Problem 83 - A Graphical Approach to Precalculus with Limits

Question

In Exercises \(81-83 \text { , each function has a graph with an endpoint (a translation of the point }(0,0) .)\) 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.) $$y=-0.5 \sqrt{x+10}+5$$

Step-by-Step Solution

Answer

Domain: [-10, ∞); Range: (-∞, 5]

Step 1: Understanding the Basic Function

The basic function here is \( y = \sqrt{x} \). Its graph is a curve starting from the point (0,0) and extending to the right, representing all non-negative values of \(x\). Its domain is \([0, \infty)\) and its range is \([0, \infty)\).

Step 2: Analyzing the Function Transformation

The given function is \( y = -0.5 \sqrt{x + 10} + 5 \). This implies several transformations to the basic function. The expression \(x+10\) inside the square root suggests a horizontal shift to the left by 10 units. The coefficient \(-0.5\) indicates a reflection over the x-axis and vertical shrinkage, and the \(+5\) implies a vertical upward shift by 5 units.

Step 3: Locating the Endpoint

For the function \( y = -0.5 \sqrt{x+10}+5 \), the endpoint is found by setting \( x + 10 = 0 \), which results in \( x = -10 \). This means the endpoint on the graph of the function is at \( (-10, 5) \).

Step 4: Determining the Domain

The domain of the function consists of all possible input values \( x \) that do not result in an undefined expression. Since we have a square root, \( x + 10 \geq 0 \) must hold, leading to \( x \geq -10 \). Thus, the domain is \([-10, \infty)\).

Step 5: Determining the Range

The range of the function consists of all possible output values \( y \). Starting from the graph’s endpoint at \( y = 5 \), the reflection inverts the graph function, moving it down from 5, never exceeding 5. Hence, the range is \((-fty, 5]\).

Key Concepts

Function Transformation

Function transformation involves changing the position, shape, or orientation of a basic function on the coordinate plane. For the square root function, understanding these transformations is key to predicting how graphs will behave under certain modifications.

In the context of the given function, several transformations occur:

• The basic square root function is initially centered at the origin, \((0, 0)\).
• When transformed as \(y=-0.5 \sqrt{x+10}+5\), it undergoes a horizontal shift to the left by 10 units. This is evident from the \(x+10\) indicating the x-values are adjusted by -10.
• There’s a vertical shift upward by 5 units, which is signified by the "+5" outside the square root.
• A reflection over the x-axis occurs due to the negative coefficient -0.5, flipping the curve upside down.
• The coefficient also causes a vertical shrink, reducing the steepness of the original graph.

Understanding these transformations allows us to modify the basic function systematically, giving us insight into its endpoint and overall behavior. Transformations like these are foundational in more complex function manipulations.

Square Root Function

The square root function, represented as \(y=\sqrt{x}\), is among the basic functions used to understand more complex mathematical functions. It is characterized by its unique half-parabolic shape that starts from the origin and extends infinitely to the right.

• The domain of the simple square root function is \[0, \infty)\], as square roots are defined only for non-negative values of \(x\).
• The range is also \[0, \infty)\], since the output value \(y\) cannot be negative.

These characteristics change when we apply transformations like translating or reflecting. Recognizing these basic properties helps in analyzing modifications to function behavior, such as horizontal and vertical shifts, as well as reflections, like those observed in \(y=-0.5 \sqrt{x+10}+5\). This function reverses and translates the basic square root graph, which impacts both its domain and range significantly, emphasizing the effect of transformations on function properties.

Graphing Functions

Graphing functions is a critical skill in understanding how functions behave visually, and it involves plotting points based on a function's equation on the Cartesian plane. For transformed functions like \(y=-0.5\sqrt{x+10}+5\), understanding the endpoint and shape of the graph is crucial.

Here's how this function is graphed:

• First, identify the endpoint by solving \(x+10=0\), setting \(x=-10\).
• Next, determine the starting point on the y-axis: \(y=5\). This is derived from approaching the root equation through transformations.
• The function is reflected over the x-axis, implying the graph moves downward as \(x\) increases.
• From the endpoint \((-10, 5)\), plot points to show the curve's nature, ensuring the downward trend due to the reflection is illustrated.

Graphing these points defines the function's domain and range visually, solidifying the function's behavior from the transformational effects. By moving from the abstract equation to a visual representation, students can elucidate intricate details of function transformations and obtain a deeper understanding of the relationship between domain and range.