Lesson 3-2

Example 1
Graph f(x) = x2 and g(x) = -x2. Describe how the graphs of f(x) and g(x) and are related.
	
x	f(x) = x2	g(x) = -x2	
-2	4	-4	
-1	1	-1	
0	0	0	
1	1	-1	
2	4	-4	

To graph both equations on the same axis,
let y = f(x) and y = (x).

The graph of g(x) is a reflection of the graph of f(x) over the x-axis. The symmetric relationship can be stated algebraically by g(x) = -f(x), or f(x) = -g(x). Notice that the effect of multiplying a function by -1 is a reflection over the x-axis.

Example 2
Use the parent graph y = x2  to sketch the graph of each function.

a.	y = x2 + 1
	This function is of the form y = f(x) + 1.	
Since 1 is added to the parent function y = x2,
the graph of the parent function moves up 1 unit.

b.	y = (x - 2)2
	This function is of the form y = f(x - 2).	
Since 2 is being subtracted from x before
being evaluated by the parent function,
the graph of the parent function y = x2
slides 2 units right.

c.	y = (x + 1)2 - 2
	This function is of the form y = f(x + 1)2 -2.	
The addition of 1 indicates a slide 1 unit
left, and the subtraction of 2 moves the
parent function y = x2 down two units.

	

Example 3
Graph each function. Then describe how it is related to its parent graph.

a.	g(x) = 3x3
	The parent graph is g(x) = x3.	
g(x) = 3x3 is a vertical expansion
by a factor of 3.

b.	h(x) = -0.75x3 - 2
	h(x) = -0.75x3 - 2 reflects the parent	
graph, h(x) = x3, over the x-axis,
compresses it vertically by a factor
of 0.75, and shifts the graph down
2 units. Notice that multiplying by
a negative number greater than
negative one compresses the graph
vertically.

Example 4
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 - 4	
	The graph of y = f(x) is a translation of y = x2.
The parent graph has been translated right
1 unit and down 4 units.

b.	y = |f(x)|	
	|f(x)| = |(x - 1)2 - 4|
This transformation reflects any portion of
the parent graph that is below the x-axis so
that it is above the x-axis.

c.	y = f(|(x)|)	
	f(|(x)|) = (|(x)| - 1)2 - 4
This transformation results in the portion
of the parent graph on the left of the y-axis
being replaced by a reflection of the portion
on the right of the y-axis.

	

Example 5
During a baseball game, a batter pops a ball straight up so that the ball’s initial velocity coming off the bat is 80 mph. The height of the baseball in one second intervals is recorded in the table below. Write the equation of the related parabola that models the data.	Time
(seconds)	Height
(feet)	
	0.0	4	
	0.5	40	
	1.0	68	
	1.5	88	
	2.0	100	
	2.5	104	
	3.0	100	

A graph of the data reveals that a parabola is the best model for the data. The parent graph of a parabola is the graph of the equation y = x2. To write the equation of the related parabola that models the data, we need to compare points located near the vertex of each graph. An analysis of the transformation these points have undergone will help up determine the equation of the transformed parabola.

From the graph, we can see that parent graph	
has been turned upside-down, indicating that
the equation for this parabola has been
multiplied by some negative constant c.
Through further inspection of the graph and
its data points, we can see that the vertex of
the parent graph has been translated to the
point (2.5, 104). Therefore, an equation that
models the data is y = c(x - 2.5)2 + 104.

To find c, compare points near the vertex of the graphs of the parent function f(x) and the graph of the data points. Look at the relationship between the differences in the y-coordinates for each set of points.

x	f(x)	
-2.5	6.25	
-2.0	4.00	
 -1.5	2.25	
-1.0	1.00	
-0.5	0.25	
0	0.00	
0.5	0.25	
Time
(seconds)	Height
(feet)	
0	4	
0.5	40	
1.0	68	
1.5	88	
2.0	100	
2.5	104	
3.0	100	

These differences are in a ratio of 1 to -16. This means that the graph of the parent graph has been expanded by a factor -16. Thus, an equation that models the data is y = -16(x - 2.5)2 + 104.