Problem 37 - Intermediate Algebra

Question

Graph each of the functions. $$f(x)=3(x-2)^{3}-1$$

Step-by-Step Solution

Answer

The function is a vertically stretched, shifted right by 2, and shifted down by 1 transformed cubic graph.

Step 1: Understand the Basic Function

The function given is a cubic function of the form \( f(x) = 3(x-2)^3 - 1 \). The basic cubic function is \( g(x) = x^3 \), which we will transform to graph \( f(x) \).

Step 2: Apply Horizontal Shift

The \( (x-2) \) inside the cubic term means we will shift the graph of \( g(x) = x^3 \) horizontally to the right by 2 units. To visualize this, consider how each point \( (x, y) \) on \( g(x) = x^3 \) moves to \( (x+2, y) \) on the horizontally shifted version \( g(x-2) = (x-2)^3 \).

Step 3: Apply Vertical Stretch

The coefficient 3 applies a vertical stretch to the transformed function. This means for every point \( (x, y) \) on \( (x-2)^3 \), the new \( y \)-value will be \( 3y \). This stretches the graph vertically by a factor of 3.

Step 4: Apply Vertical Shift

Subtracting 1 from the entire function, \( 3(x-2)^3 - 1 \), shifts the graph vertically downward by 1 unit. After the stretch and horizontal shift, each point's \( y \)-value is decreased by 1.

Step 5: Confirm Transformations on Key Points

Take a few key points on the original \( g(x) = x^3 \) graph, such as \( (-1, -1), (0, 0), (1, 1) \), transform them by first shifting by +2 (\( (1, 1), (2, 0), (3, 1) \)), then vertical stretch by 3 (\( (1, 3), (2, 0), (3, 3) \)), and finally shift them down by 1 (\( (1, 2), (2, -1), (3, 2) \)) for points on \( f(x) = 3(x-2)^3 - 1 \).

Key Concepts

Cubic Function Transformation

When graphing cubic functions like \( f(x) = 3(x-2)^3 - 1 \), understanding transformations is crucial. The basic cubic function is \( g(x) = x^3 \). This function is characterized by its smooth curve and symmetry about the origin. However, in the given function \( f(x) \), several transformations have been applied.

• **Horizontal Shift:** The expression \( (x-2) \) indicates a shift of the graph of \( g(x) = x^3 \) to the right by 2 units.
• **Vertical Stretch:** The coefficient \( 3 \) multiplies the function, stretching it vertically.
• **Vertical Shift:** The subtraction of 1 moves the entire graph down by 1 unit.

Each transformation modifies the graph in incremental stages, producing the final output graph of \( f(x) = 3(x-2)^3 - 1 \). It’s essential to apply the transformations in the correct sequence: first horizontal shift, then vertical stretch, and lastly vertical shift.

Vertical Stretch and Shift

The concept of vertical stretch and shift is key when you transform a function like \( f(x) = 3(x-2)^3 - 1 \). A vertical stretch is caused by the coefficient \( 3 \). This means that the values of \( y \) on the graph are 3 times their original height from the \( x \)-axis.Consider that each point \( (x, y) \) on the graph of \( g(x-2) = (x-2)^3 \) becomes \( (x, 3y) \). This change means the graph appears narrower because the stretches increase the \( y \)-value for every point.After stretching, the vertical shift comes into play. Subtracting 1 from the function indicates that every point on the previously stretched graph shifts down by 1 unit. For example, if you consider a point \( (x, 3y) \), it transforms into \( (x, 3y - 1) \). This shift downward adjusts the graph, moving the curve vertically along the \( y \)-axis.

Horizontal Shift in Graphing

Horizontal shifts are a fundamental principle in function transformations, especially with cubic functions. In \( f(x) = 3(x-2)^3 - 1 \), you see a classic horizontal shift.The term \( (x-2) \) nestled inside the cubic power causes this shift. Specifically, it moves the baseline graph of \( g(x) = x^3 \) to the right by 2 units.To visualize this, picture each key point \( (x, y) \) on the standard cubic graph as moving rightward by 2 units on every \( x \)-coordinate. For instance, a point originally at \( (0, 0) \) becomes \( (2, 0) \), and similarly, \( (-1, -1) \) becomes \( (1, -1) \). This step makes the whole graph shift along the \( x \)-axis without any alteration in the \( y \)-values initially.