Problem 20 - Algebra 2

Question

Graph each function. \(y=\sqrt[3]{x+2}-7\)

Step-by-Step Solution

Answer

The graph of the function \(y=\sqrt[3]{x+2}-7\) is a slightly distorted 'S' shape that is shifted 2 units to the left and 7 units down compared to the basic cubic root function.

Step 1 - Understand the parent function

The parent or basic function in this case is \(y=\sqrt[3]{x}\). This function looks like a slightly distorted 'S' and passes through the origin (0,0).

Step 2 - Acknowledge the transformation

In the equation \(y=\sqrt[3]{x+2}-7\), the +2 within the root means that the graph should shift 2 units to the left, and the -7 outside the root represents a shift downward by 7 units. This is because the transformation of a function \(y=f(x-h)+k\) translates the graph of function f(x) horizontally by h units and vertically by k units, where h = 2 and k = -7 in this case.

Step 3 - Graphing the transformed function

Begin by graphing the cubic root function \(y=\sqrt[3]{x}\). Then, shift the entire graph 2 units to the left and 7 units down to obtain the graph of function \(y=\sqrt[3]{x+2}-7\). It will have the same shape as the cubic root function but it will be relocated.

Key Concepts

Graph Transformations

When we talk about graph transformations, we are referring to changes in the position, shape, size, or orientation of a graph. These transformations help us understand how altering the equation of a function affects its visual representation. One common type of transformation involves translations, where the graph is shifted horizontally or vertically. Another type is scaling, which can stretch or compress the graph.

There are also reflections, which flip the graph over an axis. By understanding how each transformation works, you can easily alter and predict the appearance of graphs. In our exercise, the transformation of the cubic root function involves both horizontal and vertical shifts, making its graph move in a new position while maintaining its shape.

Function Shifts

Shifting a function refers to moving the entire graph of a function either horizontally, vertically, or both, without altering its shape. This is a way of adjusting the function's position on the graph. There are specific notations used in equations to denote these shifts.

• Horizontal shifts occur when we add or subtract a constant from the input variable, moving the graph left or right. In the equation \( y = \sqrt[3]{x+2} - 7 \), the "+2" inside the root causes a horizontal shift of 2 units to the left.
• Vertical shifts happen when we add or subtract a constant to the entire function, moving the graph up or down. Here, the "-7" outside the root causes a vertical shift of 7 units downward.

Understanding these shifts helps not only in graphing but also in translating real-world problems into mathematical functions. Recognizing patterns from function shifts can make it easier to predict and visualize changes.

Parent Function

A parent function is the simplest form of a function type, without any transformations such as shifts or scaling. It's basically the building block from which more complex functions can be derived through transformations. For cubic root functions, the parent function is \( y = \sqrt[3]{x} \).

This function exhibits a distinctive 'S' shape that passes through the origin (0,0) and adores a continuous curve without any sharp corners or breaks. Recognizing the parent function is essential as it helps you understand the fundamental shape and behavior of the graph. Once you know the parent function, you can apply various transformations to develop new functions and understand how they relate to the original graph.

• Helps in identifying basic graph shape.
• Acts as a reference for transformations.
• Facilitates understanding of functional behavior.

Thus, mastering the concept of the parent function paves the way for more advanced exploration of equations and their graph interpretations.