A
spreadsheet can calculate the values of a function and generate a
quick graph. In this example, we will graph the function: “Y = x² = x”.
This week, we will continue the topic quadratic equations, started last week. We will also discuss quadratic inequalities and graphing.
There are several tools available for graphing.
A
spreadsheet can calculate the values of a function and generate a
quick graph. In this example, we will graph the function:
“Y
= x² = x”.
In column A, put a few x values that you want on the graph. In column B, enter the function, replacing “x” with the address of the box immediately to the left. The expression “=A2^2+A2” is in box B2. If you are used to the notation using “^” for exponentiation, you have a head start translating formulas into spreadsheets. You only have to type the expression once, then copy and paste the expression into the rest of the boxes in the column that will get the “Y” values.
The next step is to draw the graph. Click on the “Insert” pull-down menu, then select “Chart”. Specify the range so that it includes both the x and y values for all the points you want on your graph. When you get the list of chart formats, pick the “scatter chart with lines and symbols”.
Change the size of the spread sheet's window and position the spreadsheet so that everything that you want to include in your report is visible. Then, hold the ALT key while you press the “print scrn” key (this limits the screen capture to the active window). You can paste the image directly into Word (but not into Outlook Express); however the pasted image is uncompressed and the resulting file can be huge. If you paste the image into Paint, you can save it in GIF format, for 90% compression (ignore the warning about loosing color).
Another
graphing tool on your computer is Paint itself. On the view pull-down
menu, make sure that the “Status Bar” option is checked.
This option will display (in the lower right corner) the X and Y
coordinates of the cursor. You can manually draw a graph from a list
of points and save the image.
As I mentioned in the syllabus, the GCalc site: http://humblestar.net/GCalc/ draws graphs from your entry. To use this site, in the long box above the graph, enter just the right hand side of the desired equation (do not enter "y="), and then press ENTER on the keyboard. Then capture the screen image.
Another way to draw graphs is through Aleks. Although the graphing tool is not available separately, you have access to the tool while working on any problem that involves drawing a graph. When the tool is in place, you can use it to graph anything you want.
The graph of the general quadratic formula, ax² + bx + c, is a vertical parabola. If a is positive, the parabola opens upward; if a is negative, the parabola opens downward. (One way to remember this is that people who think positive are happy and an upward parabola looks like a smile.) The greater the absolute value of a, the narrower the parabola appears. When a is zero, the graph is a straight line, but the formula is not a true quadratic. Beside the direction in which the parabola opens, there are certain other numbers that help in drawing the graph.
The axis of the quadratic function is the vertical line, x = -b/2a. The parabola is symmetric around this axis, so we can draw one side and then take its mirror image to obtain the other side. The intersection of the parabola with the axis is the vertex, which is the highest or lowest point.
If the vertex is above the x axis and the parabola opens upward, there will be no intercepts. Where intercepts exist, they are at the “solution” points, x=[-b ± √{b²-4ac}]/2a.
A quadratic inequality has the format ax² + bx + c < 0. Even if the quadratic cannot be factored, it can still be rewritten, using the quadratic formula, as
(x
+ (b/2a) + √(b²-4ac))(x
- (b/2a) + √(b²-4ac))< 0
2a
2a
If a>0, this becomes
x > (-b/2a) -
√(b²-4ac)and
2a
x
< (-b/2a) + √(b²-4ac)
2a
The
quadratic inequality in the example states that the generic parabola,
opening upward, is below the x axis. We know the intercepts by
solving for ax² + bx + c = 0. Since we are only in the x values,
and, we know that x must be between the intercepts, all we need to do
is to draw the connecting line.
There is a great explanation at http://www.purplemath.com/modules/ineqquad.htm.
If your dictionary is like mine(Webster's Ninth New Collegiate Dictionary) it defines a parabola as a curve such that each point is at the same distance from a fixed line as from a fixed point. Fortunately, page 699 of our textbook translates this definition into the more familiar, "curve with the same shape as y= ax² + bx +c, where a, b, and c are constants".
Another definition of parabola is the path of a thrown object. The link between the definitions is gravity.
Acceleration
from gravity is usually given as “32 feet per second per
second”. This means that if you drop an object, the speed
starts at zero and increases by 32 feet per second each second that
it falls. After two seconds, for example, it is falling at 64 feet
per second. The average speed, while it is falling for t seconds, is
the starting speed (zero) plus the ending speed (32 t) divided by
two.
speed = 32t/2 = 16t.
The distance traveled during the fall is
distance = speed•time = 16t².
This acceleration is the source of the “-16t²” term in the formula for the height of a thrown ball on page 301 of our text. It is the same (with very slight variation, all over the earth.
An inequality is a statement about the relative values of two expressions, for example
1. X < Y
An "equivalent inequality" is true if (and only if) the original inequality is true, no matter what the values of the variables are.
2. 2X < 2Y (equivalent)
3. -2X <-2Y (not equivalent)
4.2X² < 2Y² (not equivalent)
Note that #4 has the same truth value as #1 when X is positive; however, it is still not equivalent to #1.
As noted on page 124 of our text, adding the same constant to both sides of an inequality or multiplying both sides by the same positive number results in an equivalent inequality
The solution to example 4 on page 566 of the text states that “we do not multiply by the LCD.” Why not?
Can you give an example of a set of inequalities with no solution? Is there a term for this situation?
How many times does the graph of a parabola intersect the Y axis? How many times does it intersect the X axis?
During the filming of a car advertisement, the automobile was dropped. How quickly did the car go from zero to 60 miles per hour?