Turning Points
In this section you will learn how to read and describe the graph of a polynomial function in terms of increasing and decreasing.
Where a graph changes, either from increasing to decreasing, or from decreasing to increasing, is called a turning point.
We will then explore how to determine the number of possible turning points for a given polynomial function of degree n.
Read through the notes carefully, taking notes of your own. Then, when you watch the video you’ll have things to look out for. After that, try the practice problems. If you’re confused, go back through your notes and don’t neglect the importance of integrating prior learning and previously held knowledge!
Polynomial Functions
The graph to the left is of a polynomial function of degree four. Let’s explore how we look at the graph, to establish common language.
The graphs of all polynomial functions are what is called smooth and continuous. This means that there are not any sharp turns and no holes or gaps in the domain.
Slope: Only linear equations have a constant slope. These polynomial functions do have slopes, but the slope at any given point is different than the slope of another point near-by. The slope of a linear equation is the same at any point. Regardless of what points you choose from linear equation, the slope formula always provides the same slope.
But our quartic function doesn’t have a constant slope. For all x values from negative infinity up to -0.52, the slope of this quadratic is negatives. It is going down. But, from -0.52 to 0.649 the slope is positive. At each point the slope is different, but all points have a positive slope in this interval.
Since the slope is different at all consecutive points, we can say that the graph is decreasing from negative infinity to -0.52.
Direction: It is easy to say that this graph is, “going up both ways.” That would mean on the left and right it is going up. That’s actually not true, though. A graph is read from left to right. On the left, this graph is actually going down, it is decreasing, until it gets to x = -0.52. It turns out, for reasons you’ll learn in calculus, that at x = -0.52, the slope is zero. From -0.52 to 0.649, the graph increases, before decreasing again.
Where the graph changes from decreasing to increasing, or from increasing to decreasing, are points called turning points.
The graph above has three turning points. They’re noted on the graph. The coordinates are (-0.52, -2.65) and (0.694, 0.311) and (2.076, -3.039).
Number of Turning Points
A polynomial of degree n, will have a maximum of n – 1 turning points. For example, a suppose a polynomial function has a degree of 7. The maximum number of turning points it will have is 6. A quadratic equation always has exactly one, the vertex. A linear equation has none, it is always increasing or decreasing at the same rate (constant slope).
Let’s see some examples of a polynomial of degree 5. Let’s look at turning points, both actual and maximum, but also x – intercepts and direction.
Remember, a turning point is defined as the point where a graph changes from either (A) increasing to decreasing, or (B) decreasing to increasing. So in the first example in the table above the graph is decreasing from negative infinity to zero (the x – values), and then again from zero to positive infinity. It never switches from negative to positive. It switches from negative to zero and zero to negative, but zero is not a positive number.
In the next section we will explore something called end behavior, which will help you to understand the reason behind the last thing we will learn here about turning points. A polynomial function of degree 5 will never have 3 or 1 turning points. It will be 4, 2, or 0. A polynomial of degree 6 will never have 4 or 2 or 0 turning points. It will be 5, 3, or 1.
Let’s summarize the concepts here, for the sake of clarity.
Increase v Decreasing Algebraically
From the graph of f(x) = x5 (use desmos.com to graph it), we can see that it is increasing when the inputs are negative. We will need to be able to tell if a function is increasing or decreasing over an interval algebraically, without a graph. Here’s now to do that.
- Take two consecutive inputs (relatively close), like x = -5 and x = – 4.
- Plug in those values into the function to find the outputs.
- f(-5) = – 3125, and f(-4) = – 625.
- If the outputs increase for increasing inputs, the function is increasing.
- Input increase: -5 < -4 and outputs increase: -3125 < -625.
- If the outputs decrease while the inputs increase, the function is decreasing.
- For f(x) =x4, inputs increase: -5 < – 4, but outputs decrease: 3125 > 625.
In the next section we will explore something called end behavior, which will help you to understand the reason behind the last thing we will learn here about turning points. A polynomial function of degree 5 will never have 3 or 1 turning points. It will be 4, 2, or 0. A polynomial of degree 6 will never have 4 or 2 or 0 turning points. It will be 5, 3, or 1.
Let’s summarize the concepts here, for the sake of clarity.
Summary
- Polynomial functions of a degree more than 1 (n > 1), do not have constant slopes.
- Increasing: The graph is going up, when read from left to right.
- Each point on the curve that is going up is positive.
- Decreasing: The graph is going down, when read from left to right.
- Each point on the curve that is going down is negative.
- A turning point is where a graph changes from increasing to decreasing, or from decreasing to increasing.
- The maximum number of turning points for a polynomial of degree n is n –
- The total number of turning points for a polynomial with an even degree is an odd number.
- A polynomial with degree of 8 can have 7, 5, 3, or 1 turning points
- The total number of points for a polynomial with an odd degree is an even number.
- A polynomial of degree 5 can have 4, 2, 0 turning points (zero is an even number).
Homework review video:
