Topic: Functions and Graphs II
Subtopic: Parabolas II - Graphical Approach
Overview
In this lesson we continue to explore parabolas, given in both the standard form y=a(x-h)2+k and the general form y=ax2+bx+c. This lesson covers analyzing these equations graphically and producing the graphs electronically. We are also able to use this information to solve some basic optimization application problems (e.g., optimize profits in a small business).
Be sure to play/explore with your grapher what happens to the basic parabola y=x2 when you put a negative in front, a coefficient other than 1 in front, add/subtract a number to the x2 (as in x2 ± #), add/subtract a number to the x (as in (x ± #)2). Watch for the effects on the shape of the graph, vertex, and axis intercept points.
Objectives
By the end of this topic you should know and be prepared to be tested on:
- 12.2.1 Electronically produce the graph of a parabola from its equation either in standard or general form
- 12.2.2 Electronically analyze the graph of a parabola finding its vertex, y-intercept point, and x-intercept points using features of your grapher
- 12.2.3 Solve application problems requiring optimizing a quadratic function by electronically finding the vertex and analyzing the significance of its coordinates
Terminology
Define: point vs. value, minimum point, maximum point, extrema point, optimization problem
Text Notes
Text:
Intro & Inter Algebra for CS 3ed by Blitzer, sect. 11.3 (cont'd)
- examples 5-8 IMO the text goes too far and enters the realm of college algebra. Don't freak out! Take advantage of technology to find the extrema points electronically rather than the text's algebraic approach. Further ...
- example 6. You should recognize that the graph of the function is an upside down parabola and to find the "maximum height" you need to find the coordinates of the vertex (maximum point) of the parabola. Use a grapher and the MAXIMUM feature to do this instead of computing algebraically. Recognize that the x-coordinate of the vertex gives the horizontal distance and the y-coordinate gives the height of the shot put.
- example 7. Finding the actual quadratic profit function is studied more in a college algebra class. But once you have the profit P(x) quadratic function then you should be able to produce its graph, electronically find the vertex using the minimum/maximum feature of your grapher, and interpret what the x and y coordinates represent (such as y dollars of profit earned from the sales of x unites of product).
- mid-page 777. You can SKIP the "table method" in this class. You are not expected to be able to produce a table on your calculator in this class. Everything you need to accomplish in this section can be and should be done graphically instead.
- example 8. Again, the focus at this level of class (IMO) should not be on translating the word problem into a quadratic equation, but instead electronically analyzing a given quadratic equation to solve an application. Concentrate on graphing the given equation electronically, finding the vertex, and understanding what the coordinates mean in terms of the application.
- SKIP all of sections 11.4-11.5.