Problem 29 - A Graphical Approach to Precalculus with Limits

Question

The graph of \(y=-\frac{1}{4}|x+2|-3\) can be obtained from the graph of \(y=|x|\) by shifting horizontally _______ units to the __________ vertically shrinking by applying a factor of _______ reflecting across the __________ -axis, and shifting vertically _________ units in the direction.

Step-by-Step Solution

Answer

Shift left 2 units, shrink by factor 4, reflect across x-axis, shift down 3 units.

Step 1: Identify Horizontal Shift

Start with the parent function \( y = |x| \) and consider the transformation function \( y = -\frac{1}{4}|x+2|-3 \). The expression \( x+2 \) inside the absolute value indicates a horizontal shift. In the form \( y = |x-h| \), \( h \) is the horizontal shift. Here, \( h = -2 \), meaning the graph is shifted 2 units to the left.

Step 2: Identify Vertical Shrink

The coefficient outside the absolute value, \(-\frac{1}{4}\), indicates a vertical shrink and a reflection. Normally, if \( a \) (in \( y = a|x| \)) is less than 1, the graph vertically shrinks by a factor of \( \frac{1}{a} \). Here, the factor is \( \frac{1}{4} \), meaning the graph is vertically shrunk by a factor of 4.

Step 3: Determine Reflection

The negative sign in front of \( \frac{1}{4} \) indicates that the graph is reflected across the x-axis. This means the entire graph is flipped upside down.

Step 4: Identify Vertical Shift

The constant term at the end of the function, -3, shows the vertical shift. Specifically, because it is negative, the graph moves 3 units downwards.

Key Concepts

Absolute Value Functions

Absolute value functions are fundamental in mathematics. The absolute value of a number is its distance from zero on the number line, without considering direction. The function is written as \( y = |x| \) and is known for its distinctive V-shaped graph. This shape is symmetrical and centered at the origin (0,0) on a graph. The vertices of the absolute value graph are always at the point where the input, \( x \), equals zero.

Absolute value functions are quite straightforward. They always produce positive outputs, which explains the unique V-shape. This is because, regardless of whether \( x \) is positive or negative, \(|x|\) is always positive. Such functions are commonly used in diverse areas like error measurement, data processing, and graphs analysis.

Vertical Stretch and Shrink

When we talk about stretching or shrinking a graph vertically, we refer to how the graph's height changes. In an absolute value function, a vertical stretch or shrink is driven by the coefficient placed before \(|x|\).

If that coefficient \( a \) is greater than 1, it indicates a vertical stretch. The graph becomes taller as each point is pulled away from the x-axis. Conversely, if \( a \) is between 0 and 1, the graph shrinks vertically, causing it to flatten.

A quick example: In the function \( y = \frac{1}{4}|x| \), because \( \frac{1}{4} \) is less than 1, this indicates a vertical shrink by a factor of 4. As a result, the graph appears more compressed vertically compared to the parent graph \( y = |x| \).

Horizontal and Vertical Shifts

Shifts in a graph mean moving the entire graph up, down, left, or right. For absolute value functions, this involves adjusting positions horizontally or vertically. These shifts do not change the shape of the graph, only its position.

Horizontal shifts are determined by the value inside the absolute value brackets. If we have \( y = |x-h| \), the graph shifts horizontally \( h \) units. If \( h \) is positive, the shift is to the right; if \( h \) is negative, it moves to the left.

For vertical shifts, look for the constant added or subtracted outside the absolute value bars. In \( y = |x| + k \), if \( k \) is positive, the graph shifts up; if negative, it moves downward.

In the transformed function \( y = -\frac{1}{4}|x+2|-3 \), the graph shifts 2 units to the left due to \( x+2 \) and 3 units downwards due to the \(-3\) at the end.

Reflection across Axes

Reflecting a graph changes its orientation across an axis. For absolute value functions, reflections are crucial transformations. The graph \( y = |x| \) can be flipped over the x-axis or y-axis, altering its appearance.

A reflection over the x-axis occurs when a negative sign is placed in front of the function. It effectively inverts the entire graph, changing all positive outputs into negative, and all negative outputs to positive. This is evident in the function \( y = -\frac{1}{4}|x| \), where the graph is reflected upside-down across the x-axis compared to \( y = |x| \).

However, reflections over the y-axis are slightly less common in absolute value functions because the symmetry of \( |x| \) means it already mirrors across the y-axis without an additional negative sign. Understanding these reflections helps in visualizing how graphs transform and behave differently on being manipulated.