Problem 29 - Applied Calculus

Question

Sketch several members of the family \(y=x^{3}-a x^{2}\) on the same axes. Discuss the effect of the parameter \(a\) on the graph. Find all critical points for this function.

Step-by-Step Solution

Answer

Parameter \( a \) changes the critical point and shape of the graph. Critical points: \( x = 0 \) and \( x = \frac{2a}{3} \).

Step 1: Understand the Function

The given function is \( y = x^3 - ax^2 \). It is a cubic function where the parameter \( a \) influences the shape and position of the graph. Our goal is to investigate how changing \( a \) affects the graph and identify any critical points.

Step 2: Derive Critical Points

To find the critical points, we take the derivative of \( y \) with respect to \( x \). The derivative is \( y' = 3x^2 - 2ax \). Set the derivative equal to zero to find critical points: \( 3x^2 - 2ax = 0 \). Factor out \( x \): \( x(3x - 2a) = 0 \). So, \( x = 0 \) or \( x = \frac{2a}{3} \) are the critical points.

Step 3: Sketching the Graph

To sketch the graph, consider different values of \( a \). For example, set \( a = 0 \), \( a = 1 \), and \( a = -1 \). For each, substitute back into the function: - For \( a = 0 \), the function becomes \( y = x^3 \).- For \( a = 1 \), \( y = x^3 - x^2 \).- For \( a = -1 \), \( y = x^3 + x^2 \). Draw these functions on the same set of axes to compare the changes.

Step 4: Analyze the Effect of 'a'

The parameter \( a \) shifts the position of the critical point \( x = \frac{2a}{3} \). As \( a \) increases or decreases, the position of this point moves along the x-axis. Also, the term \( -ax^2 \) affects the concavity and shape of the parabola produced by the cubic function, creating different turning points depending on \( a \).

Key Concepts

Graph Sketching

Graph sketching is a crucial skill in understanding how mathematical functions behave visually. For the given cubic function, \( y = x^3 - ax^2 \), sketching involves identifying how different values of the parameter \( a \) influence the graph. Start by realizing that each of these functions will have a general cubic shape. The term \( -ax^2 \) will adjust the curve, adding complexity.To accurately sketch:

• Choose a variety of \( a \) values, such as \( a = 0, 1, -1 \).
• Begin sketching the curve for \( a = 0 \), which looks like a standard cubic function \( y = x^3 \).
• For \( a = 1 \), you reduce the growth at larger \( x \) values, forming \( y = x^3 - x^2 \).
• With \( a = -1 \), it modifies the curve to \( y = x^3 + x^2 \), introducing a different turning behavior.

Graph sketching helps visually capture these shifts, providing a deeper insight into function behavior.

Critical Points

Critical points are where the slope of the function is zero, meaning the derivative is zero. Finding these points can provide key insights into the behavior of the function. For the cubic function \( y = x^3 - ax^2 \), the derivative is given by \( y' = 3x^2 - 2ax \).To find critical points:

• Set the derivative \( 3x^2 - 2ax = 0 \) and solve the equation.
• Factorizing gives \( x(3x - 2a) = 0 \), leading to solutions \( x = 0 \) and \( x = \frac{2a}{3} \).

This means that, irrespective of \( a \), there is always a critical point at \( x = 0 \). The other critical point, \( x = \frac{2a}{3} \), varies with the value of \( a \), indicating how transformable the function's peaks and troughs are based on this parameter.

Parameter Influence

The parameter \( a \) plays a significant role in transforming the shape and position of the function \( y = x^3 - ax^2 \). The influence of \( a \) can be seen in several ways as it directly impacts the graph:

• Position of Critical Points: As \( a \) changes, the critical point \( x = \frac{2a}{3} \) moves accordingly. This movement can shift the maxima and minima of the function along the x-axis, altering where the function's turning points occur.


• Concavity and Shape: The term \(-ax^2\) distinctly alters the graph's curvature. For \( a > 0 \), it introduces a downward bend forcefully affecting the graph around the positive x-axis. Conversely, if \( a < 0 \), the curve gets an upward add-on, affecting the negative side more strongly.

Understanding these parameter effects can provide valuable insight into predicting and manipulating the behavior of cubic functions, especially in practical applications where such modifications may be desired.