Problem 61 - Precalculus : Building Concepts and Connections

Question

Use the given information to (a) sketch a possible graph of the polynomial function; (b) indicate on your graph roughly where the local maxima and minima, if any, might occur; (c) find a possible expression for the polynomial; and \((d)\) use a graphing utility to check your anstroers to parts \((a)-(c)\) The real zeros of the polynomial \(h(x)\) are \(x=3\) and \(x=0.5,\) each of multiplicity \(1,\) and \(x=\sqrt{2},\) of multiplicity 2. As \(|x|\) gets large, \(h(x) \rightarrow+\infty\)

Step-by-Step Solution

Answer

The possible polynomial function is \(h(x) = (x-3)^1*(x-0.5)^1*(x-\sqrt{2})^2\), upon sketching, this function doesn't seem to have any local maxima or minima as it appears to be increasing in both ends

Step 1: Plot the real zeros

Start by identifying and plotting the real zeros on a graph which are \(x=3\), \(x=0.5\), and \(x=\sqrt{2}\). The multiplicity of each zero needs to be considered as well. Multiplicity basically tells how many times the function will cross the x-axis at that zero. If multiplicity is 1, the graph crosses the axis. If multiplicity is 2, the graph touches the axis and turns around.

Step 2: Use End Behavior to Sketch Graph

As given, as \(|x|\) gets large, \(h(x) \rightarrow+\infty\). This means that both sides of the graph point upwards. Hence, the graph will be increasing at the far right and left. Connect the dots created by the zeros and end behavior to sketch a graph.

Step 3: Determine Local Maximum and Minimum

In the graph, look for points where the function switches from increasing to decreasing (local maxima), or from decreasing to increasing (local minima). Note that not all polynomial functions have local maximum or minimum.

Step 4: Find the Polynomial Expression

Given that \(h(x)\) is a polynomial function with the given zeros and multiplicities, a possible function can be built as follows: \(h(x)=(x-3)^1*(x-0.5)^1*(x-\sqrt{2})^2\). Multiply it out to get a possible polynomial function.

Step 5: Verify with a Graphing Utility

Input the supposed polynomial function into a graphing utility to confirm that its graph fits the given characteristics. That can be used to check the accuracy of answers to parts (a)-(c)

Key Concepts

Real Zeros

Real zeros of a polynomial are values of \(x\) where the function equals zero, meaning the graph crosses or touches the x-axis at these points. In our example, the polynomial has real zeros at \(x = 3\), \(x = 0.5\), and \(x = \sqrt{2}\). Identifying these zeros is crucial because they indicate where the graph changes direction or crosses the x-axis.
Real zeros are essential in formulating the polynomial expression itself, as each zero contributes a factor of the form \( (x - zero) \). Here, each zero's multiplicity will play a role in how the graph behaves at these points. If a zero has a multiplicity of 1, the graph will simply cross the x-axis at that zero. If the multiplicity is greater, like 2 in our case for \(x = \sqrt{2}\), the graph touches the x-axis but does not cross it, creating a slight 'bounce' at that point.

Multiplicity

Understanding multiplicity is vital because it influences the shape and behavior of the polynomial graph at its zeros. Multiplicity refers to how many times a particular zero appears in the polynomial.
Let's break it down with our example:

• Zeros with multiplicity 1, such as \(x = 3\) and \(x = 0.5\), cause the graph to cross directly through the x-axis at these points. This crossing signifies that the change in sign of the polynomial is occurring sharply.
• For \(x = \sqrt{2}\), which has a multiplicity of 2, the graph behaves differently. Instead of crossing, the curve 'bounces' off the x-axis at this zero, indicating that there's no change in sign at this stage.

The multiplicity of zeros helps determine how the polynomial function's graph transitions from one section to another, enhancing your ability to sketch the graph accurately.

End Behavior

End behavior describes what happens to a polynomial's graph as \(x\) approaches positive or negative infinity. This can help us understand the general direction the arms of the graph will take.
In our example, we're given that as \( |x| \) becomes large, \( h(x) \rightarrow +\infty\). This specific detail informs us that as \(x\) moves toward extremely large or small values, the graph moves upwards, indicating that both ends of the graph rise.
Understanding end behavior is crucial when sketching the graph of a polynomial. It helps you predict the rising or falling nature at the graph's tails, guiding you in connecting the portions of the graph formed by zeros and peaks or troughs accurately.

Local Maxima and Minima

Local maxima and minima are critical points where a polynomial function changes direction. A local minimum is a point where the function goes from decreasing to increasing, creating a low point in the graph, while a local maximum is where it switches from increasing to decreasing, forming a peak.
Finding local maxima and minima involves observing the graph, often at turning points between the real zeros. However, remember that not every polynomial will have these points; they depend on the specific configuration of zeros and coefficients.

• If a graph turns downward after a zero and then back up, you might have a local minimum.
• If a graph peaks and then descends, you may find a local maximum.

While our example doesn't explicitly outline these, sketching and verifying with a graphing utility can reveal whether any such critical points exist. This not only helps in finding these local extremities but also in understanding the overall geometry of the polynomial function.