Problem 56 - Precalculus : Building Concepts and Connections

Question

This set of exercises will draw on the ideas presented in this section and your general math background. Sketch the graph of a cubic polynomial function with exactly two real zeros. There can be more than one correct answer.

Step-by-Step Solution

Answer

A cubic function like \(f(x) = -(x - 1)(x - 2)^2\) or any other variant with two real zeros would have a graph that crosses the x-axis at one point and touches the x-axis (without crossing it) at another, giving it exactly two real zeros.

Step 1: Understand Real Zeros of Polynomial Functions

Real zeros of polynomial functions are the x-values where the function equals zero - in other words, where the graph intercepts the x-axis. For a cubic function to have exactly two real zeros, the graph of the function should cross the x-axis twice. A zero can be of multiplicity 1 (where the graph crosses the x-axis) or a higher multiplicity (where the graph touches, but doesn't cross the x-axis).

Step 2: Sketching the Cubic Polynomial Graph

The exact form of the cubic function isn't mentioned, so let's assume a general cubic function \(f(x) = a(x - r)(x - s)^2\), where 'r' and 's' are the zeros and 'a' is the coefficient that determines the direction of the 'S'. The graph should now be sketched as an 'S' curve which touches x axis at point 's' without crossing it, and intersects (crosses) the x-axis at point 'r'. This ensures only two real zeros for the function.

Step 3: Finalizing the Sketch

Different sets of 'r' and 's' values will result in different correct answers, but each should have the common characteristic where the graph crosses the x-axis at one point only and touches it at another. For example, if we plot \(f(x) = -(x - 1)(x - 2)^2\), it will cross the x-axis at 'x = 1' and touch it at 'x = 2' with only two real zeros.

Key Concepts

Real Zeros

Real zeros of a cubic polynomial function are the points where the graph intersects the x-axis. Simply put, these are the x-values that make the polynomial equal to zero. For a cubic function, which is a polynomial of degree three, you can have up to three real zeros. But it's crucial to remember that graph behavior at these zeros might vary based on multiplicity.

• When the graph crosses the x-axis, each crossing indicates a real zero with an odd multiplicity.
• When the graph just touches the x-axis without crossing, it represents a real zero with an even multiplicity.

This understanding helps when sketching a function with specific real zeros, such as the requirement of exactly two in this exercise. Here, you would expect the graph to cross the x-axis at one point and merely touch it at another, justifying two real zeros.

Multiplicity of Roots

The multiplicity of roots in a polynomial function reflects how many times a particular root is repeated. Understanding multiplicity is vital because it guides the shape and behavior of the graph near its zeros.

• If the multiplicity is odd, the graph will cross the x-axis at the zero.
• If the multiplicity is even, the graph will touch the x-axis and then bounce back.

For instance, consider the function given in the example, such as a generic cubic function expressed in the form of \(f(x) = a(x - r)(x - s)^2\). Here, the zero at \(x = r\) has a multiplicity of 1. This is why the graph crosses the x-axis at this point. Conversely, the zero at \(x = s\) has a multiplicity of 2, indicating the graph touches but does not cross the x-axis. This insight into multiplicity allows for precise graph sketching, understanding the number of times a zero is counted.

Graph Sketching

Graph sketching is all about visually representing the polynomial function, particularly highlighting key features like zeros and their behavior determined by multiplicity. A cubic polynomial graph typically assumes the shape of an 'S-curve.' Depending on the leading coefficient's sign, this 'S' can face upwards or downwards. For the function \(f(x) = a(x - r)(x - s)^2\), here's what to keep in mind:

• The graph is more like an 'S,' coming from top left to bottom right if the coefficient \(a\) is negative.
• It intersects the x-axis at \(x = r\), reflecting a real zero where it crosses.
• The graph just touches the x-axis at \(x = s\), showing a real zero with even multiplicity.

When graph sketching, adjusting values of \(r\), \(s\), and \(a\) affects the graph's position and direction. This flexibility implies multiple correct representations can satisfy the condition of having exactly two real zeros, as specified in the original exercise.