Interpreting Graphs

I find graphical solutions to be the quickest method to approach many arduous tasks when a graphing utility is involved. Solve a system? Graph and locate the intersection. Find real roots? Graph and look for intercepts. Identify asymptotes? Graph and see where that weird squiggle is.

My honors Pre-Calculus students on the other hand, are not convinced. They thought I was torturing them when I asked them to graph to find real roots. I knew they studied all the other methods of solving polynomials last year and this year I wanted to focus on the more interesting task of complex roots. Several students announced they would rather do synthetic division than pull roots off a graph. I consider that torture! But even with graphing calculators they had such a hard time estimating the values of intercepts, and I promised to only give integer solutions so they knew to round.

We have since moved on to rational functions, yet the same issues are still cropping up. On the test I’m grading, one question is: “Describe the graph of f(x)=\dfrac{2x^2-8}{x^2+8x+12} Include domain, range, holes and asymptotes (vertical and horizontal).” It is certainly a question that involves synthesizing a number of ideas, but they have graphing calculators, are good at factoring and we did plenty of problems just like this. I shouldn’t be getting multiple vertical asymptotes, domains like -6<x<6 or ranges of y<1. I don't know what to do.

How do you help students pull important information out of a graph?

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3 thoughts on “Interpreting Graphs

  1. Jill W. says:

    Personally, I prefer not to graph as well, simply because it doesn’t seem as accurate. Maybe I just wasn’t taught how to use the calculator to find the exact answers. I always end up looking at the graph and then guessing from there. Any advice?

    • Tina C. says:

      That’s exactly what I want kids to do- use the graph to come up with a good estimate, then verify using trace or the original function. It’s faster than checking all the factors!

  2. It seems like the even bigger issue here is that the two approaches should reinforce each other — they should learn to see the graph as another way of getting at the behavior of the function, along with things like plugging in numbers, factoring, and other algebraic techniques. Not to mention flexibility in problem-solving, so they can choose the approach that will be the easiest for their given situation.

    Maybe I’d start with some matching exercises, given functions and graphs, figure out which goes with which. Have some functions that share the same asymptotes and other that share the same roots so that they have to notice which parts of the function’s algebraic definition tell them about which features of the graph. My conjecture would be that the reason they resist the graphing is that they don’t understand the relationship between the graph and some of the properties that they understand from the point of view of algebra. I know when I taught algebra 2, one of my first-day review pieces to help me learn where they were was to ask “What is the relationship between the quadratic formula and the graph of a parabola?” or something along those lines, where I was always stunned (yes, even after several years of the same results) to see how few of them could tell me that the formula gives the location of the x-intercepts.

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