Function FamiliesLesson 1-5

Warm-upF(x) = 3x + 3 G(x) = x/3 - 11. F(6)2. G(21)3. F(-4)4. G(-9)5. F(0)6. G(3)Did you notice any relationship between the F

functions and the G functions?

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.

Warm-upFor the following graph, find domain, range,

maximum, minimum, zeros (roots), y-intercepts, intervals of increase and decrease, and the end behavior.

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.

Important DefinitionsX-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-axis

Y-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

Important DefinitionsIncreasing 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.

PracticeWhat is the domain, range, maximum,

minimum, and end behavior of each of the following?

1. 2.

3. (-3, 5), (-5, 2), (4, -3), (7, 0)

6 Function FamiliesLinear: y = xQuadratic: y = x2

Cubic: y = x3

Absolute Value: y = |x|Square root: y = √xRational: y = 1/x

Linear FunctionsCharacteristics 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

Graph of Linear Function

Quadratic FunctionsCharacteristics of a quadratic function (parabola)

Of the form y = x2

Domain: 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

Graph of Quadratic Function

Cubic FunctionsCharacteristics 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 infinity

Interval of increase: all real numbers or (-∞, ∞)

Graph of Cubic Function

Absolute Value FunctionsCharacteristics of an absolute value function

Of 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

Graph of Absolute Value Function

Square root FunctionsCharacteristics 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, ∞)

Graph of Square Root Function

Rational FunctionsCharacteristics 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,∞)

Graph of Rational Function

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.

Example of Vertical Translationy = x2 y = x2 - 4

Example of Reflection

Does a vertical translation affect our following characteristics?DomainRangeX-InterceptY-InterceptMaximumMinimumInterval of IncreaseInterval of DecreaseEnd Behavior

Does a reflection affect our following characteristics?DomainRangeX-InterceptY-InterceptMaximumMinimumInterval of IncreaseInterval of DecreaseEnd Behavior