Algebra II Recipe: Functions and Their Graphs

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

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.

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.

1. Substitute the given values in place of x.
2. Follow order of operations to find the numerical value of the function.