Problem 81 - A Graphical Approach to Precalculus with Limits

Question

each function has a graph with an endpoint (a translation of the point (0,0) .) Enter each into your calculator in an appropriate viewing window, and, using your knowledge of the graph of \(y=\sqrt{x}\), determine the domain and range of the function. (Hint: Locate the endpoint.) $$y=10 \sqrt{x-20}+5$$

Step-by-Step Solution

Answer

Domain: \([20, \infty)\), Range: \([5, \infty)\).

Step 1: Understand the base function

The base function in question is the square root function, given by \( y = \sqrt{x} \). The graph of this function starts at the point (0,0), and is defined for all \( x \geq 0 \). The graph is only in the first quadrant of the coordinate plane.

Step 2: Analyze the Transformation

The given function is \( y = 10\sqrt{x - 20} + 5 \). This is a transformation of the base function \( y = \sqrt{x} \). The transformation \( x - 20 \) shifts the graph 20 units to the right, and the transformation \( +5 \) shifts the graph 5 units upward. Additionally, the coefficient 10 vertically stretches the graph.

Step 3: Determine the Endpoint

The expression under the square root \( x - 20 \) must be non-negative, meaning \( x \geq 20 \). Thus, the endpoint of the function \( y = 10\sqrt{x - 20} + 5 \) will be at \( x = 20 \). Plugging \( x = 20 \) into the function gives \( y = 10\sqrt{0} + 5 = 5 \). Therefore, the endpoint is at (20, 5).

Step 4: Identify the Domain

Since the square root function \( \sqrt{x - 20} \) is defined for \( x \geq 20 \), the domain of the given function \( y = 10\sqrt{x - 20} + 5 \) is \( x \geq 20 \). In interval notation, this is \([20, \infty)\).

Step 5: Identify the Range

The smallest value of \( y \) occurs at the endpoint where \( y = 5 \). As \( x \) increases beyond \( x = 20 \), the value of \( y = 10\sqrt{x - 20} + 5 \) can increase without bound because the square root function gradually increases. Hence, the range is \( y \geq 5 \). In interval notation, this is \([5, \infty)\).

Key Concepts

Function Transformations

Function transformations help us understand how the graph of a function changes with modifications. Essentially, these transformations can alter the position, size, or shape of a graph.

• **Translation**: A function can be shifted horizontally or vertically. In the function \( y = 10 \sqrt{x - 20} + 5 \), the entire graph is moved 20 units to the right (due to \( x - 20 \)) and 5 units up (due to \( +5 \)). This means the original starting point \((0,0)\) of \( y = \sqrt{x} \) is now at \((20, 5)\).
• **Scaling**: The coefficient before the square root, 10, is a vertical stretch. It makes the graph steeper, meaning for each value of \( x \), \( y \) increases faster than in the unscaled base function.

Understanding transformations is key because it allows us to predict the graph's new shape and position based on changes to the function's expression. Graphs can be rescaled, mirrored, or translated, all based on the type of transformation applied.

Domain and Range

In any function, the domain and range are critical. The domain refers to all possible input values (or \( x \)-values) for which the function is defined.

• The domain of \( y = 10\sqrt{x - 20} + 5 \) is based on the expression inside the square root, \( x - 20 \). Since the square root is only defined for non-negative values, \( x \) must be 20 or greater. Thus, the domain is \([20, \infty)\).

The range, on the other hand, refers to all possible output values (or \( y \)-values) the function can produce.

• Since the function's starting \( y \)-value is 5, found at the endpoint \((20, 5)\), and \( y \) increases as \( x \) increases, the range is \([5, \infty)\). The graph can rise indefinitely beyond 5.

Knowing domain and range helps students understand restrictions and potential outputs the function can take, which is vital when analyzing graphs.

Square Root Function

The square root function, denoted as \( y = \sqrt{x} \), provides a basic framework for understanding other, more complex functions.

• Its graph starts from the point \((0,0)\), where \( x \geq 0 \), and increases gradually. This creates a distinct curved line in the positive quadrant of the graph.

When modifying the square root function as seen in \( y = 10\sqrt{x - 20} + 5 \), we use this basic shape and adjust it via transformations.

• The transformation shifts the start to \((20,5)\), and each transformation element alters the way this curve behaves (through translation and stretching).
• The square root's natural non-linearity means that although the change is subtle at first, \( y \) increases more rapidly as \( x \) goes further from the initial starting point.

Understanding the square root function and its behaviors is foundational for grasping how transformations affect it, leading to proper graphical analysis of transformed functions.