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=ax**. 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).

^{2}+bx+cBe sure to play/explore with your grapher what happens to the basic parabola y=x^{2} when you put a negative in front, a coefficient other than 1 in front, add/subtract a number to the x^{2} (as in x^{2} ± #), 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**

Depending on the text, they may in this section go too far (in my opinion) into the realm of precalculus. Don't freak out! There is a good chance you can skip some of the more difficult examples and problems. Also take advantage of technology! Here are some notes:

- Minimum/maximum value problems: These problems involve a right side up parabola and you have to find the y-coordinate of the vertex (the "minimum value" of the function) or involve an upside down parabola and you have to find the y-coordinate of the vertex (the "maximum value"). Take advantage of technology! Use a grapher and the MINIMUM and MAXIMUM features to find these "extrema points" instead of computing algebraically.

- Profit function problems: 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).

- Height function problems: The height function formula, h(t)=-gt
^{2}+v_{o}t+h_{o}which describes the parabolic path of an object thrown or dropped, is studied more in a college algebra class. For our purposes you should be able to produce its graph, electronically find the vertex using features of your grapher, and interpret the coordinates. You should be able to recognize that the x-coordinate of the vertex gives the horizontal distance and the y-coordinate gives the height of the object.

- Finding a quadratic function that models an application: The focus at this level of class should not be on translating the word problem into a quadratic equation, but instead electronically analyzing a given quadratic equation to solve an application. You will be given the function, not have to find it yourself. Concentrate on graphing the given equation electronically, finding the vertex, and understanding what the coordinates mean in terms of the application

- SKIP any "table method" if it comes up. 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.

**Supplemental Resources**

It is recommended that you be able to use a calculator (handheld or software) to solve the optimization problems covered in this lesson. If you have a handheld graphing calculator then these sites may help:

- For TI-84 (TI=Texas Instruments) see:

Prof. Milner's Graphing Calculator Tutorial Videos: Find a relative minimum point and,

Prof. Milner's Graphing Calculator Tutorial Videos: Find a relative maximum point

- For TI-86 or TI-89 see:

Prof. Keely's Calculator Guide: Finding Extrema Points

- For other models (or makes) try searching on one of the sites recommended here: FAQs - Graphing Calculator Tutorials

**READ:** Once you are able to use your grapher to locate an extrema point read my Lecture - An Application of Quadratic Functions and try to follow along on your own grapher.