04-2c+-+Domain,+Range,+Maxima+and+Minima

If we insert any real value of x into a quadratic equation, the result is a real number. Therefore the domain of any quadratic function is all real numbers, but the range of the function will depend on the vertex and the direction in which the parabola opens.

A parabola that opens upward has a minimum value while a parabola that opens downward has a maximum value. Be careful when you describe a maximum or minimum --- these terms refer to the y-value at the vertex, not the ordered pair that describes the vertex.



Here is an example:

Consider the function f(x) = -3x2 + 2x -4. What is the maximum or minimum value, and what is the domain and range of the function? The axis of symmetry is x =  The y-value is the maximum value of the function,.
 * Does the function have a maximum or a minimum? Since a is negative, the parabola will open downward, so the function has a maximum.
 * What is the x-value of the vertex? Remember that the vertex lies on the axis of symmetry, so we need to find the equation of the axis of symmetry.
 * [[image:http://latex.codecogs.com/gif.latex?x=-%5Cfrac%7Bb%7D%7B2a%7D=%5Cfrac%7B2%7D%7B2%5Cleft%20(%20-3%20%5Cright%20)%7D=%5Cfrac%7B1%7D%7B3%7D align="center"]]
 * [[image:http://latex.codecogs.com/gif.latex?x=-%5Cfrac%7Bb%7D%7B2a%7D=%5Cfrac%7B2%7D%7B2%5Cleft%20(%20-3%20%5Cright%20)%7D=%5Cfrac%7B1%7D%7B3%7D align="center"]]

The domain is all real numbers,. The range is all real numbers less than or equal to ; in set notation this is written as



media type="file" key="maxima and minima.swf"

Although we have been able to calculate the maximum and minimums of the functions, there are times when using the graphing calculator makes more sense. Let's walk through an example of using the TI

Consider the function f(x) = x2 - 6x + 3

Since a is positive, the graph opens upward <span style="display: block; font-family: Verdana,Geneva,sans-serif;">We find the line of symmetry (which includes the x-value of the vertex) by using the formula and substituting -6 for b and 1 for a; the line of symmetry is x = 3, and the x-value of the vertex is 3.

<span style="display: block; font-family: Verdana,Geneva,sans-serif;">If we evaluate the function at x = 3, we find the y-value of the vertex to be -6. So, the minimum value of the function is -6, the domain is all real numbers, and the range is all real numbers greater than or equal to -6.

<span style="display: block; font-family: Verdana,Geneva,sans-serif;">We can graph this function on the TI83. <span style="display: block; font-family: Verdana,Geneva,sans-serif;">Open the emulator and enter the function in the Y= area <span style="display: block; font-family: Verdana,Geneva,sans-serif;"> <span style="display: block; font-family: Verdana,Geneva,sans-serif;">Now click GRAPH <span style="display: block; font-family: Verdana,Geneva,sans-serif;">We can look at the table view to verify our calculation of maximum and minimum :

<span style="display: block; font-family: Verdana,Geneva,sans-serif;">We can use the calculator to find the maximum or minimum value of a function. Press 2nd TRACE to bring up the <span style="font-family: Verdana,Geneva,sans-serif;">﻿ CALCULATE menu. Move the cursor to 3 to activate the MINIMUM calculation. <span style="display: block; font-family: Verdana,Geneva,sans-serif;">

<span style="display: block; font-family: Verdana,Geneva,sans-serif;">Click ENTER and you will be taken to the graph. Notice the little blinking spider on the curve. You are being prompted for the left bound; this means that the calculator will be looking for a minimum to the right of this point. Click Enter.

<span style="display: block; font-family: Verdana,Geneva,sans-serif;">The next screen will ask for the right bound. Be certain that the spider is to the right of the minimum point. If it is not, use the arrow keys to move it anywhere to the right of the minimum. You will be prompted to vertify that this is the right bound. Click Enter.

<span style="display: block; font-family: Verdana,Geneva,sans-serif;">The next screen asks you to guess; just click Enter to pass over it. The last screen will show the calculated minimum value,

<span style="display: block; font-family: Verdana,Geneva,sans-serif;">You might be wondering why you are being asked for left and right bounds when it is obvious that there is only one minimum point. When you work with higher level equations there can be more than one maximum and/or minimum value to the function, so the calculator wants to be sure that it is searching in the right range for your needs. <span style="display: block; font-family: Verdana,Geneva,sans-serif;">Back to Properties of Quadratic Functions in Standard Form