Problem 22 - A Graphical Approach to College Algebra

Question

Use transformations of graphs to sketch the graphs of \(y_{1}, y_{2},\) and \(y_{3}\) by hand. Check by graphing in an appropriate viewing window of your calculator. $$y_{1}=\sqrt[3]{x}, \quad y_{2}=2-\sqrt[3]{x}, \quad y_{3}=1+\sqrt[3]{x}$$

Step-by-Step Solution

Answer

Transformations involve reflections and vertical shifts of the parent function graph.

Step 1: Understand the Parent Function

The parent function is \(y=\sqrt[3]{x}\), which represents the cube root of \(x\). This function passes through the origin \((0,0)\) and is symmetric with respect to the origin. The graph resembles an S-shaped curve, which increases from bottom left to top right through the origin.

Step 2: Transform for \(y_1\)

Since \(y_1 = \sqrt[3]{x}\), it is identical to our parent function. There are no transformations applied, so the graph of \(y_1\) is exactly the base curve of \(y = \sqrt[3]{x}\).

Step 3: Transform for \(y_2\)

For \(y_2 = 2 - \sqrt[3]{x}\), we perform two transformations:1. Reflect \(y = \sqrt[3]{x}\) across the x-axis to get \(y = -\sqrt[3]{x}\).2. Shift the entire graph up by 2 units, resulting in \(y = 2 - \sqrt[3]{x}\). The new graph still passes through the origin; however, the origin-point reflection is moved to \((0, 2)\).

Step 4: Transform for \(y_3\)

For \(y_3 = 1 + \sqrt[3]{x}\), apply a vertical shift:1. Shift the curve of \(y = \sqrt[3]{x}\) up by 1 unit. The curve remains the same shape, only shifted vertically, moving the original point \((0, 0)\) to \((0, 1)\).

Step 5: Verification using a Graphing Calculator

Plot the functions \(y_1 = \sqrt[3]{x}\), \(y_2 = 2 - \sqrt[3]{x}\), and \(y_3 = 1 + \sqrt[3]{x}\) on a graphing calculator to verify correctness. Ensure the reflections and shifts are accurately depicted as described in the transformations.

Key Concepts

Parent Function

Every time we talk about transforming functions, it's crucial to start with the parent function. In this case, our parent function is the cube root function, expressed mathematically as \( y = \sqrt[3]{x} \). Let’s dive into what the cube root function looks like and why it’s important.
Why is it called a "parent"? It’s because all other transformed forms stem from this basic function. Think of it like the starting point or the building blocks of similar functions.
Here's what the graph of \( y = \sqrt[3]{x} \) looks like:

• The graph passes through the origin \((0, 0)\).
• It appears as an S-shaped curve running through the origin, gradually rising from bottom left to top right.
• The graph is symmetrical with respect to the origin.

Knowing the parent function helps us understand and predict the impact of various transformations like shifts and reflections. The magic is in seeing how these transformations work in the context of graph transformations.

Cube Root Function

The cube root function is the star of our transformations exercise. This function, \( y = \sqrt[3]{x} \), naturally forms an S-shaped curve. It’s fascinating to see this not just mathematically but visually on a graph.
This is how we see the cube root function being transformed in our exercise:

• For \( y_2 = 2 - \sqrt[3]{x} \), the graph undergoes two significant changes:
• First, there's a reflection across the x-axis. In simple terms, the S-shaped curve that goes upward now flips downward.
• Next, we move the curve up vertically by 2 units. This means every point on this reflected curve rises by 2.
• Next for \( y_3 = 1 + \sqrt[3]{x} \):
• This transformation involves just a vertical shift upward by 1 unit.
• The S-shaped curve remains untouched in its form, just shifted vertically so that the origin goes from \((0, 0)\) to \((0, 1)\).

Transformations like these are powerful tools in mathematics as they allow us to manipulate graphs with a stroke of creativity. Understanding each effect fundamentally involves visualizing these moves on a graph.

Graphing Calculator

A graphing calculator is a crucial technological tool for visualizing functions and their transformations. It's like having a small laboratory in your hands where you can experiment with graphs and see results instantly.
Here’s how you can use a graphing calculator effectively for our exercise:

• Firstly, plot the parent function \( y_1 = \sqrt[3]{x} \). This will show the characteristic S-curve we’re analyzing and transforming.
• Next, enter \( y_2 = 2 - \sqrt[3]{x} \) to observe the reflection and vertical shift in action. Watch how the graph flips and then lifts.
• Then, input \( y_3 = 1 + \sqrt[3]{x} \) and see how a simple upward shift changes the position of the same curve.
• Use the zoom and window settings to ensure the curves are displayed correctly. The appropriate viewing window can make the difference between a clear or confusing graph.

The beauty of using a graphing calculator lies in its ability to confirm what we know theoretically. It's like having an educational assistant - it confirms, displays, and sometimes even surprises us with the visual wonders of math.