Inverse Relationships
I am going to show an inverse relationship in three different ways. The relationship I am showing could come from dividing pizza at a party. If my mom buys me 8 pizzas for my birthday party no matter how many friends I invite, the amount of pizza each kid can eat is an inverse relationship.
This is a inverse graph. You can tell the graph is inverse because the graph is decreasing. However, the graph does not decrease at a constant rate so the graph has a curve to it. When the graph curves there is no y-intercept or x-intercept, but as the graph continues in both directions but never actually touches the x or y axises.
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I always like to check to see if a table is linear first. After I checked an saw there was no constant rate of change, I looked for a constant relationship between the x and y variables. When I multiplied the corresponding x and y variables together I did get a constant value each time. You can see that xy=8 in this case because 8 was the constant number of pizzas my mom was buying me. When there is no constant rate of change, but there is a constant relationship between the independent and dependent variable we have an inverse relationship.
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This is the equation that matches the graph and table above. I discovered the form for an inverse equation is any of the fact families of xy=k, where k is the constant. So our fact family would be:
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Parent Problem
The route for a one day charity bike ride covers 50 miles. The participants all travel this route at different average speeds.
- Make a table and graph to show how riding time changes as the average speed increases. Show speeds from 4 to 20 miles per hour with intervals of 4 mph.
- Write an equation for the relationship of riding time and average speed.
- Tell how the riding time changes as the average speed increases from 4-8mph and 8-12mph and 12-16mph.
- How do you know this is a non-linear relationship?
I designed this problem after the one we did in class about going on vacation. My parents didn't need any of my help and they remembered how to calculate slope from the linear problem. Distance, speed and time seem to be a really real world problem because they didn't even ask me how to get started. They just did the problem.
Reflection
I learned that inverse relationships never cross either axis. All my work with relationships so far have always had a x and a y intercept. Zero with division makes it impossible to have an intercept. Because you cannot divide by 0 there will never be an intercept. This is very interesting, and Mrs. Wallace says we will study more about it in high school. I find this interesting and want to look into it more.