Problem 78 - A Graphical Approach to College Algebra

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.) $$y=-2 \sqrt{x+15}-18$$

Step-by-Step Solution

Answer

Domain: \([-15, \infty)\); Range: \(( -\infty, -18 ]\).

Step 1: Understand the Base Function

The given function is based on the square root function. Recall that the graph of the basic function \( y = \sqrt{x} \) starts at the origin (0,0) and extends infinitely left and upwards.

Step 2: Identify the Transformations

The function given is \( y = -2\sqrt{x+15} - 18 \). We must first break down the transformations: \( x+15 \) shifts the graph 15 units to the left. The multiplication by -2 reflects the graph over the x-axis and stretches it vertically by a factor of 2. Finally, the -18 shifts the graph downward by 18 units.

Step 3: Determine the New Endpoint

The translation to the left by 15 means the new starting point in terms of the x-coordinate is at \( x = -15 \). Substitute \( x = -15 \) into the function to find the y-coordinate of the new endpoint: \[y = -2 \sqrt{-15 + 15} - 18 = -2 \times 0 - 18 = -18\]Thus, the endpoint of the graph is at \((-15, -18)\).

Step 4: Determine the Domain

The domain of the function is determined by the x-values for which the expression under the square root is non-negative. Since the graph starts at \( x = -15 \) and extends rightward, the domain is \( x \geq -15 \). In interval notation, this is \([-15, \infty)\).

Step 5: Determine the Range

The range is determined by the possible y-values the function can take. Starting at \( y = -18 \) at \( x = -15 \) and moving towards positive infinity, the graph moves downward, so the lowest y-value is \(-\infty\). Therefore, the range is \( y \leq -18 \), or in interval notation, \(( -\infty, -18 ]\).

Key Concepts

Function Transformations

Understanding function transformations is crucial when graphing square root functions. The function \( y = -2\sqrt{x+15} - 18 \) is a transformed version of the base function \( y = \sqrt{x} \), which we all start learning about from its basic graph that reflects a simple, "arched" line moving rightward starting at the origin (0,0).

Function transformations modify this base graph in specific ways:

• Horizontal Translations: The term \( x+15 \) indicates a shift of the graph left by 15 units. Generally, for a function \( f(x) \), \( f(x+c) \) results in a shift to the left by \( c \) units.
• Reflections and Vertical Stretches: Multiplying by \(-2\) reflects the graph over the x-axis. This stretch also doubles the vertical distance from the x-axis.
• Vertical Translations: The \(-18\) at the end moves the graph down by 18 units.

By breaking down these changes step by step, we see how a simple root function changes into a more complex one. These transformations are like adjusting the image on a screen - you can flip, stretch, or move it around to get what you need.

Domain and Range

When analyzing a function graphically, determining its domain and range gives us crucial insights into its behavior. This is especially true for square root functions like \( y = -2\sqrt{x+15} - 18 \), which have specific constraints due to the nature of square roots.

• Domain: The domain represents all the possible x-values the function can intake. Given the condition inside our square root, \( x+15 \) must be non-negative, leading us to solve \( x+15 \geq 0 \). This guides us to \( x \geq -15 \). So, the domain is all x-values starting from -15 and going to infinity, described as \([-15, \infty)\) in interval notation.
• Range: The range concerns the resulting y-values after calculating the function over its domain. Here, due to the transformation and reflection \(-2\sqrt{x+15}-18\), the function takes a downward path starting at -18. Consequently, the highest point on the y-axis begins at -18 and moves downward toward negative infinity, represented as \(( -\infty, -18 ]\).

Understanding these helps predict how the function behaves without even graphing it: where it starts, what it covers, and crucially, what it doesn't.

Graphical Representation of Functions

Graphing functions assists in visualizing mathematical concepts, allowing us to see transformations, growth patterns, and endpoints straight on the graph paper or screen. For a square root function like \( y = -2\sqrt{x+15}-18 \), graphing can simplify understanding.

Firstly, take note of the starting point or endpoint. In this function, after applying the transformations, the starting point is at \((-15, -18)\). This point can be referred to as where the graph starts from its initial journey.

Secondly, observe the direction and shape of the function's path or curve. Normally, \( y=\sqrt{x} \) shoots upwards with increasing x. But, due to the transformation \(-2\) in our example, the graph is a downward curve.

• It is crucial to include these endpoints while graphing as the viewer can visually capture the restricted nature of the domain and range.
• The curve becomes a reflection over the x-axis, detailing how it stretches downward.

Graphing, therefore, is not only about plotting but also about interpretation and understanding the broader impact of each transformation on function behavior. This visual mechanism allows predictions and checks against expected domain and range calculations.