Problem 44 - Thomas Calculus

Question

Graph the functions in Exercises \(37-46 .\) 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 graph of \( y = -4 \sqrt{x} \) has no symmetry, and it is decreasing on the interval \( (0, \infty) \).

Step 1: Identify the Function Type

The given function is \( y = -4 \sqrt{x} \), which involves the square root of \( x \). This type of function is a transformation of the basic square root function \( y = \sqrt{x} \) with a vertical scaling by \( -4 \), meaning it is reflected in the x-axis and stretched.

Step 2: Determine Domain

Since the square root function requires the radicand (the term inside the square root) to be non-negative, the domain of \( y = -4 \sqrt{x} \) is \( x \geq 0 \).

Step 3: Plot Key Points

Choose some key values for \( x \), such as 0, 1, 4, to find their corresponding \( y \) values. For instance, when \( x = 0 \), \( y = 0 \); when \( x = 1 \), \( y = -4 \); when \( x = 4 \), \( y = -8 \).

Step 4: Sketch the Graph

Plot the points (0,0), (1,-4), and (4,-8) on a coordinate plane, then draw a smooth curve passing through these points depicting the function \( y = -4 \sqrt{x} \). The curve opens downward due to the negative sign.

Step 5: Analyze Symmetry

The function \( y = -4 \sqrt{x} \) does not exhibit symmetry with respect to the y-axis (even symmetry) or the origin (odd symmetry), because these would require both positive and negative x values, and the function is only defined for \( x \geq 0 \).

Step 6: Determine Increasing/Decreasing Intervals

To find where the function is increasing or decreasing, note the derivative \( y' = \frac{-2}{\sqrt{x}} \) for \( x > 0 \), which is always negative, indicating that the function is decreasing on its entire domain \( (0, \infty) \).

Key Concepts

Square Root Function

A square root function typically takes the form \( y = a \sqrt{x} \), where \( a \) is a constant. This kind of function involves the variable \( x \) under a square root, creating a curve that either rises or falls depending on the sign and size of \( a \). In our case, with the function \( y = -4 \sqrt{x} \), the coefficient \(-4\) indicates a few transformations:

• Reflection: The negative sign before the \(4\) means the graph is reflected over the x-axis, flipping it upside down compared to the basic \( y = \sqrt{x} \).
• Vertical Stretch: The \(4\) implies the graph is stretched vertically. This transformation makes the curve steeper than the basic square root function.

A key aspect of square root functions is their restricted domain. Since you cannot take the square root of a negative number and expect a real result, \( x \) must be greater than or equal to 0, i.e., the domain is \( x \geq 0 \). The graph rises or falls along the y-axis depending on the specific transformations applied.

Function Symmetry

Symmetry in functions refers to the balance on either side of a central line or point. A function can exhibit different types of symmetry:

• Even Symmetry: If a graph is symmetric about the y-axis, it means \( f(x) = f(-x) \) for all \( x \) in the domain.
• Odd Symmetry: If a graph is symmetric about the origin, it means \( f(-x) = -f(x) \).

For the function \( y = -4 \sqrt{x} \), neither type of symmetry is present. Because this function's domain only includes non-negative values \( x \geq 0 \), it cannot satisfy the symmetry conditions for either even or odd symmetry:

• Exact Calculation: Without negative \( x \)-values in its domain, we're unable to compare \( f(x) \) and \( f(-x) \).

Thus, the graph of this function distinctly lacks symmetry, making it different from functions defined over the entire real line.

Increasing and Decreasing Intervals

In terms of intervals, a function can either be increasing, where it rises as you move from left to right, or decreasing, where it falls. To determine this for \( y = -4 \sqrt{x} \), we look at the derivative, \( y' \).

• Derivative: The derivative \( y' = \frac{-2}{\sqrt{x}} \) helps us identify intervals of increase or decrease.
• Behavior: Here, the derivative is negative for all \( x > 0 \), meaning the slope of the tangent line along the graph is always negative.
• Conclusion: This negative derivative implies the function is consistently decreasing over its domain.

Thus, for the entire range \( (0, \infty) \), the function \( y = -4 \sqrt{x} \) is decreasing. There are no increasing intervals; the function starts at the highest point when \( x = 0 \) and continues to fall as \( x \) increases.