Lauren Callahan and Cassie McClenaghan

Section 2.1: Functions  Relation- when the value of one variable is related to the value of a second variable  Function- the common link between 2 relations is each input corresponds to exactly one output INPUT OUTPUT XYZXYZ The set X is the domain. The corresponding Y is the range. The relation shown is a function because the input corresponds to only one output. It is not a function if at least one input has more than one output. f(x) is another term for ‘y’ To find the value of a function ‘f’ defined by f(x)= 2x 2 - 3x we take f(3) and solve the function by putting 3 where x is. For example, F(3)=2(3) 2 - 3(3)= 18-9=9

Section 2.2: Graph of a Function  Vertical Line Test- A set of points in the xy-plane is the graph of a function, if and only if, every vertical line intersects the graph in at most one point  A graph of a function is the collection of points (x,y) that satisfies the equation y=f(x).

Section 2.3: Properties of Functions EVEN AND ODD FUNCTIONS  A function is even, if and only if, whenever the point (x,y) is on a graph where (-x,y) is also on the graph. f(-x)=x  A function is odd, if and only if, whenever the point (x,y) is on a graph where (-x,-y) is also on the graph. f(-x)=-f(x)  To determine whether ‘f’ is even, odd, or neither, we replace x by –x in the equation INCREASING, DECREASING, AND CONSTANT GRAPHS  A function is increasing if for any x1 and x2, f(x1)<f(x2)  A function is decreasing if for any x1 and x2, f(x1)>f(x2)  A function is constant if for all choices of x, the values of f(x) are equal LOCAL MAXIMA AND MINIMA  A function has a maxima(greatest value) and a minima(smallest value) in a set of numbers. AVERAGE RATE OF CHANGE  The average rate of change of from a to be is defined as: f(b)-f(a)/b-a

Section 2.4: Library of Functions; Piecewise Defined Functions  1. Constant Function- f(x)=b  2. Identity Function- f(x)=x  3. Square Function- f(x)=x 2  4. Cube Function- f(x)=x 3  5. Square Root Function- f(x)=√x  6. Reciprocal Function- f(x)=1/x  7. Absolute Function- f(x)= |x|  8. Cube Root Function- f(x)= 3 √x  Piecewise Function- when functions are defined by more than one equation

Section 2.5: Graphing Techniques-Transformations Vertical Shift Up- if a positive number k is added to the right side of the equation y=f(x), the new equation would be y=f(x)+k Vertical Shift Down- if a positive number k is subtracted to the right side of the equation, the new equation would be y=f(x)-k Horizontal Shift Right- if in the equation y=f(x), x is replaced with (x-h), the new equation would be y=f(x-h) Horizontal Shift Left- if in the equation y=f(x), x is replaced with (x+h), the new equation would be y=f(x+h) Vertical Stretch- if the right side of a function y=f(x) is multiplied by positive number a, the graph of the new function is y=af(x). (multiply each y-coordinate by a) Horizontal Stretch- if the function y=f(x) is multiplied by a positive real number a, the graph of the new function is y=f(ax). (multiply each x-coordinate by a) X-axis Reflection- if the right side of the function y=f(x) is multiplied by -1, the graph of the new function is y=-f(x) Y-axis Reflection- if in the function y=f(x) the x is replaced by –x, then the new equation would be y=f(-x)

Section 2.6: Mathematical Models-Building Functions  Using the concepts from you can build and analyze functions.  Remember the distance formula