Reversing the Process

An Algebraic Analysis of a Polynomial Inequality Given Its Graph.

You are asked to find the points that will make the inequality true.

We will now work backward. We will interpret a graph and identify the polynomial inequality.

At the right is a graph with some highlighted key points, so we will begin by analyzing the graph to determine:

~ The original function

~ The behavior of the function around the zeros

~ The solution to the inequality (assume it to be included in the red region)

Example 8: Algebraic Solution

The graph below

1. holds the solution to the inequality

2. contains the zeros of the function

3. describes the actual graph of the function

Step 1: Determine the zeros of the function

The image above highlights the function's zeros.

The zeros of the function are when the graph crosses the x-axis.

They are (-3, 0), (-1, 0) (1, 0) and (4,0)

x = -3, x = -1, x = 1, and x = 4

Step 2: Use the zeros to determine the factors of the function

If x = -3 then the factor must be (x + 3) if x + 3 = 0, then x = -3

If x = -1, then the factor must be (x + 1)

If x = 1, then the factor must be (x - 1) if x - 1 = 0, then x = 1

If x = 4, then the factor must be (x - 4)

Step 3: Define the function

Before we define the function, look at the below graph.

At x = -3, x = -1, and x = 1, the graph crosses the x-axis.

At x = 4, the graph touches and turns. It is a double root.

So, the factor (x - 4) must be (x - 4)2

We can now define the function

y = (x + 3)(x + 1) (x - 1) (x - 4)2 or y = (x + 3) (x2 - 1) (x - 4)2

As you read the information below, interact with the graph (right) by clicking on the icons.

Step 4: Determine the behavior of the function around the critical points.

If we look at the red areas and the function's graph, that is, its actual behavior, we can graph the results on a number line.

Before that, let's look at how the regions are enclosed.

~ The line is dashed, which means the inequality doesn't include the "equal to" sign.

~ The zeros are not included, and we would use parentheses instead of brackets.

Region I - Red region and the graph is below the x-axis --> negative on the interval

(-∞, -3)

Region II - White region and the graph is above the x-axis --> positive on the interval (-3, -1)

Region III - Red region and graph is below the x-axis --> negative on the interval (-1, 1)

Region IV - White region and graph is above the x-axis --> positive on the interval (1, ∞)

Inequality Graphing

With this information from the function's graph, we can complete the number line.

Step 5: Number line graph

<------------------3)++++++++++++++ (-1 ----------------- 1) +++++++++(4+++++++>

Step 6: Determine the solution set for the polynomial inequality

Because we are looking at the red region, the solution is all values of x that would result in the function's value less than 0, where x < 0.

(-∞, -3) u (-1, 1)

The inequality: (x + 3) (x2 - 1) (x - 4)2 < 0

We do not have to write this inequality in standard form.

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