Sec. 3.2: Families of Graphs Objective: 1.Identify transformations of graphs studied in Alg. II 2.Sketch graphs of related functions using transformations.

Pre-Calculus Section 3.2: Families of Graphs All parabolas that we studied in algebra II are related to the basic graph of y = x 2. This makes the graph of y = x 2 the ______________ of the family of parabolas. A family of graphs is a group of graphs that share _______ or _______ similar characteristics. A parent graph is a basic graph that is _____________ to create other members in a family of graphs. Some different parent graphs are shown below: parent graph onemore transformed

Reflections and translations can affect the appearance of the graph. A ___________ flips the figure over a line called the axis of symmetry or line of symmetry. Examples: 1. Graph f(x) = x 3 and g(x) = -x 3. Describe how the graphs of g(x) and f(x) are related. Notice, multiplying a function by a ______ will make the graph reflect over the x-axis. When a constant term C is added to or subtracted from a parent function, the result is a ____________ of the graph ___ _or ________. (EX: y = f(x) + c ) reflection y = x 3 y = - x 3 **The graphs are reflections of each other over the x-axis. translation updown

2. Use the parent function y = x 3 to sketch the graph of the function y = x 3 – 1. When a constant C is added or subtracted from the x before evaluating the function, the result is a _____________ of the graph _______ or _______. (EX: y = f(x + c) ) Since -1 is added to the parent function y = x 3, the graph will shift down 1 unit. translation left right y = x 3 So each y-value shifts down 1 unit. Parent Function:

3. Use the parent function y = x 3 to sketch the graph of the function y = (x + 2) 3 Since 2 is being added to x before being evaluated by the parent function, the graph will slide left 2 units. y = x 3 Parent Function: So each x-value will slide 2 units left.

We can also have a graph that combines the two translations. 4. Graph the function y = (x – 1) A ___________ has the effect of shrinking or enlarging a figure. When the leading coefficient of a function is not 1, then the function is __________ or _____________. (EX: y = 3f(x) ) y = x 3 Parent Function: The parent function will slide 1 unit right and shift up 3 units. dilation expandedcompressed

5. Graph each function. Then describe how it is related to the parent graph. A. g(x) = 4x -1 Since the parent function is multiplied by -1, the graph is a reflection across the x-axis. The graph will finally shift up 3 units. Each box = 0.5

The following table summarizes the relationships in families of graph. Change to Parent Function y = f(x), c > 0 y = -f(x) Graph is reflected over the x-axis. y = f(-x) Graph is reflected over the y-axis y = f(x) + C Graph is translated up or down. y = f(x + C) Graph is translated rt. or lt. y = C f(x), c > 1 Graph is expanded vertically. y = C f(x), 0 < c < 1 Graph is compressed vertically. y = f(cx), c > 1 Graph is compressed horizontally. y = f(cx), o < c < 1 Graph is expanded horizontally.

Other transformations may affect the appearance of the graph. We will look at two cases that change the shape of the graph. Examples: 6. Observe the graph of each function. Describe how the graphs in parts (B) and (C) relate to the graph in part (A). A. f(x) = (x + 1) 2 – 2 Highest exponent is a 2, so graph must be a U-shape figure. Parent graph is : y = x 2 The graph will slide 1 unit left and shift 2 units down.

B. |f(x)| =| (x + 1) 2 – 2| C. f(|x|) = (|x| + 1) 2 – 2 This transformation reflects any portion of the parent function that is below the x-axis, so that it is above the x-axis. So the graph we di in part(a) that is below the x-axis will reflect above the x-axis, and everything else stays the same. This transformation takes the portion of the graph on the left of the y-axis being replaced by a reflection of the portion on the right of the y-axis. This means the right side of the graph is the same and the left side changes to the reflection of the right side.