Problem 113 - Precalculus Essentials

Question

Begin by graphing the cube root function, \(f(x)=\sqrt[3]{x} .\) Then use transformations of this graph to graph the given function. $$r(x)=\frac{1}{2} \sqrt[3]{x+2}-2$$

Step-by-Step Solution

Answer

The graph of \( f(x) = \sqrt[3]{x} \) is shifted to the left by 2 units, downwards by 2 units, and vertically stretched by a factor of 0.5 to obtain the graph of \( r(x) = \frac{1}{2} \sqrt[3]{x+2}-2 \).

Step 1: Graph the Cube Root Function

To graph the cube root function \( y = f(x) = \sqrt[3]{x} \), a set of points (x, y) can be plotted on the Cartesian coordinate system where x is a real number and y is the cube root of x.

Step 2: Horizontal Shift

The function \( r(x) = \frac{1}{2} \sqrt[3]{x+2}-2 \) is derived by shifting the original function \( f(x) = \sqrt[3]{x} \) 2 units to the left. This transformation affects the x-values of the function. The graph of \( r(x) \) can be obtained by taking the graph of \( f(x) \) and shifting each point 2 units to the left.

Step 3: Vertical Shift

The graph is further transformed by moving it down by 2 units. This transformation affects the y-values of the function. The graph of \( r(x) \) can be obtained by taking the previous graph and shifting every point down by 2 units.

Step 4: Vertical Stretch

The function \( r(x) = \frac{1}{2} \sqrt[3]{x+2}-2 \) involves a vertical stretch by a factor of 0.5. The graph of \( r(x) \) can be obtained by taking the previous graph and stretching it vertically by a factor of 0.5, meaning every y-coordinate of the graph of \( f \) is halved.

Key Concepts

Cube Root Function

The cube root function, represented as \( y = \sqrt[3]{x} \), is a fundamental mathematical function. This function calculates the value that, when raised to the power of three, gives the number \( x \). The graph of the cube root function provides key insights into its nature:

• The domain and range of \( \sqrt[3]{x} \) are all real numbers \( (-\infty, \infty) \).
• The graph passes through the origin, \( (0,0) \), indicating symmetry around it.
• It gradually increases, showcasing a curved path in both the positive and negative directions.

Understanding the cube root function is essential before applying transformations such as shifts or stretches.

Horizontal Shift

A horizontal shift involves moving the graph of a function left or right on the coordinate plane. For the function \( r(x) = \frac{1}{2} \sqrt[3]{x+2} - 2 \), the \( +2 \) inside the cube root indicates a shift.

• Here, each x-coordinate is adjusted by subtracting 2.
• This results in the entire graph of \( \sqrt[3]{x} \) moving 2 units to the left.

Horizontal shifts do not affect the y-values directly. They modify the point location on the x-axis, allowing us to reposition the function as needed.

Vertical Shift

A vertical shift adjusts the position of a graph upwards or downwards on the y-axis. In our function \( r(x) = \frac{1}{2} \sqrt[3]{x+2} - 2 \), the \( -2 \) at the end triggers this transformation:

• Each y-coordinate is reduced by 2, shifting the graph downward.
• This affects the vertical placement but not the shape of the curve.

Vertical shifts add or subtract a constant from the y-values, allowing us to elevate or lower the graph efficiently.

Vertical Stretch

A vertical stretch modifies the graph by scaling the y-coordinates, changing the graph's "thinness" or "thickness." In \( r(x) = \frac{1}{2} \sqrt[3]{x+2} - 2 \), the coefficient \( \frac{1}{2} \) is responsible for this transformation:

• The y-values are multiplied by \( \frac{1}{2} \), effectively halving them.
• This makes the graph appear flatter compared to its original form.

Vertical stretching doesn't change the direction of the curve. Instead, it either extends or compresses it vertically, altering its overall steepness.