Problem 77 - Applied Calculus

Question

How will the graph of \(y=(x+3)^{3}+6\) differ from the graph of \(y=x^{3}\) ? Check by graphing both functions together.

Step-by-Step Solution

Answer

The graph of \(y=(x+3)^3+6\) is shifted 3 units left and 6 units up from \(y=x^3\).

Step 1: Understanding the Base Function

Begin with the base function, which is \(y=x^3\). This is a standard cubic function that passes through the origin (0,0) and is symmetrical about the origin. It has inflection points and is an odd function.

Step 2: Analyzing Horizontal Shift

The function \(y=(x+3)^3+6\) has a term \((x+3)\) inside the parentheses. This indicates a horizontal shift. The graph of the function \(y=x^3\) is shifted to the left by 3 units because the term \(x+3\) replaces \(x\).

Step 3: Calculating Vertical Shift

Next, notice the \(+6\) outside the cube. This changes the vertical position of the function. The graph of \(y=x^3\) is shifted upwards by 6 units due to this term.

Step 4: Graphing Both Functions

Graph both functions, \(y=x^3\) and \(y=(x+3)^3+6\), on the same grid. You should observe that the second graph is the same shape as the first but translated 3 units to the left and 6 units upwards.

Key Concepts

Cubic Functions

Cubic functions are a type of polynomial function with the highest degree of 3, hence they take the form of \(y = ax^3 + bx^2 + cx + d\). The simplest form, known as the cubic parent function, is \(y = x^3\). This function has some distinctive characteristics:

• Symmetry and Origin: It is symmetric about the origin since it is an odd function. For every point \((x, y)\), there exists a point \((-x, -y)\).
• Shape: It has a curve that starts from the bottom left, passes through the origin, and then proceeds to the top right, resembling a signature 'S' shape.
• Inflection Point: The origin \((0,0)\) is also an inflection point. This is where the graph changes concavity.

A cubic function's graph can be shifted and transformed in various ways to form new functions. These transformations include horizontal and vertical shifts, which are common and useful for altering the graph's position without changing its shape.

Horizontal Shift

In graph transformations, horizontal shift is when a graph moves along the x-axis, either to the left or the right, without any alterations to its vertical position or shape. This happens when a constant is added or subtracted from the \(x\) variable inside the function. For example, in the function \(y = (x+3)^3\), the graph shifts horizontally:

• Direction: A horizontal shift is determined by the sign of the constant. If the constant is positive, the graph moves to the left; if negative, it moves to the right. Here, \(x+3\) indicates a leftward shift by 3 units.
• Understanding: Think of it as adjusting the graph's starting point. If you started at \(x=0\) initially, you now start at \(x=-3\) after the shift.

Horizontal shifts are essential for modifying the placement of a graph on the coordinate plane, particularly when working with overlays of functions or comparing multiple graphs.

Vertical Shift

Vertical shifts in graph transformations involve moving a graph up or down along the y-axis, again without changing its shape or horizontal placement. This is achieved by adding or subtracting a constant from the entire function. In the case of \(y = (x+3)^3 + 6\), the graph undergoes a vertical shift:

• Direction: Unlike horizontal shifts, vertical shifts are intuitive: a positive constant lifts the graph upwards, while a negative constant shifts it downwards. The \(+6\) means the graph moves upward by 6 units.
• Effect on Function: Although the vertical shift changes the y-values of the function, the x-values and the fundamental shape remain constant. This means \(y = x^3\) becomes \(y = x^3 + 6\), thereby raising all points on the graph by 6 units.

These transformations allow a function to align better with real-world data or particular requirements, without altering the function's inherent behavior or characteristics.