3.5 Graphing Techiques: Transformations

Example: For the function below. Graph using transformations.

f(x)=-(x-3)^2+4

First we must examine the base function y=x^2

Graph using plotting points. We can use the standard set of x-values to find ordered pairs. Substitute the standard set of x-values into the base function to get the base graph.

x	y
-2	(-2)^2=4
-1	(-1)^2=1
0	(0)^2=0
1	(1)^2=1
2	(2)^2=4

The graph below shows the points plotted and the line that connects them. The domain is (- infty, infty) and the range is [ 0, infty )

Analyze the transformations.

f(x)=-(x-3)^2+4

The -3 inside the square shifts the graph right 3 units.

The – in the front of the base reflects the graph over the x-axis.

The +4 outside the square shifts the graph up 4 units.

You can see the graph after the transformations.

The domain is (- infty, infty) and the range is ( - infty, 4 ]

Here is a video example of a transformation of a square function.

Example: For the function below. Graph using transformations. Find the y-intercept. State the horizontal asymptote and the domain and range.

$f(x)=-2^{x+2}+2$

First we must examine the base function y=2^x

Graph using plotting points. We can use the standard set of x-values to find ordered pairs.

x	y
-2	2^(-2)=1/4
-1	2^(-1)=1/2
0	2^0=1
1	2^1=2
2	2^2=4

The graph below shows the points plotted and the line that connects them. This graph has a horizontal asymptote at y=0. The domain is (- infty, infty) and the range is (0, infty)