Problem 52 - Beginning and Intermediate Algebra

Question

If the following transformations are performed on the graph of \(f(x)\) to obtain the graph of \(g(x),\) write the equation of \(g(x)\). \(f(x)=\sqrt{x}\) is shifted down 6 units.

Step-by-Step Solution

Answer

The equation for \(g(x)\) is: \(g(x) = \sqrt{x} - 6\).

Step 1: Identify the transformation

We are given that the graph of \(f(x)\) is shifted downward by 6 units to form the graph of \(g(x)\). This means we will subtract 6 from the output of the function, \(f(x)\).

Step 2: Apply the transformation to \(f(x)\)

We will subtract 6 from the output of the function, so our transformation will look like this: \(g(x) = f(x) - 6\).

Step 3: Substitute \(f(x)\) into the transformation

Given that \(f(x) = \sqrt{x}\), we can substitute this into our equation for \(g(x)\): \(g(x) = \sqrt{x} - 6\).

Step 4: Simplify the equation for \(g(x)\)

We already have the equation for \(g(x)\) in a simple form: \(g(x) = \sqrt{x} - 6\).

Step 5: Verify the new equation

Our new function, \(g(x)\), should be a downward shift of \(f(x)\) by 6 units. Let's check: 1. The original function, \(f(x) = \sqrt{x}\), starts at the point (0, 0). 2. The transformed function, \(g(x) = \sqrt{x} - 6\), starts at the point (0, -6). This confirms that the graph of \(g(x)\) represents a downward shift of the graph of \(f(x)\) by 6 units.

Step 6: Final Answer

The equation for \(g(x)\) is: \(g(x) = \sqrt{x} - 6\).

Key Concepts

Graphing Functions

Graphing functions is a fundamental concept in mathematics that involves plotting the coordinates of a function onto a graph. This visual representation helps to understand the behavior and characteristics of a function. To graph a function:

• Identify the function equation, like \(f(x) = \sqrt{x}\).
• Determine key points (x, y) by plugging x-values into the function to get y-values.
• Plot these points on a coordinate grid.
• Join the points smoothly to reflect the continuous nature of the function.

Graphs provide insights into a function's intercepts, domain, range, and even its symmetry. For instance, the graph of \(f(x) = \sqrt{x}\) starts from the origin (0,0) and extends to the right, reflecting that it only takes non-negative x-values. Also, graphing helps to easily visualize transformations, like shifts, stretches, or reflections applied to a function.

Vertical Shifts

Vertical shifts involve moving the graph of a function up or down on the coordinate plane without altering its shape. This is achieved by adding or subtracting a constant from the function.

• To shift the graph upward, add a positive constant, \(g(x) = f(x) + c\).
• To shift downward, subtract a positive constant, \(g(x) = f(x) - c\).

For example, given \(f(x) = \sqrt{x}\) and using \(g(x) = f(x) - 6\), the function is shifted 6 units downwards. This transformation affects the y-values, lowering them by 6, while the x-values remain unchanged. Vertical shifts don't affect the domain and symmetry of the function but alter the range. In our case, the range changes from \([0, \infty)\) for \(f(x)\) to \([-6, \infty)\) for \(g(x)\). Recognizing these shifts by transforming the function formula is crucial for understanding the behavior of graphs.

Square Root Functions

Square root functions are a type of radical function represented by \(f(x) = \sqrt{x}\). These functions produce outputs that are non-negative because they express the principal square root.
The key characteristics of square root functions include:

• Domain: Non-negative values of x (\([0, \infty)\)).
• Range: Non-negative outputs (\([0, \infty)\)).
• Graph: Starts at the origin (0,0), and curves upwards to the right, resembling a narrow swoosh.
• Continuous and smooth, increasing gradually as x increases.

Square root functions can undergo various transformations such as shifts, stretches, or compressions. When a vertical shift is applied, as seen in the function \(g(x) = \sqrt{x} - 6\), this doesn't change the direction or shape of the curve but simply moves it up or down on the graph. Recognizing these transformations helps in accurately plotting and interpreting the function's graph.