Problem 44 - College Algebra with Modeling and Visualization

Question

Use transformations to explain how the graph of \(f\) can be found by using the graph of \(y=x^{2}\) \(y=\sqrt{x},\) or \(y=|x| .\) You do not need to graph \(y=f(x)\). \(f(x)=|4-x |\)

Step-by-Step Solution

Answer

The graph of \( f(x) = |4-x| \) is found by shifting the graph of \( y=|x| \) 4 units to the right.

Step 1: Identify the base function

The function given is \( f(x) = |4-x| \). Notice that it is based on the function \( y=|x| \). This is because the absolute value operation is involved, which corresponds directly to \( y = |x| \).

Step 2: Replace \(x\) in the base function

In the base function \( y = |x| \), replace \( x \) with \( 4-x \). This reflects the transformation applied to the base function. The expression \( |4-x| \) means the graph of \( y=|x| \) has been horizontally transformed.

Step 3: Horizontal Reflection

The transformation \( |4-x| \) can be rewritten as \( |-(x-4)| \). This indicates that first, the graph of \( y=|x| \) is reflected across the vertical axis because of the negative sign inside the absolute function, leading to \( y=|-x| \) which is identical to \( y=|x| \).

Step 4: Translate the graph horizontally

The term \( x-4 \) signifies a horizontal shift. Since it is \( x-4 \), it shifts the graph to the right by 4 units. Thus, the graph of \( y = |x| \) is translated 4 units to the right to become \( y=|x-4| \), which retains the same absolute value V-shape but moves rightward on the x-axis.

Step 5: Final Function Transformation

Combining the horizontal reflection and translation, the transformation to the graph for the function \( f(x) = |4-x| \) is effectively a horizontal shift of the \( y = |x| \) graph, 4 units to the right with no vertical changes. The negative sign affects the reflection, but results in the same shape as original. Therefore, the fully transformed graph is \( y = |(x-4)| \), equivalent to \( y = |4-x| \).

Key Concepts

Absolute Value Function

The absolute value function is a fascinating concept in mathematics, essentially taking a number and outputting its non-negative value. With the notation \( y = |x| \), the graph is created by essentially ignoring the negative signs of any input values. The graph of \( y = |x| \) characteristically resembles the letter 'V', centered at the origin (0,0) on the coordinate plane.

• The absolute value equation \( y = |x| \) ensures all output values are positive or zero.
• This is because an absolute value effectively measures the "distance" from zero, without considering direction.
• So, \(|-3| = 3\) and \(|3| = 3\), displaying the function's symmetry with respect to the y-axis.

The absolute value function naturally influences transformations like reflections and translations, which we will explore in relation to the function \( f(x) = |4-x| \).

Horizontal Shifts

Horizontal shifts in graphs occur when the entire graph of a function moves left or right along the x-axis.
For a function \( y = |x| \), a common horizontal shift is expressed by replacing \( x \) with \( x - c \) or \( x + c \). The value \( c \) determines the direction and magnitude of the shift.

• If \( c \) is positive, \( y = |x - c| \) shifts the graph rightwards by \( c \) units.
• If \( c \) is negative, \( y = |x + c| \), shifts the graph leftwards by \( |c| \) units.

In the function \( f(x) = |4-x| \), observe the expression \( x-4 \). It means the graph of \( y = |x| \) is shifted to the right by 4 units. This might seem counterintuitive at first because it reads \( x-4 \).
However, this subtraction indicates that the movement on the x-axis is "back" by 4 units (to the positive side).

Horizontal Reflections

Horizontal reflections involve flipping the graph of a function across a vertical line, usually the y-axis. This transformation switches the direction of the graph on the x-axis.

In the case of \( f(x) = |4-x| \), note that replacing \( x \) with \(-x\) doesn’t change the shape of the graph of an absolute value function, due to its inherent symmetry. However, writing it as \(|4-x|\) or \(|-(x-4)|\) indicates the initial reflection, even if it doesn’t visually alter the graph.

• Reflections are typically seen in equations like \( y = |-x| \).
• For absolute value graphs, this negation reflects the graph across the y-axis, but maintains the same 'V' shape.

Though the reflection doesn’t change the appearance for \( f(x) = |4-x| \), the process highlights the negative component inside the equation as a common element of transformation steps.