Talk about Algebra: The Intercepts

The skill in question goes like this:  Given an equation in standard form, identify the intercepts and provide their meaning.

Let me try this example (which comes from our class notes) to help explain this and provide means to talk about it:

Photo Credit:  Cedar Point Amusement Park

Suppose you have $500 to organize a family reunion at Cedar Point.  Tickets to the park cost $20 for adults and $15 for kids under the age of 12.  (Disclaimer:  I have no idea what the prices are, but work with me.)  So that means that as we consider how many kids and how many adults are going, we have a limited number of tickets we can buy.  If we have a lot of kids, that means less adults.  More adults?  Less kids.  In the end the total we can spend (the sum, the two numbers added up) on tickets is (equals) $500.  So the equation might look like this:

20a + 15k = 500, where a is the number of adults and k is the number of kids.

Now the Intercepts come in as the extreme points in our example.  Literally an intercept is where a graph crosses either the X axis or the Y axis on the co-ordinate plane.  But fundamentally, the intercept is where one of the two variables (quantities) is 0, and we have to figure out the other one.

So the question is this:  How many kids can we sent into the park if all the adults decide to do something else with their day?  That would be a = 0, and after a little algebra we see the answer is 500 / 15 = 33.3, or 33 kids.  We might also ask, how many adults can go if we simply forget the kids and let them hang out at the hotel pool (with proper lifeguarding of course)?  That would be k = 0, and so 500 / 20 = 25 adults.

The actual intercepts would be the points (0, 33.3) and (25, 0), and their 'meaning' would be "If there are no kids, 25 adults can go, and if there are no adults, 33 kids can go, with a few dollars left over".

If you or your student would like to contribute another such example of two things that can be set up in a form of inverse relation (that is one goes up as the other goes down), please share it by clicking on the "Comment" button bellow.  Students attached to an example can pick up 2 Brownie points, but please only leave a First Name and class period, do not leave a last name.

Discliamer:  Inverse variation is actually defined by a slightly different formula and I only use the term here to suggest "a case where as one goes up the other goes down".  In true inverse variation there are no intercepts because the graph is a hyperbola and not a line.