Problem 60 - 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 only points at which the graph of the polynomial \(f(s)\) crosses the s-axis are (-1,0) and \((2,0),\) and the only point at which it just touches the s-axis is \((0,0) .\) The function is positive on the intervals \((-\infty,-1)\) and \((2, \infty)\)

Step-by-Step Solution

Answer

A possible expression for the polynomial function is \(f(s) = a*(s+1)(s^2)(s-2)\) where 'a' is a positive constant. The function has a local maximum at (-1,0), local minimum at (2,0), and touches the s-axis at (0,0).

Step 1: Sketching the Graph

Since the polynomial crosses the s-axis at (-1,0) and (2,0) and it only touches the s-axis at (0,0), plot these points on the s-axis. Based on the intervals of positivity (-∞,-1) and (2, ∞), the graph is above the s-axis to the left of -1 and to the right of 2. Therefore draw your polynomial function emerging from below the s-axis at (-1,0), touching the s-axis at (0,0) and again emerging from below at (2,0).

Step 2: Indicating Possible Local Maxima and Minima

With the sketch, a local minimum occurs at (2,0) where it crosses the s-axis and heads upwards and a local maximum occurs at (-1,0) where it crosses the s-axis downwards. The point (0,0) does not give a local maximum or minimum as it merely touches the s-axis.

Step 3: Formulate the Polynomial

A possible polynomial equation for the given conditions could be \(f(s) = a*(s+1)(s^2)(s-2)\) where 'a' is a positive constant since the function is positive for s < -1 and s > 2

Step 4: Verification using Graphing Utility

Plot the function \(f(s) = a*(s+1)(s^2)(s-2)\) using any graphing utility. Adjust 'a' until the graph matches the previously sketched graph, crossing the s-axis at (-1,0) and (2,0), touching at (0,0), and is positive for \(s < -1\) and \(s > 2\).

Key Concepts

Local Maxima and Minima

When studying polynomial functions, it is important to identify key features like local maxima and minima. These are the points where the function reaches its highest or lowest value locally.
In the given exercise, local maxima and minima occur:

• A local maximum at (-1,0): This is where the graph crosses the s-axis and the function value increases from this point.
• A local minimum at (2,0): Here, the function value decreases before reaching this point and increases afterward.

The point where the function merely touches the s-axis, such as (0,0), does not represent either a maximum or minimum. Local extrema give insights into the polynomial's behavior and help in drawing an accurate graph.

Polynomial Expression

A polynomial expression is an algebraic expression consisting of terms that are summed together, with each term including a variable raised to a non-negative integer power. It's crucial to find a possible polynomial expression that fits the given graph conditions. In this task, you're constructing the function based on the intercepts and behavior of the graph.
The proposed polynomial expression for the graph was:\[f(s) = a*(s+1)(s^2)(s-2)\]
Considerations for this expression include:

• Roots of the polynomial, such as \(-1, 0, \) and \(2,\) where the function crosses or touches the s-axis.
• The degree of the polynomial, which gives hints about the general shape and limits the function's growth.
• The constant 'a', which affects the vertical stretch or compression of the graph.

Choosing a proper expression is a balance between matching the given roots and maintaining the function's specified regions of positivity.

Graphing Utility Verification

Graphing utilities are powerful tools for verifying the accuracy of your polynomial sketches and expressions. They provide visual confirmation of the function's behavior over chosen intervals.
Here's how to use it effectively:

• Input the proposed polynomial expression: Using a graphing calculator or software, insert \(f(s) = a*(s+1)(s^2)(s-2)\) and observe the plotted results.
• Adjust the constant 'a' to ensure the graph correctly crosses at (-1,0) and (2,0) and touches at (0,0).
• Check the intervals: Verify that the graph is positive on \((-∞,-1)\) and \((2,∞)\).

By matching the plotted graph with your expected sketch, graphing utilities confirm the accuracy of your polynomial expression and its behavior.