Problem 71 - 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 that 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 of \(f(x) = (x-c)^3\) horizontally along the x-axis by \(c\) units.

Step 1: Analyze the function

We are given the function \(f(x) = (x-c)^3\). This is a cubic function where \(c\) shifts the graph horizontally. Different values of \(c\) will move the graph to the right or left along the x-axis.

Step 2: Graph for positive c-values

For part (a), plot the function for \(c = 0, 2, 4, 6\) in the viewing rectangle of \([-10, 10]\) by \([-10, 10]\). Each increase in \(c\) shifts the graph 2 units right. Thus, you'll have graphs with turning points at \((0,0)\), \((2,0)\), \((4,0)\), and \((6,0)\).

Step 3: Graph for negative c-values

For part (b), plot the function for \(c = 0, -2, -4, -6\) in the same viewing rectangle. Each decrease in \(c\) shifts the graph 2 units left. Therefore, these graphs have turning points at \((0,0)\), \((-2,0)\), \((-4,0)\), and \((-6,0)\).

Step 4: Draw Conclusion about the shift

Observe how varying \(c\) changes the horizontal position of the graph without altering its shape. Positive values of \(c\) move the graph right by \(c\) units, while negative values move it left by \(c\) units.

Key Concepts

Horizontal Shifts

In the realm of function graphing, horizontal shifts are an essential concept to grasp. When we tackle an equation like \( f(x) = (x-c)^3 \), the variable \(c\) plays a pivotal role in moving the graph sideways on the coordinate plane. This movement, or shift, happens without altering the graph's shape, size, or orientation.

• If \(c\) is positive, the entire cubic graph slides to the right by \(c\) units. This means a graph of \((x-2)^3\) has its turning point 2 units to the right of the origin \((2, 0)\).
• Conversely, if \(c\) is negative, the graph shifts to the left. Thus, \((x+2)^3\) moves to \((-2, 0)\). The structure stays the same, only the position changes.

Recognizing this pattern is crucial for understanding how different function transformations affect a graph's placement in a plane. Keep in mind that only the horizontal position changes with \(c\), offering a straightforward visual cue when sketching cubic graphs.

Cubic Functions

Cubic functions, described by the equation \( f(x) = x^3 \), are a staple in algebra due to their distinctive properties and shape. The standard cubic function features an S-shaped curve, originating at the origin (0,0) and stretching indefinitely in both directions on the x-axis.

Here are key characteristics of cubic functions:

• Shape: The graph of a cubic function is called a 'cubic curve.' It has an inflection point where the graph changes concavity, typically at the origin in the basic form \( f(x) = x^3 \).
• Direction: The function moves upwards in the positive x-direction and downwards in the negative x-direction.
• Symmetry: Cubic functions, especially \( f(x) = x^3 \), exhibit odd symmetry, meaning they are symmetric about the origin. This symmetry provides a predictable pattern when plotting the curve.

Understanding cubic functions is crucial for analyzing complex mathematical relationships as they serve as building blocks for more advanced calculus concepts. They illustrate non-linear behavior, offering an invaluable geometric perspective on polynomial functions.

Function Transformations

Function transformations cover a wide variety of changes we can apply to standard functions to alter their appearance and position on a graph. By manipulating the variables and constants in equations, we can achieve translations, reflections, dilations, and rotations. For our focus, let's see how transformations affect the cubic function \( f(x) = (x-c)^3 \).

• Translations: The concept of translating a function, like the discussed horizontal shift, involves moving the graph without rotating or resizing it. In \( (x-c)^3 \), the graph slides horizontally based on \(c\): right for positive \(c\), and left for negative \(c\). This does not affect the graph's basic S-shaped structure.
• Affine Transformations: While not as relevant for our current example, don't overlook that transformations can include scaling and reflecting the graph, changing its orientation or size. However, \((x-c)^3\) is purely about position.

Recognizing and applying these transformations helps in predicting how a function behaves and looks graphically. Gaining confidence in manipulating these changes empowers you to handle various mathematical challenges with greater ease.