Problem 19 - Precalculus : Building Concepts and Connections

Question

Identify the underlying basic function, and use transformations of the basic function to sketch the graph of the given function. $$H(s)=-|s|-3$$

Step-by-Step Solution

Answer

The function H(s)=-|s|-3 is a basic absolute value function |s|, which has been reflected about the x-axis and shifted down by 3 units.

Step 1: Identify the basic function

The basic function here is the absolute value function (|s|) which is a V-shaped curve and is always positive. It touches the x-axis at the origin, i.e., at point (0,0).

Step 2: Identify the transformations

The function H(s)=-|s|-3 has two transformations. The negative sign in front of the absolute value function flips the V-shaped curve so that it opens downwards. The '-3' after the absolute value function indicates a vertical downward shift of the function by 3 units.

Step 3: Apply the transformations and sketch the graph

We start sketching the basic function, the V-shaped curve. The negative sign flips the basic function so now it opens downwards. Then, we apply the second transformation by moving the graph downward by 3 units. This results in the point (0,0) moving to (0,-3).

Key Concepts

Absolute Value Function

The absolute value function is fundamental in understanding many mathematical transformations. It is typically expressed as \(|x|\), where \(x\) represents any real number. The graph of an absolute value function is a V-shaped curve that is symmetrical about the vertical axis. This function outputs only non-negative values, making the vertex of the graph touch the x-axis at the origin, or at

• (0,0)

. Understanding this property is crucial, as this symmetry will help you predict how the graph will look after incorporating transformations such as reflections or shifts.

The typical form of the absolute value function is quite similar to that of a linear function, with a significant difference being that it creates a sharp turn rather than extending infinitely in either direction.

Vertical Shift

When we talk about graph transformations, a vertical shift is one of the simplest concepts to grasp. It involves moving the entire graph of a function upwards or downwards along the y-axis without altering the shape of the graph. This movement is achieved by adding or subtracting a constant from the function.

In the context of our exercise, the term "-3" in the equation \(H(s) = -|s| - 3\) directs us to perform a vertical downward shift. This means we move every point of the reflected V-shaped graph down by 3 units. The consequence of this transformation is that the vertex of the graph shifts from the origin (0,0) to (0,-3).

This kind of transformation does not affect the symmetry or the opening direction of the graph; it merely changes its vertical position.

Reflection

The concept of reflection in graph transformations involves changing the direction that a graph "faces". Specifically, for an absolute value function, reflection is commonly associated with the coefficient in front of the absolute value term. If the coefficient is negative, the graph reflects over the x-axis.

In our problem statement, the equation \(H(s) = -|s| - 3\), the negative sign in front of the absolute value symbol induces a reflection. Typically, the graph of \(|s|\) opens upwards, forming a standard V-shape. However, the negative sign flips this V-shape, resulting in a graph that opens downwards instead.

Reflection is a crucial concept because it significantly alters the orientation of the graph without affecting its symmetry about a vertical line. This means although the graph of \(-|s|\) looks different from \(|s|\), it maintains the same distance and shape from the y-axis on both sides.