Problem 57 - Precalculus with Limits

Question

In Exercises 55-62, write an equation for the function that is described by the given characteristics. The shape of \(f(x) = x^3\), but shifted 13 units to the right

Step-by-Step Solution

Answer

The equation of the function, which has the same shape as \(f(x) = x^3\) but is shifted 13 units to the right, is \(f(x) = (x-13)^3\).

Step 1: Identify Original Function

The given function is \(f(x) = x^3\).

Step 2: Apply Shift Principle

To shift a function 'h' units to the right, we replace \(x\) with \(x-h\) in the original function.

Step 3: Determine New Function

Following the principle in the previous step, to shift the function 13 units to the right, replace \(x\) with \(x-13\) in the original function. Hence, the new function, after being shifted 13 units to the right, is \(f(x-13) = (x-13)^3\).

Key Concepts

Transformation of Functions

Understanding how to transform functions is a fundamental aspect of algebra that allows us to graphically represent changes in equations. A transformation involves altering a function's graph either by shifting, stretching, compressing, or reflecting it. These transformations can happen either along the x-axis (horizontally) or the y-axis (vertically), and often include a combination of such changes.

When we speak of transforming functions, we typically do so by modifying the function's algebraic formula in a particular way that corresponds to the desired change in the graph. For instance, we can shift the entire graph to the right or the left by adding or subtracting a number to the variable x within the function. It's like picking up the whole graph and moving it without altering its shape. Similarly, vertical shifts involve adding or subtracting a number outside the function.

To maintain the clarity of an explanation when dealing with shifting functions, here are a few key points to remember:

• Horizontal Shifts: Replace x with (x - h) to move the graph h units to the right, or with (x + h) to move it h units to the left.
• Vertical Shifts: Add or subtract a number outside of the function to move it up or down.

Cubic Functions

A cubic function is a specific type of function characterized by its defining feature, the cube of the variable, which gives it the form \(f(x) = ax^3 + bx^2 + cx + d\), where a, b, c, and d are constants. The most basic cubic function is \(f(x) = x^3\). Its graph is known for the distinctive \('S'\)-shaped curve, symmetric about the origin when \(a\) is positive. This symmetry reflects the fact that every cubic function possesses an inflection point where the curvature of the graph changes direction.

When working with cubic functions, it's essential to grasp these characteristics:

• They can intersect the x-axis up to three times or as few as once, depending on their factors.
• They always have one point of inflection.
• Increasing the coefficient \(a\) stretches the graph vertically, making it steeper.
• Decreasing \(a\) to a negative value reflects the graph over the x-axis.

If the cubic function is defined simply as \(f(x) = x^3\), it represents the purest form without real or complex roots or constant terms, creating a clean, smooth curve that passes through the origin.

Horizontal Shifts

A horizontal shift is a type of transformation that moves a function along the x-axis without altering its shape. When you apply a horizontal shift to a function, you either add or subtract a constant from the x-variable before it is used in the function's rule.

Shifting Right: To move a graph to the right by h units, you replace every occurrence of x in the function's formula with (x - h). For example, if you have the function \(f(x) = x^3\), shifting it 13 units to the right would result in the transformed function \(f(x) = (x - 13)^3\). This is what happens when you replace x with (x - 13) in the original function.

Shifting Left: Conversely, to move a graph to the left, you would replace x with (x + h).

These shifts do not affect the 'steepness' of the graph or alter its orientation; they simply reposition it along the horizontal plane. In a cubic function, a horizontal shift maintains the property of one point of inflection, and the general \('S'\)-shaped curve but shifts where the curve intersects the x-axis and y-axis.