 2016. Since its upload, it has received 45 views.
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Date Created: 02/08/16
Algebra for Calculus Study Guide for Exam 1 Created By: Sarah Cross Professor Tracy Laux Section 2.1 Terms Function- Any equation whose graph passes the vertical line test, and where each x-value corresponds with only one y-value. Domain- the set of all possible inputs that do not make the function undefined. How far the graph extends on the x-axis. Range- The set of all possible outputs. How far the graph extends along the y-axis. Skills Evaluating Functions Practice Problems p.192 # 35,36,37,38 Net Change- Equation: f (b) - f (a) Practice Problems for Net Change p. 193 # 39,40,41,42 Difference Quotient- Equation: f(a+h) – f(a)/ h Practice Problems p.193 # 43, 44, 45, 46, 47, 48, 49, 50 Piecewise Defined Functions Practice Problems p.191 # 31, 32, 33, 34 Section 2.2 Terms Linear Function: y = mx + b *slope is represented by the variable m * Y-intercept is represented by the variable b Constant Function: f(x) = b *The graph of a constant function is a horizontal line Power Functions: f(x) = x^n Root Functions: f(x) = x^1/n Skills Become Familiar with the Graphs of Common Functions: f(x) = x^2, f(x) = x^3, f(x) = √x, f(x) = │x│ Power Functions that are EVEN take on the shape of a PARABOLA Power Functions that are ODD take on the shape of the f(x) = x^3 graph Graphing Piecewise Functions < or > is depicted on a graph by an OPEN hole; not included in the graph ≤ or ≥ is depicted on a graph by a CLOSED hole; included in the graph Graph of the Greatest Integer Function This problem is done exactly the same as a piecewise function, except all of the functions are constant (horizontal lines). Vertical Line Test The vertical line test is done on graphs to determine whether or not it is a function. It is done by drawing a vertical line through the graph. If the line touches the graph in more than one place, it is NOT a function. Section 2.3 All that is needed for this section is the ability to use your graphing calculator, and the ability to interpret graphs. Section 2.4 Terms Average Speed = distance traveled/ time elapsed Average rate of change = change in y/ change in x = f (b) – f (a)/ b – a Net Change: f (b) – f (a) Difference Quotient: used to calculate instantaneous rates of change Skills Ability to calculate Net Change, and Average Rate of Change Difference Quotient If a function is increasing on an interval, the average rate of change between any two points is positive. If a function is decreasing on an interval average rate of change between any two points is negative. Linear Functions Have CONSTANT Rate of Change Section 2.5 Skills Identifying Linear Functions (expressed as f(x) = ax +b) Graphing Linear Functions Slope and Rate of Change Slope: y2 – y1/ x2 – x1 OR f(x2) – f(x1) / x2 – x1 “The difference between “slope” and “rate of change” is simply a difference in point of view” (Stewart 228). Making a Linear Model from a Rate of Change Making a Linear Model from a Slope Making a Linear Model involving Speed Section 2.6 Transformation of Functions Skills A constant added onto a function causes the graph to shift vertically. If the constant is positive it moves up and if it is negative it moves down. (p. 234) Vertical Shifting Horizontal Shifting Combined Horizontal and Vertical Shifts Reflecting Graphs If the equation reads y= -f (x) graph is reflected over x-axis If the equation reads y= f(-x) graph is reflected over y-axis Vertical Stretching and Shrinking To vertically stretch or shrink a graph, multiply the y-value by the number you are stretching/ shrinking by. If the function is vertically stretched/shrunk, the equation will look like… y=cf(x) o If the c value is greater than 1, vertically stretch the graph by multiplying y-values by c. o If the c value is greater than 0, but less than 1, vertically shrink the graph by multiplying the y-values by c. Horizontal Stretching and Shrinking To horizontally stretch or shrink a graph, you must multiply the x- values of the coordinate points by 1/c. o The equation looks like… y = f (cx) o If the c value is greater than 1 the graph shrinks by multiplying the x-values by 1/c o If the c value is greater than 0, but less than 1 stretch the graph by multiplying the x-values by 1/c. Even and Odd Functions Even and Odd functions deal with symmetry. o ODD functions are symmetrical over the ORIGIN o EVEN functions are symmetrical over the Y-AXIS o Functions CANNOT be symmetrical over the x-axis because it would fail the vertical line test. *Study Guide made in reference to College Algebra 7E, by James Stewart *
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