(a) (-∞, -8) U (-4, 0) U (5, 9)
To find the intervals of decreasing, you just pay attention to where the graph is angled downward, when looking at it from left to right.
I usually do this by drawing lines on the x value where there is a local max/min and looking to see what the values are at that line.
You then make an ordered pair out of the two numbers.
This is the same process for finding both the increasing and decreasing intervals.
(b) -4, 5
A common mistake for this problem could be that you put the maximum and minimum x values in for this function, but <u>the problem is only asking for the maximums</u>.
If you look at the graph and find the point, the x-values are what you will use for this problem.
(c) +
For these kinds of problems, I draw an imaginary vertical line through the middle of the function.
- If the far right and left parts of the function are going up, then the function is positive.
- If the far right and left parts of the function are going down, then the function is negative.
- If the far left part of the function is going down while the far right part of the function is going up, then the function is positive.
- If the far left part of the function is going up while the far right part of the function is going down, then the function is negative.
These positives/negatives belong to the leading coefficient of f.
(d) 6
The degree of f relies on the number of times the function touches the x-axis.
<u>Multiplicity:</u> the number of times any given x-intercept appears in the equation
- If an x-intercept passes through the x-axis, then that intercept has a multiplicity of 1
- If an x-intercept bounces off of the x-axis, then that intercept has a multiplicity of 2
- If an x-intercept passes through the x-axis, but flattens out when it touches the x-axis, then that intercept has a multiplicity of 3
Adding up the multiplicities of all of the x-intercepts gives you the degree of the function.
In this problem, there are x-intercepts that simply pass through the x-axis at -9, -6, 8, and 10, and an x-intercept that bounces off of the x-axis at 0, which makes it a function of degree 6.
