Problem 11 - College Algebra and Calculus: An Applied Approach

Question

Use the graph of \(y=x^{3}\) to sketch the graph of the function. $$f(x)=(x+1)^{3}-4$$

Step-by-Step Solution

Answer

The graph of the function \(f(x)=(x+1)^{3}-4\) is a transformation of the graph \(y=x^{3}\), obtained by shifting the graph of \(y=x^{3}\) one unit to the left and four units down. The transformed graph passes through the point (-1,-4) instead of the origin.

Step 1: Graphing the Initial Function

Initially, it would be crucial to comprehend the function \(y=x^{3}\)'s basic form. This cubic function reveals an increasing trend. The graph would pass through the origin, as there is no constant added or subtracted.

Step 2: Applying the Transformations

Now, apply the transformations to \(y=x^{3}\) as indicated in the function \(f(x)=(x+1)^{3}-4\). The \(+1\) within the parenthesis implies a horizontal shift 1 unit to the left. The \(-4\) outside of the parentheses suggests a vertical shift 4 units downward.

Step 3: Sketching the Transformed Graph

The graph \(f(x)=(x+1)^{3}-4\) is then obtained by translating each point in the original graph \(y=x^{3}\) 1 unit to the left and 4 units down. Hence, instead of passing through the origin, this graph will now pass through the point (-1,-4).

Key Concepts

Function Transformations

Function transformations involve taking a basic graph and altering it in various ways to produce a new graph. In the context of cubic functions, which are equations of the form \(y = x^3 + bx^2 + cx + d\), transformations can include

• translations (shifts)
• stretches or compressions
• reflections

For the cubic function \(f(x)=(x+1)^3-4\), we focus mainly on translations. These are changes that do not alter the shape but move the graph horizontally or vertically.
Most transformations can be understood through changes directly applied to the basic cubic function \(y = x^3\). Whenever we have an expression like \((x+a)^3\), it infers a change in the x-direction, often a horizontal shift. When we see a term like \(-4\) added outside the cube, that indicates a vertical shift.
By recognizing and applying these transformations, it becomes easier to graph complex functions by shifting simple, well-known graphs.

Graphing

Graphing is essentially plotting points on a coordinate grid to visualize mathematical functions. For cubic functions like \(y = x^3\), the process begins by identifying key characteristics. The basic graph starts at the origin \((0, 0)\) and extends symmetrically through the first and third quadrants, displaying a sharp increase in y-values as x moves away from zero in both positive and negative directions.
To graph function transformations like \(f(x)=(x+1)^3-4\), first consider the graph of \(y = x^3\). Then apply the transformations: each point \((x, y)\) on \(y = x^3\) moves to a new location based on the translations indicated. This approach helps maintain the overall shape while relocating it according to the transformation rules.
By using graphing techniques, students can better understand how mathematical equations translate into visual forms in a coordinate plane.

Horizontal and Vertical Shifts

Horizontal and vertical shifts are transformations that move a graph either left or right, or up or down. For the function \(f(x)=(x+1)^3-4\), we apply both directions of shift.

• The term \((x+1)\) indicates a horizontal shift. A positive number in \((x+a)\) actually shifts the graph to the left. For \(x+1\), it moves one unit to the left.
• The \(-4\) at the end represents a vertical shift, moving the graph four units downward.

This combination of shifts replaces the graph's original position with a new one. For \(f(x)=(x+1)^3-4\), the graph of \(y = x^3\) initially passing through the origin \((0, 0)\) is now shifted to \((-1, -4)\).
Recognizing these shifts is crucial for accurately sketching graphs and understanding their behavior. Horizontal shifts change the inputs (x-values), while vertical shifts alter the outputs (y-values), effectively repositioning the curve without changing its shape.