2 Warm-up F(x) = 3x + 3 G(x) = x/3 - 1 F(6) G(21) F(-4) G(-9) F(0) G(3) Did you notice any relationship between the F functions and the G functions?
3 Warm-upWithout looking back at your notes, define domain and range in your own words.Using your definitions, what is the domain and range of the following graph? Assume that it doesn’t continue past this picture.
4 Warm-upFor the following graph, find domain, range, maximum, minimum, zeros (roots), y-intercepts, intervals of increase and decrease, and the end behavior.
5 What is a function family? A function family is a group of functions that all share the same characteristics. For example, all lines share the characteristics that they have a domain and range of all real numbers, they are continuous, and they have a constant rate of change.
6 Important Definitions X-intercepts/roots – any location where the value (output) of the equation is equal to 0. In a graph, this is where the graph crosses the x-axisY-intercepts – when the value of x = 0, we find our y-intercept. In a graph, this is where the graph crosses the y-axis.Domain – all possible x-valuesRange – all possible y-valuesMaximum – the ordered pair of the highest point on the graphMinimum – the ordered pair of the lowest point on the graph
7 Important Definitions Increasing intervals – the x-values of the graph between which the graph is going UP.Decreasing intervals – the x-values of the graph between which the graph is going DOWN.Constant interval – the x-values of the graph between which the graph is a STRAIGHT LINE.End Behavior – what is happening when the x-values are becoming more negative or more positive out of the graph.
8 PracticeWhat is the domain, range, maximum, minimum, and end behavior of each of the following?3. (-3, 5), (-5, 2), (4, -3), (7, 0)
9 6 Function Families Linear: y = x Quadratic: y = x2 Cubic: y = x3 Absolute Value: y = |x|Square root: y = √xRational: y = 1/x
10 Linear Functions Characteristics of a linear function Of the form y = xDomain: all real numbersRange: all real numbersWill have one root (x-intercept) and one y-interceptHas no maximum or minimum valueEntire function is increasingEnd behavior in opposite directions
12 Quadratic Functions Characteristics of a quadratic function (parabola) Of the form y = x2Domain: all real numbersRange: y ≥ 0 for parent graph.Minimum of 0 at the vertex in the parent graph.Can have 0, 1, or 2 roots (x-intercepts) and 1 y-intercept. Has 1 root in the parent graph – the vertex.End behavior in the same direction, up.Interval of decrease x < 0; Interval of increase x > 0
14 Cubic Functions Characteristics of a cubic function Of the form y = x3 Domain: all real numbersRange: all real numbersWill have neither a minimum nor a maximum value.Has 1 x-intercept (root) and 1 y-intercept: the origin (0,0)End behavior in opposite directions: to negative infinity as x approaches negative infinity; to positive infinity as x approaches positive infinityInterval of increase: all real numbers or (-∞, ∞)
16 Absolute Value Functions Characteristics of an absolute value functionOf the form y = |x|Domain: all real numbersRange: y ≥ 0 for parent graph.Will have a minimum at the vertex: (0, 0)Has 1 root (x-intercept) and 1 y-intercept: (0, 0)End behavior in the same direction, up.Interval of decrease: x < 0; Interval of increase: x > 0
18 Square root Functions Characteristics of an absolute value function Of the form y = √xDomain: x ≥ 0 for the parent graph.Range: y ≥ 0 for parent graph.Minimum value at the vertex: (0, 0)1 root (x-intercepts) and 1 y-intercept: (0, 0)End behavior to positive infinity.Interval of increase: x > 0 or [0, ∞)
20 Rational Functions Characteristics of a rational function Of the form y = 1/xDomain: x ≠ 0 for the parent graph.Range: y ≠ 0 for parent graph.Will have neither a maximum nor a minimumHas neither a root (x-intercept) nor a y-intercept in the original function. Instead, has a vertical asymptote that on the y-axis and a horizontal asymptote on the x-axis.End behavior to 0 on both sides of the graph.Interval of decrease: all real numbers except x ≠ 0 or(-∞, 0) U (0,∞)
22 TransformationsWhat happens when you add or subtract a constant from a parent function?The function shifts up or down the amount of your constant.What happens when you make a parent function negative?The function is reflected across the x-axis.
23 Example of Vertical Translation y = x2 y = x2 - 4