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Algebra II Recipe: Functions and Their Graphs
A. Definitions
  1. relation - a set of ordered pairs of input and output values
  2. domain - a set of input values, the x-values, and the input variable (x) is the independent variable.
  3. range - a set of output values, the y-values, and the output variable (y) is the dependant variable
  4. function - a relation that has exactly one output for each input OR one y-value for each x-value. The relation is not a function if an input value has more than one output.
  5. function notation - f(x) = -x2 - 3x + 5 or g(x) = 2x + 6
B. Ways to Identify Functions
  1. mapping - given a relation, match each x-value with with its y-value. If an x-value gets "mapped" to more than one y-value, then the relation is not a function. The relation {(-3, 3) (1, 1) (4, 4) (1, -2)} is not a function because the x-value 1 gets "mapped" to a y-value of 1 and -2
  2. vertical line test - a relation can be shown as a graph. If a vertical line touches the graph at exactly one point as it passes over the graph, then the relation is a function. A circle is not a function, but a parabola is a function.
C. Graphing Equations with One and Two Variables
  1. Graphing y = (any number)
    • It's a horizontal line through that number on the y-axis.
    • All points ont he horizontal line will have that same y-value.
  2. Graphing x =(any number)
    • It's a vertical line through that number on the x-axis.
    • All points on the vertical line will have that same x-value.
  3. Graphing an equation with x and y
    • Construct a table of values, choosing at least 5 x-values in the table.
    • Substitute the x-values into the equation to find the y-values and to complete the table.
    • Connect the points.
    • If the connection makes a line, the equation is linear.
    • If the connection makes a curve, the equation is NOT linear.
D. Evaluating Functions
  1. Substitute the given values in place of x.
  2. Follow order of operations to find the numerical value of the function.
When x = 2, evaluate f(x) = 3x + 2
When x = -4, evaluate f(x) = 2x2 - 1

G Redden

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