Problem 116 - 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. $$ h(x)--\sqrt[3]{x-2} $$

Step-by-Step Solution

Answer

To graph \(h(x) = \sqrt[3]{x-2}\), simply shift the graph of \(f(x) = \sqrt[3]{x}\) to the right by 2 units. This is a horizontal shift resulting from subtracting a constant (2) from the x.

Step 1: Understanding the Base Function

First, graph the base function, \(f(x) = \sqrt[3]{x}\). The graph will pass through the points \((0,0), (1,1), and (-1,-1)\), because \(0^3 = 0\), \(1^3 = 1\), and \((-1)^3 = -1\). The graph cuts the x-axis at x=0, rises to the right and falls to the left.

Step 2: Understanding Transformations

Now, to understand how to transform the cubic root function into \(h(x) = \sqrt[3]{x-2}\), remember that when a constant is subtracted from the input of a function, the graph of the function shifts right by that many units. This is a horizontal shift. In this case, the constant value is 2, therefore, the graph of \(f(x) = \sqrt[3]{x}\) will shift 2 units to the right to become \(h(x) = \sqrt[3]{x-2}\).

Step 3: Graphing the Transformed Function

Again start by marking the point where the function intersects with both axes, but this time it is shifted to (2,0) instead of (0,0). The other points also shift accordingly e.g.,(3, 1), (1, -1). The graph of \(h(x)\) will look identical to the graph of \(f(x)\), but it will appear to be moved 2 units to the right.

Key Concepts

Cube Root Transformation

The process of modifying the basic cube root function, represented as f(x) = \(\sqrt[3]{x}\), involves alterations that adjust its shape, position, and orientation on a graph. To understand cube root transformations, imagine the standard cube root graph, which is distinct with its characteristic 'S' shape. This curve crosses the origin and moves upward to the right and downward to the left.

When transforming a cube root function, you might modify the equation by including constants or coefficients, which can stretch, compress, reflect, or translate the curve. For instance, adding a constant within the cube root, as in h(x) = \(\sqrt[3]{x - c}\), results in a horizontal shift of the graph by 'c' units to the right if 'c' is positive, or to the left if 'c' is negative. Understanding these alterations helps in predicting the movement of the curve on the coordinate plane and assesses its new intercepts, slope, and progression.

By mastering cube root transformations, students can quickly sketch new functions by making precise modifications to the base graph without redrawing the curve from scratch each time. It's like having a template that you can adjust as needed to fit the given equation.

Horizontal Shift

A horizontal shift is a type of function transformation that moves a graph to the left or right along the x-axis. The direction and magnitude of the shift are determined by the addition or subtraction of a constant in the function's formula. For the function f(x) = \(\sqrt[3]{x}\), a horizontal shift can be represented by h(x) = \(\sqrt[3]{x - h}\), where 'h' denotes the shift.

If we subtract a positive constant 'h', as seen in the textbook exercise where h(x) = \(\sqrt[3]{x - 2}\), the entire graph shifts 'h' units to the right. Conversely, if 'h' were negative, the graph would shift 'h' units to the left. Visualizing horizontal shifts is immensely helpful when you need to plot a graph quickly and accurately without plotting numerous points.

Tip: Always remember that adding to 'x' inside the function's formula results in a move left, and subtracting from 'x' leads to a move right, quite opposite to what you might initially expect. It's one of those counterintuitive rules that make a big difference in graphing!

Function Transformations

Function transformations are alterations made to a function's formula that affect the graph's appearance and position. These changes can include translations (shifts), reflections over the axes, and stretching or compressing the graph either vertically or horizontally. Common transformations involve adding, subtracting, multiplying, or dividing the function by constants, or changing the variable 'x' in various ways.

In the cube root function f(x) = \(\sqrt[3]{x}\), transformations can be applied to shift the graph horizontally as seen with h(x) = \(\sqrt[3]{x - 2}\), or vertically by manipulating the output, such as f(x) + k = \(\sqrt[3]{x}\) + k. They can also affect the shape of the graph; for example, multiplying the cube root function by a constant > 1 will stretch it vertically, while a number between 0 and 1 will compress it.

Effective use of function transformations helps students to navigate through complex equations and visualize their graphs efficiently. It's like editing a picture; once you know how the tools work, you can adjust any image to look exactly how you need it to.