Problem 118 - College Algebra

Question

Begin by graphing the cube root function, \(f(x)-\sqrt[3]{x} .\) Then use transformations of this graph to graph the given function. $$ g(x)-\sqrt[3]{-x+2} $$

Step-by-Step Solution

Answer

After graphing the cube root function, a reflection over the y-axis was performed and then the function was horizontally shifted to the left by 2 units to graph the function \(g(x) = \sqrt[3]{-x + 2}\).

Step 1: Graphing the Cube Root Function

Firstly, plot the base function, the cube root function \(f(x) = \sqrt[3]{x}\). Plot some points such as (-8,-2), (-1,-1), (0,0), (1,1), and (8,2) and use these points to sketch the graph. Remember, the graph of the cube root function increases for all real values of \(x\), passing through the points plotted.

Step 2: Reflection of the Cube Root Function

The function \(g(x) = \sqrt[3]{-x}\) is a reflection of the base function \(f(x) = \sqrt[3]{x}\). This means that every positive \(x\) value becomes negative, and every negative \(x\) value becomes positive in the graph. Reflect the original cube root graph over the y-axis to achieve this. The reflected graph is otherwise identical to the original cube root function, but reversed left-to-right.

Step 3: Horizontal Shift of the Reflected Cube Root Function

In the given function, \(g(x) = \sqrt[3]{-x + 2}\), observe that '+2' has been added to the '-x' inside the cube root. This results in a horizontal shift (translation) to the left by 2 units. Shift the reflected graph from Step 2, two units to the left.

Key Concepts

Graph Transformations

Understanding graph transformations can help you easily manipulate functions and visualize their changes. For the cube root function, consider its base form as the starting point. The primary goal is to see how adjusting the equation transforms the graph:

• Scaling: Making the graph wider or narrower.
• Shifting: Moving the graph up, down, left, or right.
• Reflecting: Flipping the graph over the x-axis or y-axis.

In our case, we deal with reflections and shifts that simplify how the function modifies its appearance on the graph. These transformations follow very logical rules, so understanding them can make graphing functions intuitive.

Reflection Over Y-axis

The reflection over the y-axis involves flipping the graph to the opposite side. For functions such as the cube root, this can affect the entire graph's position across the y-axis. Imagine reflecting the base cube root function \( f(x) = \sqrt[3]{x} \), which is centered along the origin.

• Reflecting the base function gives us:\( g(x) = \sqrt[3]{-x} \)
• In this reflected graph, any point (x, y) on the original graph moves to (-x, y).

This specific transformation flips any positive input value to its negative counterpart, and vice versa. This is a crucial step when moving on to additional transformations.

Horizontal Shift

A horizontal shift changes the position of a graph along the x-axis. When looking at transformations involving shifts, the process involves:

• Identifying the shift in the equation; for instance, in the given function \( g(x) = \sqrt[3]{-x+2} \), the "+2" inside the cube root indicates a shift.
• The sign of the number inside determines the direction. A positive sign results in a shift to the left.

In our equation, '+2' causes the graph to move 2 units to the left. Think of this as moving the entire graph sideways without altering its shape or the reflection applied earlier.

Plotting Points

Plotting points is an important part of understanding graph transformations. By plotting specific coordinates on the graph, you see how the graph's shape and position change through transformations. Start with a basic set of known points for the cube root function, such as:

• (-8, -2)
• (-1, -1)
• (0, 0)
• (1, 1)
• (8, 2)

Once you transform the function with reflections and shifts, adjust these points accordingly:

• Reflect them over the y-axis to match the reflected cube root function.
• Shift them horizontally based on the transformation specifics (i.e., two units left for this exercise).

As you plot these recalculated points, you can sketch the transformed function easily, following its new path defined by these key coordinates. This methodically helps to visualize how each transformation step alters the graph.