Problem 14 - Thomas Calculus

Question

Graph the functions in Exercises \(7-18 .\) What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing. $$ y=-4 \sqrt{x} $$

Step-by-Step Solution

Answer

The function is decreasing on the interval \( (0, \infty) \) with no symmetry.

Step 1: Understand the Function

The function given is \( y = -4 \sqrt{x} \). This is a transformation of the basic square root function \( y = \sqrt{x} \). The negative sign indicates that it is reflected over the x-axis, and the coefficient 4 indicates a vertical stretch.

Step 2: Identify Symmetries

The function \( y = -4 \sqrt{x} \) does not have any symmetry over the y-axis or the origin. However, due to the negative sign, the function is symmetrical to the line y=0 if flipped.

Step 3: Determine Increasing/Decreasing Intervals

The square root function \( y = \sqrt{x} \) is naturally increasing. However, since \( y = -4 \sqrt{x} \) involves a vertical reflection, the function is decreasing for all \( x > 0 \). Since the function is only defined for \( x \geq 0 \), it decreases on the interval \( (0, \infty) \).

Step 4: Graph the Function

Plot the basic points of the function. For example, when \( x = 0 \), \( y = 0 \) and when \( x = 1 \), \( y = -4 \cdot 1 = -4 \). When \( x = 4 \), \( y = -4 \cdot 2 = -8 \). Graph these points and connect them in a smooth curve downward from the origin.

Key Concepts

Symmetry in Graphs

Symmetry in graphs refers to how a graph can be folded or reflected along certain lines so that one side of the graph is a mirror image of the other. This function, \( y = -4 \sqrt{x} \), is a transformation of the square root function. By default, the square root function, \( y = \sqrt{x} \), does not exhibit symmetry about the y-axis or origin because of its domain from zero onwards.
However, when transformed into \( y = -4 \sqrt{x} \), the graph is reflected over the x-axis due to the negative sign. This means that it mirrors the increasing nature of the basic square root function but in a downward fashion. While you won't find symmetry about the y-axis or the origin, you can think of the function being vertically flipped.

• This reflection changes how we view the graph compared to its parent function.
• Such transformations affect the overall appearance but do not introduce symmetry in conventional terms.

Increasing and Decreasing Intervals

Understanding where a function is increasing or decreasing is crucial when analyzing graphs. For the function \( y = \sqrt{x} \), it's naturally increasing as \( x \) increases. In simple terms, as you move right along the x-axis, the function rises.
However, our transformed function \( y = -4 \sqrt{x} \) behaves differently due to the vertical reflection caused by the negative sign. This flips the increasing behavior into a decreasing one.
Thus, as \( x \) moves from 0 to positive infinity, \( y \) steadily decreases. We can summarize this as:

• The function is decreasing on the interval \((0, \infty)\).
• It decreases because of the transformation induced by the negative multiplier.
• There are no intervals of increase for this particular function.

Contrast this behavior with functions whose natural inclination is to increase, showcasing how transformations can dramatically alter a function's graphical properties.

Square Root Function Transformations

Transformations? They're magical and powerful tools in graphing! Transformed functions look different from their parent functions due to changes like shifts, reflections, stretches, or compressions. The function \( y = -4 \sqrt{x} \) is a marvelous example of a square root function transformation.
Let's break it down:

• The basic square root function is simply \( y = \sqrt{x} \), graphed as a gentle upward curve starting at the origin.
• By adding \(-4\) in front, we reflect this curve downwards about the x-axis, multiplying each y-value by \(-4\).
• This makes the function not only inverted but also stretched vertically, exaggerating its descent.

In essence, the transformation turns an increasing function into a decreasing one, visually distinct and mathematically intriguing. These transformations make understanding each component, like the negative sign and multiplication, essential in grasping how these graphs evolve from their origins.