Last time, we looked at end behavior and nested exponents. Now we’ll explore a method for sketching a graph of a polynomial.
First of all, we need to know that each factor in the factored form of a polynomial corresponds to an x-intercept on the graph. For instance, look at this polynomial and its graph:
The x-intercepts are -5, -2, and 5: exactly one for each factor (excluding the 1/20, which I add to make the whole graph visible). You can get the x-intercept by setting each factor equal to zero. It’s logical that if there are three factors, there will be three roots and the function will be cubic.
On the other hand, you can have a cubic with only 2 x-intercepts, like this: In this case, the (x+2) is repeated and has a multiplicity of 2. The graph looks like this:
The curve is tangent to the x-axis at the repeated root, -2. You may recognize this phenomenon from quadratics, for instance the graph of . In fact, any time a factor has a multiplicity of 2, the graph will appear like a parabola tangent to the x-axis. To show this further, we can even look at a quintic equation:
Notice the parabolic shape near x = 4. How can we explain this behavior?
Let’s look back at the factored equation which gave us this graph. We can imagine each factor as a hand pulling the graph in different directions. The interactions (product) of the individual pulls creates the value at a given point on the graph. When one factor becomes zero, its “hand” pulls the graph so completely that no matter how large the product of the other factors, the final value will be 0.
The factor has a unique pull on the value of the function. As x approaches 4 from either direction, (x-4) approaches 0. Think about a value for x that is really close to 4 – that is, the difference in x is infinitesimally small. Then the value of (x-4) is also infinitesimally small, and squaring that makes an even smaller number! This is quite a small number indeed, and it exhibits its control of the graph by bringing it down close to zero.
As for the parabolic shape, this can be explained in a somewhat similar manner. As (x-4) approaches zero, the product of the other numbers has a specific sign: positive or negative. The is always positive, being a square; therefore, it cannot change the sign of the final value. So when it hits zero, it rebounds off the x-axis and remains positive.
Next time, we’ll discover what happens when a factor is cubed!