We've already looked at some key features of certain functions and their graphs.
Now we're going to look at one more key feature.
Increasing Functions
If $f(x)$ gets bigger as $x$ gets bigger, $f(x)$ is said to be increasing.
Decreasing Functions
If $f(x)$ gets smaller as $x$ gets bigger, $f(x)$ is said to be decreasing.
Intervals of Increase and Decrease
Not all functions are always increasing or always decreasing.
Determining Important Features of a Function (Analyzing)
a. Vertex (if there is one).
b. Does the graph open upward or downward?
c. $x$-intercept(s) and $y$-intercept.
d. Domain and range.
e. Determine the $x$-values for which the function is positive and negative.
f. Determine the $x$-values for which the function is increasing and decreasing.
Example
Analyze the function $f(x)=-2x^2+9x+26.$
a. Vertex (if there is one).
b. Does the graph open upward or downward?
c. $x$-intercept(s) and $y$-intercept.
d. Domain and range.
e. Determine the $x$-values for which the function is positive and negative.
f. Determine the $x$-values for which the function is increasing and decreasing.
A Strange Case:
Consider the function $f(x)=-x+1$ graphed below.
Question: On what interval if $f(x)$ increasing?
Another Strange Case:
Consider the function $f(x)=x-1$ graphed below.
Question: On what interval if $f(x)$ decreasing?