Problem 17 - 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. $$S(x)=-4|x|$$

Step-by-Step Solution

Answer

The function \(S(x)=-4|x|\) is a transformation of the base function \(|x|\), involving a reflection in the x-axis and a vertical stretch by a factor of 4. Therefore, the graph of the given function is a V-shaped graph that intersects the origin (0,0) and opens downwards.

Step 1: Identify the Basic Function

The basic function here is \(|x|\), which is the absolute value function. The graph of \(|x|\) is a V-shaped graph that intersects the origin (0,0) and opens upwards.

Step 2: Recognize the Transformation

The given function is \(S(x) = -4|x|\). This function represents a vertical stretch and a reflection in the x-axis of the basic function \(|x|\). The negative sign indicates a reflection across the x-axis while the '4' is a multiplier that stretches the graph vertically by a factor of 4.

Step 3: Sketch the Transformed Function

When sketching the graph, start with the basic function \(|x|\), reflect it in the x-axis to account for the negative sign, and then stretch it vertically by a factor of 4. The resulting graph will still be V-shaped, but it will be steeper than the basic function and will open downwards.

Key Concepts

Function Transformations

Transformation in functions refers to the various ways we can modify a function’s graph in relation to its basic shape. These transformations allow us to change the position, orientation, and shape of the graph based on certain mathematical operations.
When dealing with function transformations, your starting point is typically a basic function like your simpler absolute value function, \(|x|\), which forms a V-shape graph centered at the origin.
Transformations include a range of operations such as shifting, stretching, compressing, and reflecting. For this exercise, you specifically look at how the transformations affect the absolute value function, resulting in changes like vertical stretching and reflecting across axes.

• Translation: This shifts the graph horizontally or vertically without altering its shape.
• Reflection: Flips the graph over a specified axis, effectively changing its direction.
• Stretching and Compressing: Alters the steepness or size of the graph, making it either taller (stretch) or flatter (compress).

These transformations allow you to move and distort your graph to better fit the new function's requirements.

Vertical Stretch

A vertical stretch affects the height of a graph, expanding it away from the x-axis. It is achieved by multiplying the function by a positive number greater than one. In \(S(x) = -4|x|\), the factor is 4, meaning the graph of \(|x|\) is stretched vertically by a factor of 4.
This operation exaggerates the peaks and valleys of the graph, making the V-shape steeper. To visualize the effect of vertical stretching:

• Imagine pulling the top of the V upwards, resulting in a more pronounced and sharper ascent and descent.
• The graph retains its original V-shape but the sides become steeper.

The vertical stretch does not affect the position of the vertex at the origin, rather simply enhances the steepness of each arm of the V, making them reach further along the y-axis.

Reflection Across the x-axis

Reflection across the x-axis involves flipping the graph over the x-axis, which inverts its direction. In the given function \(S(x) = -4|x|\), the negative sign before the coefficient -4 signifies that the graph is reflected across the x-axis.
This transformation:

• Changes every y-value of the graph to its opposite (i.e., positive to negative).
• Inverts the V-shape so that it opens downward instead of upward.

Reflection is a powerful transformation as it not only changes the direction in which the graph opens but also magnifies the graph's effect by affecting all y-values evenly. After this reflection, the absolute value function, originally opening upwards, now distinctly opens downwards and appears flipped along the x-axis.