Parabolas and the Real World
Regressions
A regression is a statistical analysis assessing the association between two variables. It is used to find the relationship between two variables. These two variables are called the independent and dependent variable. The independent variable, located on the x-axis, is usually time or distance, while the dependent variable, located on the y-axis is typically money or height. A regression is a function that maps the relationship whether it be linear, quadratic, trigonometric, exponential, or another mathematical relationship.
To better understand how graphing technology can create regression take a look at the following website: How to Make a Regression
This website describes the methods of Least Square to make a linear regression. This method also works for quadratic regressions and follows the same principles. Luckily for us we have the TI-Nspire that will analyze points and make a regression function. This allows us to focus on the function and concentrate on the real-world application of the quadratic equations.
Regressions on the TI-Nspire
Using TI-Nspire, we can create our own regressions. For example with the handhelds help we can find the equation of the "perfect free throw". All you will need is a basketball, hoop and video camera. By video taping a free throw from a profile view we can see the parabolic path that the ball takes. If we freeze frame the video every half second we can take data points to find the relationship between the height and distance of the ball.
With the TI-Nspire, we can superimpose an image on graph. As seen to the right. Here we she a students attempt to make the "perfect" shot. Careful attention must be given to the size of the picture on the graph itself. Make sure that the intersection of the backboard and rim is as close to 10 feet tall and 15 feet away as you can make it. By selecting menu and then point. We can create points close to the path of the basketball.
When you have a healthy amount of points (at least 5 for good results), import their values in a data and spread sheet page. With the column properly labeled height and distance, you can now properly make a quadratic regression by pressing
- menu
- statistics
- Stat Calculations
- Quadratic Regression
When completed we can enter the regression function into the TI-Nspire to see the "perfect free-throw" and begin to analyze the graph.
With the regression equation we can now begin to answer various questions by analyzing the graph.
- How high is the ball when it is released?
- What is the ball's maximum height?
- Does the ball reach its maximum height right between the shooter and hoop?
- If the hoop was not there how far would the basketball go before hitting the ground?
Using TI-Nspire to Quick Fix
As you can see when calculating a regression with the TI-Nspire, the quadratic regression is usually given in the form of ax2 + bx + c. With this standard form we can easily find the zeros of the parabola. Another feature of the TI-Nspire is a "quick regression". With a quick regression we can move a parabola over a certain aspect of a picture and find a regression in vertex form.
Below is how you can make a quick graph on the TI-Nspire. By creating the parent function seen in the first slide it is clear that it is not a appropriate model for the bridge. The parabola needs to be translated to the left. In the second slide you can see that using the dynamic geometry software on the Nspire we can grab and move the function to the left. What has happened to the function's equation? Is this an appropriate model for the bridge?
Clearly, the equation has changed. What used to be y = x2 is now y = (x + 0.67)2. This validates what we have learned about the Vertex Form of parabolas. The h value controls the horizontal motion of the vertex in the equation y = (x - h)2 + k. By having a h value of .67 we have moved the parabola 0.67 units to the left. However, the parabola is too narrow. By grabbing the side of the parabola we can change the width of the parabola easily. Remembering that the a value controls the width, as we pull the parabola we see the leading coefficient become smaller until it becomes 0.04. Using this method we can quickly make an appropriate model for the bridge.
Reflection
- In what type of contexts are parabolas useful?
- How can understanding the nature of parabolas be useful in solving real life problems?
- Are there advantages to one method or the other when finding models in the Nspire.
Back to home page Quadratic Modeling with the TI-Nspire
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