Question

Describe the sequence of transformations from \(f(x)=\sqrt[3]{x}\) to \(y\). Then sketch the graph of \(y\) by hand. Verify with a graphing utility. \(y=\sqrt[3]{x+1}\)

Short answer

Firstly, the transformation from \(f(x)=\sqrt[3]{x}\) to \(y=\sqrt[3]{x+1}\) is identified as a horizontal shift to the left by 1 unit. Secondly, the original function and the transformed function are sketched; the latter being a shift to the left of the former. Finally, a graphing utility is used to verify these sketches and confirm their correctness.

Step-by-step solution

Identify the Transformation

First, the transformation from \(f(x)=\sqrt[3]{x}\) to \(y=\sqrt[3]{x+1}\) needs to be identified. In this case, it's a horizontal shift to the left by 1 unit. This is because adding a value inside the function \(f(x+1)=\sqrt[3]{x+1}\) moves the graph in the opposite direction on the x-axis (i.e., it goes left by 1 unit).

Sketch the Original and Transformed Function

Start by sketching the original \(f(x)=\sqrt[3]{x}\) function. It will appear as a line rising from the third quadrant, passing through the origin and continuing into the first quadrant. Now, for the transformed function \(y=\sqrt[3]{x+1}\), shift the original graph left by 1 unit on the x-axis. The new graph should now start from (-1,0), instead of (0,0). It will still stretch from the third quadrant into the first quadrant, but the curve will be shifted left.

Verifying the Graph using Graphing Utility

Now, use a graphing utility to graph both \(f(x)=\sqrt[3]{x}\) and \(y=\sqrt[3]{x+1}\). The graphs should correspond with the hand-drawn illustrations which helps in confirming the accuracy of the transformations made at Step 2.

Key concepts

Horizontal Shift

Understanding the horizontal shift in graph transformations is crucial for analyzing functions and their graphical behaviour. The shifting principle is straightforward: if you add a constant inside the function's argument, such as moving from \( f(x) = \sqrt[3]{x} \) to \( y = \sqrt[3]{x+1} \), you are applying a horizontal shift. Specifically, adding a number inside the function (with the variable) shifts the graph in the opposite direction of a number line. So, adding 1 inside the radical means the graph shifts 1 unit to the left. This movement is referred to as a horizontal shift because it occurs along the x-axis.

It's essential for students to remember that horizontal shifts do not affect the shape of the graph; they only affect the position. The graph of the cube root function maintains its original shape, including the direction of its rise or fall, its intercepts with the axes, and the slope at any given point after the transformation. The function's output is not altered by a horizontal shift; instead, it is the input values that are adjusted.

Cube Root Function

The cube root function, denoted as \( f(x) = \sqrt[3]{x} \), is an essential function in mathematics, representing the inverse of the cubic function \( y = x^3 \). It differs from the square root function in that it involves the third root, which allows for the existence of a real cube root for negative numbers as well. This property gives the cube root function its characteristic s-shaped curve which passes through the origin, rising in the first quadrant and falling in the third quadrant.

Graphing the cube root function by hand helps students understand the broader characteristics of its shape and behavior. Key points to consider when sketching include the point at the origin (0,0), the relative increase or decrease as you move away from the origin, and the symmetry of the graph about the origin. Essentially, the cube root graph rises slower than a linear function and will always cross the horizontal and vertical axes at the same point, since \( \sqrt[3]{0} = 0 \).

Graphing Utility

A graphing utility, such as a graphing calculator or mathematical software, is an essential tool for verifying the accuracy of hand-drawn graphical transformations. When students move from the conceptual understanding to practical application, graphing utilities offer a visual confirmation and can help identify any possible errors. By inputting the cube root function and its transformed equation into the utility, students can compare the digital output with their sketch, ensuring parity between the two.

When verifying transformations, students should pay attention to key features of the functions, such as intercepts, general shape, and asymptotic behavior if applicable. The advantage of using a graphing utility lies not only in its accuracy but also in the ability to handle more complex functions and transformations with ease, providing a graphical solution that might be time-consuming or difficult to produce by hand.

Sketching Graphs

Sketching graphs is a foundational skill that enables students to conceptualize the behavior of functions. When sketching the graph of a cube root function after a horizontal shift, start with the original graph. Make sure to note down the basic shape of the curve and any key features like intercepts with the axes. Then, apply the transformation by simply shifting every point of the original graph horizontally by the required units.

To accurately sketch transformed graphs like the shifted cube root function, here are some tips:

• Use a pencil for initial sketches so you can adjust as needed.
• Draw a smooth curve that shows the general trend of the function without worrying about perfect precision.
• Mark important points, such as intercepts and points where the graph shifts direction, to anchor the overall shape.
• Label axes and points clearly to avoid confusion when evaluating the transformation.
• Check the symmetry, especially for functions like the cube root, which are symmetrical about the origin.

With practice, the skill of sketching graphs by hand will help students better understand and connect with the underlying mathematics of the functions they are studying.