Problem 77 - Brief 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 is shifted 3 units left and 6 units up from \( y = x^3 \).

Step 1: Identify the Base Graph

The base function is \( y = x^3 \). This is a cubic function, which is symmetric around the origin and passes through points such as (-2, -8), (-1, -1), (0, 0), (1, 1), and (2, 8).

Step 2: Analyze Horizontal Shift

The modified function is \( y = (x+3)^3 + 6 \). The term \((x+3)^3\) indicates a horizontal shift. Since it is \(x+3\), it means the graph of \( y = x^3 \) will shift left by 3 units.

Step 3: Analyze Vertical Shift

The "+6" outside the cubed term indicates a vertical shift upwards by 6 units. So, after shifting 3 units left, the entire graph will move vertically upward by 6.

Step 4: Graph Both Functions

First, graph the function \( y = x^3 \) using its known points and shape. Next, apply the transformations: Shift the graph left by 3 units and then, from there, shift it upward by 6 units to represent \( y = (x+3)^3 + 6 \).

Step 5: Compare Graphs

Verify visually that the graph of \( y = (x+3)^3 + 6 \) is the graph of \( y = x^3 \) moved 3 units to the left and 6 units upwards. Both graphs will still have a cubic shape, but their positions will differ according to these transformations.

Key Concepts

Cubic Function

A cubic function is a polynomial of degree 3. Its standard form is represented as \( y = ax^3 + bx^2 + cx + d \), where \( a, b, c, \) and \( d \) are constants, and \( a eq 0 \). The simplest cubic function is \( y = x^3 \), which produces a curve that passes symmetrically through the origin.One of the key features of cubic functions is their S-shaped curve, which typically span through both negative and positive y-values. The basic graph crosses the origin and points like \((-2, -8)\), \((-1, -1)\), \( (0, 0) \), \((1, 1)\), and \((2, 8)\). This symmetry around the origin means if you fold the graph along the y-axis, both sides will overlap perfectly.

• Cubic graphs can exhibit one real root or three real roots.
• They usually have a turning point known as an inflection point, located where the curve changes concavity.
• As \( x \to \pm \infty \), \( y \to \pm \infty \) for cubic functions.

Understanding the basic structure of a cubic graph helps when applying transformations like shifts.

Horizontal Shift

A horizontal shift occurs when every point on the graph of a function moves left or right. This happens when you modify the \( x \) variable within the function. In our exercise, the function changes from \( y = x^3 \) to \( y = (x+3)^3 + 6 \), indicating a horizontal shift.The term \((x+3)^3\) signifies that the graph is moving horizontally. But how do we determine the direction? If you replace \( x \) with \( x + h \), the graph shifts horizontally by \( h \) units.

• \( x + h \): Shift to the left by \( h \) units.
• \( x - h \): Shift to the right by \( h \) units.

In the equation \( (x+3) \), the expression implies a shift to the left by 3 units. Each point in \( y = x^3 \) will be moved 3 units left from its original position. So, a point originally at \( (0, 0) \) will now be at \( (-3, 0) \), maintaining the graph’s original S-shape but repositioned leftward.

Vertical Shift

A vertical shift modifies the graph by moving it up or down without changing its shape. This occurs when a constant is added or subtracted to the function as a whole. In our function \( y = (x+3)^3 + 6 \), we can see that \(+6\) results in a vertical shift.The vertical shift can be easily understood and applied:

• \(+ c\): Move the graph upwards by \( c \) units.
• \(- c\): Move it downwards by \( c \) units.

In this exercise, the graph of \( y = (x+3)^3 \) is moved up 6 units. Originally, if a graph point was at \( (-3, 0) \) — after the horizontal shift — it will now be at \( (-3, 6) \).This upward movement impacts every point tandemly, so the new graph keeps its S-shape but is balanced higher in the coordinate plane. This simplifies graphing transformations as it’s simply moving the entire function up without altering slopes or symmetry.