Problem 75 - Algebra and Trigonometry

Question

A family of functions is given. In parts (a) and (b) graph all the given members of the family in the viewing rectangle indicated. In part (c) state the conclusions you can make from your graphs. $$f(x)=(x-c)^{3}$$ (a) \(c=0,2,4,6 ; \quad[-10,10]\) by \([-10,10]\) (b) \(c=0,-2,-4,-6 ; \quad[-10,10]\) by \([-10,10]\) (c) How does the value of \(c\) affect the graph?

Step-by-Step Solution

Answer

The value of \(c\) shifts the graph horizontally without changing its shape; right for positive and left for negative \(c\).

Step 1: Understanding the Family of Functions

The given family of functions is defined by \(f(x) = (x - c)^3\), where \(c\) is a constant that shifts the graph along the x-axis. Different values of \(c\) will result in different graphs in the defined family.

Step 2: Graphing for Part (a)

For part (a), we need to graph the functions for \(c = 0, 2, 4, 6\) within the viewing rectangle \([-10, 10]\) by \([-10, 10]\). - When \(c = 0\): The function becomes \(f(x) = x^3\), which is a basic cubic graph with origin at \((0,0)\).- When \(c = 2\): The function is \(f(x) = (x-2)^3\), shifting the graph right by 2.- When \(c = 4\): The graph \(f(x) = (x-4)^3\) shifts right by 4.- When \(c = 6\): The graph \(f(x) = (x-6)^3\) shifts right by 6.

Step 3: Graphing for Part (b)

For part (b), we graph the functions for \(c = 0, -2, -4, -6\) within the same rectangle.- When \(c = 0\): Again, the function is \(f(x) = x^3\).- When \(c = -2\): The function \(f(x) = (x + 2)^3\) shifts the graph left by 2.- When \(c = -4\): The graph \(f(x) = (x+4)^3\) shifts left by 4.- When \(c = -6\): The graph \(f(x) = (x+6)^3\) shifts left by 6.

Step 4: Analyzing the Graphs

In part (c), the value of \(c\) affects the graph by horizontally shifting it along the x-axis without changing its shape. Positive values of \(c\) shift the graph to the right, while negative values shift it to the left. The cubic shape of the graph remains unchanged.

Key Concepts

Cubic Functions

Cubic functions are a special kind of polynomial function with the general form \(f(x) = ax^3 + bx^2 + cx + d\). However, in many exercises, like the one provided, cubic functions often focus on their simplest form. When graphed, these functions have a distinctive "S" shape. This basic structure makes it easier to see how transformations like shifts affect the graph.

In their simplest form, such as \(f(x) = x^3\), there's no horizontal or vertical shift, and the graph passes through the origin (0,0). The function increases on intervals to the right of the origin and decreases to the left, showing symmetry. Cubic functions have the characteristic of one real root, and interestingly, they can have zero, one, or two extreme points, depending on their specific form. The simplicity and symmetry make it straightforward to visualize shifts and other transformations.

Horizontal Shifts

Horizontal shifts occur when we alter the x-coordinates of the points on a graph, but not their y-coordinates. For the function \(f(x) = (x - c)^3\), the variable \(c\) modifies where the central "S" of the cubic graph is positioned along the x-axis.

• If \(c > 0\), the graph of the cubic function shifts to the right.
• If \(c < 0\), the graph shifts to the left.
• When \(c = 0\), there is no shift, and the graph remains at the origin.

Understanding horizontal shifts helps in predicting and analyzing how graphs of functions change with different values of \(c\). This ability to foresee the transformation is crucial when working with a family of functions.

Family of Functions

A family of functions refers to a group of functions that are related and differ only by certain parameters, such as the constant \(c\) in our example. Here, every function in the family of \(f(x) = (x-c)^3\) has the same basic cubic shape but shifts horizontally as discussed.

Studying families of functions allows us to see the impact of varying parameters on the graph's shape and position. This concept is especially useful because it translates well to more complex scenarios in algebra and calculus. Additionally, it aids in understanding function transformations more deeply and intuitively. Patterns can be recognized easily among the family members, leading to more efficient problem-solving in mathematics.