Problem 52 - A Graphical Approach to College Algebra

Question

Use translations of one of the basic functions \(y=x^{2}, y=x^{3},\) \(y=\sqrt{x},\) or \(y=|x|\) to sketch a graph of \(y=f(x)\) by hand. Do not use a calculator. $$y=(x-4)^{2}-4$$

Step-by-Step Solution

Answer

The graph of \(y = (x-4)^2 - 4\) is a parabola with vertex at (4, -4) that opens upwards.

Step 1: Identify the Base Function

The given function is \(y = (x-4)^2 - 4\). The base function here is \(y = x^2\), which is a basic quadratic function.

Step 2: Analyze the Transformation

The function \(y = (x-4)^2 - 4\) involves horizontal and vertical transformations. The term \((x-4)^2\) indicates a horizontal shift to the right by 4 units, and the \(-4\) outside the square indicates a vertical shift downward by 4 units.

Step 3: Sketch the Transformed Graph

Start with the graph of \(y = x^2\), a parabola with vertex at the origin (0,0). Apply the horizontal shift to move the vertex from (0,0) to (4,0). Next, apply the vertical shift to move the vertex from (4,0) to (4,-4). The graph of \(y = (x-4)^2 - 4\) is a parabola with vertex at (4, -4) and opens upward, maintaining the same shape as \(y = x^2\).

Key Concepts

Base Functions

Base functions are the starting point for understanding more complex graphs. They are the simplest form of a set of functions, which can be transformed to meet specific needs. For instance, the basic quadratic function is given by \[y = x^2\].This function creates a parabola that is symmetrical and opens upwards, with its vertex at the origin (0,0). By understanding base functions, you can use them as building blocks to plot more intricate graphs using transformations.

Some common base functions are:

• Linear: \(y = x\)
• Quadratic: \(y = x^2\)
• Cubic: \(y = x^3\)
• Square Root: \(y = \sqrt{x}\)
• Absolute Value: \(y = |x|\)

Recognizing these base functions helps you quickly assess and graph functions by identifying components like vertex and direction.

Graph Sketching

Graph sketching is a method that allows you to create a visual representation of a function by transforming its base function. It's an essential skill that gives insight into how a function behaves and assists in solving math problems. For the function given, \[y = (x-4)^2 - 4\],knowing that the base function is \(y = x^2\) helps plot the transformed graph.

Steps for effective graph sketching include:

• Start with the base graph.
• Apply transformations: look for shifts—both horizontal and vertical.
• Adjust the vertex and shape based on these transformations.

For instance, the function \((x-4)^2\) suggests a

Horizontal Shift

by moving the graph to the right by 4 units, while the \(-4\) indicates a

Vertical Shift

downward by 4 units. After applying these shifts, the vertex moves from (0,0) to (4, -4), maintaining the upward opening parabola.

Quadratic Functions

Quadratic functions are a category of polynomial functions that take the form \[y = ax^2 + bx + c\].They are defined by their characteristic parabolic shape. The simplest form is \(y = x^2\),where the graph is a "U" shaped curve with a vertex at (0,0). Quadratic functions are known for:

• Their symmetry: They are always symmetric around their vertex.
• Their direction: They open upwards if the leading coefficient \(a\) is positive and downwards if \(a\) is negative.
• Their vertex: The highest or lowest point, depending on opening direction.

Transformations in Quadratic Functions

In our example, \[y = (x-4)^2 - 4\],transformations shift the vertex from the origin to (4, -4). Such transformations help in adjusting the graph's position on the coordinate plane without altering its shape or symmetry. Understanding these aspects allows you to confidently sketch and analyze quadratic graphs without relying heavily on graphing software or calculators.