Problem 47 - Calculus with Concepts in Calculus

Question

Let \(f(x)=x^{3}\). a. On the same screen, plot the graphs of \(y=f(x), y=\) \(f(x-1)\), and \(y=f(x-2)\). b. On the same screen, plot the graphs of \(y=f(x), y=\) \(f(x+1)\), and \(y=f(x+2)\). c. On the basis of parts (a) and (b), how do you think one obtains the graph of \(y=f(x+c)\) from the graph of \(f ?\) Consider the two cases \(c>0\) and \(c<0\) separately.

Step-by-Step Solution

Answer

The graph of \(y = f(x+c)\) is a horizontal shift by \(c\) units: left if \(c > 0\), right if \(c < 0\).

Step 1: Understand the function

The original function given is \(f(x) = x^3\). This function represents a cubic graph that is symmetric with respect to the origin.

Step 2: Evaluate transformations for part (a)

For part (a), we need to transform the graph of \(f(x)\) to obtain \(f(x-1)\) and \(f(x-2)\).- \(f(x-1)\): This results in a horizontal translation to the right by 1 unit.- \(f(x-2)\): This results in a horizontal translation to the right by 2 units.

Step 3: Plot graphs for part (a)

On the same screen, plot three graphs:- \(y = f(x) = x^3\): the original cubic graph.- \(y = f(x-1) = (x-1)^3\): shifted right by 1 unit.- \(y = f(x-2) = (x-2)^3\): shifted right by 2 units.

Step 4: Evaluate transformations for part (b)

For part (b), apply transformations to obtain \(f(x+1)\) and \(f(x+2)\).- \(f(x+1)\): This results in a horizontal translation to the left by 1 unit.- \(f(x+2)\): This results in a horizontal translation to the left by 2 units.

Step 5: Plot graphs for part (b)

On the same screen, plot:- \(y = f(x) = x^3\): the original cubic graph.- \(y = f(x+1) = (x+1)^3\): shifted left by 1 unit.- \(y = f(x+2) = (x+2)^3\): shifted left by 2 units.

Step 6: Formulate conclusions

When \(f(x + c)\) is considered:- For \(c > 0\), the graph shifts to the left by \(c\) units.- For \(c < 0\), the graph shifts to the right by \(|c|\) units. Therefore, the graph of \(y = f(x+c)\) can be obtained by translating the graph of \(y = f(x)\) horizontally by \(c\) units.

Key Concepts

Horizontal Translation

When talking about function transformations, horizontal translation refers to moving the entire graph of a function left or right along the x-axis. This change is made without altering the shape or the orientation of the graph itself. For a function described by the equation \(f(x)\), translating the graph horizontally is achieved by adjusting the input \(x\) with an added or subtracted constant \(c\):

\item If the equation transforms to \(f(x-c)\), the graph shifts to the right by \(|c|\) units. \item Conversely, if it transforms to \(f(x+c)\), the graph shifts to the left by \(c\) units.

This principle was highlighted in parts (a) and (b) of the exercise: \(f(x-1)\) and \(f(x-2)\) result in the graph moving to the right, while \(f(x+1)\) and \(f(x+2)\) result in the graph moving to the left. Through horizontal translation, the graph retains its original form but adjusts its position along the x-axis, making it a vital concept in graph transformations.

Cubic Function

A cubic function is a polynomial function of degree three, generally expressed in the form \(f(x) = ax^3 + bx^2 + cx + d\). In our case, we're dealing with the simplified cubic function \(f(x) = x^3\).

\item The graph of this function creates a curve with points symmetric around the origin. This reflects both vertically and horizontally, forming an S-shaped curve that passes directly through (0,0). \item Unlike linear or quadratic graphs, what makes cubic functions unique is their capacity to change direction more noticeably within their plots, creating either a right or left turn depending on the leading coefficient and the specific transformations applied.

The original cubic function \(x^3\) is essential to understand before considering graph transformations, as its symmetry and curvature dictate the appearance of any translated versions.

Function Transformation

Function transformation involves adjusting the properties of a graph in various ways, like translating, reflecting, or scaling it. Each transformation affects the graph differently:

\item **Translation**: Shifts the entire graph along either the horizontal or vertical direction. For instance, replacing \(x\) with \(x-c\) causes a horizontal shift whereas adding or subtracting a constant from \(f(x)\) results in a vertical translation.\item **Reflection**: Flips the graph over a specific axis, such as when multiplying \(f(x)\) by -1, reflecting it vertically across the x-axis. \item **Scaling**: Alters the size of the graph vertically or horizontally, multiplying \(f(x)\) or \(x\) by a constant.

In this exercise's context, we observed horizontal translation, where the cubic function's graph shifts without altering its shape. Understanding how function transformations alter the graph locally, like shifting or reshaping, helps in mastering how to adapt any given function to fit specific criteria or scenarios in mathematical analysis.