Problem 49 - Algebra and Trigonometry

Question

\(45-54=\) A function \(f\) is given, and the indicated transformations are applied to its graph (in the given order). Write the equation for the final transformed graph. \(f(x)=|x| ;\) shift 3 units to the right and shift upward 1 unit

Step-by-Step Solution

Answer

The final transformed function is \( f(x) = |x - 3| + 1 \).

Step 1: Identify the Base Function

The base function given is the absolute value function: \( f(x) = |x| \). This function has a V-shape centered at the origin (0,0).

Step 2: First Transformation: Horizontal Shift

To shift the graph 3 units to the right, replace \(x\) with \(x - 3\) in the function. This will adjust the horizontal position of the vertex from (0,0) to (3,0). The new function becomes \( f(x) = |x - 3| \).

Step 3: Second Transformation: Vertical Shift

To shift the graph upward by 1 unit, add 1 to the entire function. This moves the vertex of the function from (3,0) to (3,1). The newly transformed function is \( f(x) = |x - 3| + 1 \).

Step 4: Write the Final Equation

Combining both transformations, the equation for the final transformed graph is \( f(x) = |x - 3| + 1 \). This represents first moving the graph of \( |x| \) 3 units to the right and then 1 unit upwards.

Key Concepts

Absolute Value Function

The absolute value function, represented as \( f(x) = |x| \), is a fundamental building block in mathematics. It is known for its characteristic V-shape that opens upwards and is symmetrical about the y-axis. This function essentially measures the distance of a number from zero on a number line, without considering direction.

• The vertex of the absolute value function graph is at the origin, point \((0, 0)\).
• This function is linear in nature, meaning the slope of the lines on either side of the vertex is constant.
• The standard form, \( f(x) = |x| \), means no transformation has been applied, showcasing its basic V-shape.

Recognizing and understanding the absolute value function makes it easier to perform and interpret transformations such as horizontal and vertical shifts. These transformations modify the placement of the V-shape on the Cartesian plane, without altering its slope.

Horizontal Shift

A horizontal shift of a function involves moving the graph along the x-axis, either to the left or to the right, without changing its shape. In order to shift a function horizontally, we adjust the input \(x\) directly inside the function. In the example from the exercise, shifting the absolute value function \( f(x) = |x| \) 3 units to the right involves substituting \(x\) with \(x - 3\). This results in a new function, \( f(x) = |x - 3| \). Here’s what happens:

• The entire graph shifts to the right by 3 units, so the vertex moves from point \((0,0)\) to \((3,0)\).
• The V-shape of the graph remains unchanged; the slopes on either side of the vertex are not affected.

This transformation simply relocates the function's central point along the horizontal axis, which is a powerful tool for aligning functions exactly where they are needed on a graph. Understanding the mechanics of a horizontal shift can expand your ability to visualize different function transformations.

Vertical Shift

Vertical shifts modify the vertical position of a function's graph without altering its shape or horizontal orientation. By adding or subtracting a value to the entire function, you can move the graph up or down along the y-axis.In the given exercise, the absolute value function went through a vertical shift: by adding 1 to \(f(x) = |x - 3|\), the function turns into \(f(x) = |x - 3| + 1\). This results in:

• The graph itself moves upward by 1 unit. The vertex, originally at \((3,0)\), ascends to \((3,1)\).
• The overall shape remains the same, with no change in slope or horizontal position.

The vertical shift is simple yet profound, allowing you to elevate or drop the function graphically with ease. This transformation plays a key role in positioning graphs strategically in real-world applications, ensuring they occupy the desired vertical space.