This is used with an activity in the ACCESS Algebra II class on finding patterns in the graphs of higher order functions. Created by Debbie Pate.
The chart used is attached as well as the answers.
Instructions:
We have studied a good bit about linear and quadratic equations and graphs. Now we need take a glimpse of graphs of higher order equations.
Open the Graphing Higher Order Equations Recording Chart linked below and print or copy the chart into your notebook.
We are going to fill in this chart and make conjectures about connections between six equations and their graphs.
All the graphs are one continuous curve. I am going to help you do the first one. Then you should repeat the steps for each equation and record in your notebook.
The first equation is one we are familiar with, a quadratic equation, x² + 7x + 6. The lead coefficient is the coefficient of the term with the highest power and when written in order it is the leading one. In this case we have 1x²so the lead coefficient is a 1.
The degree is the highest power. We have x². The highest power is 2 so our degree is 2.
The turning points or humps are called critical points. There are two types of critical points, relative maximums and relative minimums.
Look at the the graph on the left. There is one turning point. Is it a relative max or relative min? Relative min
In the Geogebra file, slide point A to the critical point as shown on the left. In the chart under critical point, write 1 critical point, then below it write rel. min ≈(-3.6, -6.3)
4. The x intercepts are also the solutions of the equation. How many x intercepts are there for this equation?
You can name the x intercepts as an ordered pair where y = 0 or you can name them by simply using the number where they cross the x axis. Name the x intercepts without using an ordered pair. x = -6 , x = -1
In our equation, , if we factor and solve this equation, as we learned in Algebra I, we would get the x intercepts.
5. Name the y intercepts without using ordered pairs.
6. Record all your answers in the recording chart and sketch a rough graph
Repeat for the other examples.
Study your chart and answer these questions in your notebook.
1. How does the degree compare to the number of turning points?
2. With one exception, how does the number of x intercepts compare to the degree? Name the exception. Do you have any idea what made the exception.
3. How is the y intercept related to the equation?
4. Look at each end of the graph. Describe the difference in the graphs of equations that are odd degrees and those that are even degrees.
5. How does the equation of a higher order function can help you determine the number of turning points and the number of x intercepts?
6. Describe the difference in the graphs of equations that are odd degrees and those that are even degrees. What do you think would happen if the lead coefficient was negative in each one?
