Problem 20 - College Algebra with Modeling and Visualization

Question

Write a formula for a function \(g\) whose graph is similar to \(f(x)\) but satisfies the given conditions. Do not simplify the formula. \(f(x)=|x|-3\) (a) Shifted right 4 units and downward 3 units (b) Shifted left 5 units and upward 2 units

Step-by-Step Solution

Answer

(a) \(g(x) = |x - 4| - 6\); (b) \(g(x) = |x + 5| - 1\).

Step 1: Understanding Shifts

To transform the graph of the function \(f(x) = |x| - 3\), we apply horizontal and vertical shifts based on the given conditions. A horizontal shift of \(h\) units to the right is performed by replacing \(x\) with \(x - h\) in the function, while a vertical shift of \(k\) units downward is achieved by subtracting \(k\) from the entire function's output. Conversely, a shift left by \(h\) units is done by replacing \(x\) with \(x + h\), and upward by \(k\) units is achieved by adding \(k\) to the output.

Step 2: Condition (a): Shift Right 4 Units and Downward 3 Units

For condition (a), we apply a horizontal shift to the right by replacing \(x\) with \(x - 4\), and a vertical shift downward by subtracting 3. The transformation is applied to \(f(x)\): \[g(x) = |x - 4| - 3 - 3 = |x - 4| - 6.\] Thus \(g(x)\) for condition (a) is: \(g(x) = |x - 4| - 6\).

Step 3: Condition (b): Shift Left 5 Units and Upward 2 Units

For condition (b), we apply a horizontal shift to the left by replacing \(x\) with \(x + 5\), and a vertical shift upward by adding 2. The transformation is applied to \(f(x)\): \[g(x) = |x + 5| - 3 + 2 = |x + 5| - 1.\] Thus \(g(x)\) for condition (b) is: \(g(x) = |x + 5| - 1\).

Key Concepts

Horizontal Shift

A horizontal shift involves moving the entire graph of a function left or right along the x-axis. The magnitude and direction of this shift are determined by the value added or subtracted from the variable \(x\). If the function is \(f(x) = |x|\), a shift to the right by \(h\) units modifies the function to \(f(x) = |x - h|\). This operation moves every point on the graph \(h\) units to the right. Conversely, a shift to the left by \(h\) units changes the function to \(f(x) = |x + h|\), shifting every point \(h\) units to the left.

• Example of shifting right: To shift \(f(x) = |x|\) 4 units to the right, it becomes \(f(x) = |x - 4|\).
• Example of shifting left: To shift \(f(x) = |x|\) 5 units to the left, it becomes \(f(x) = |x + 5|\).

Horizontal shifts are crucial when altering the positioning of graphs without changing their shape or orientation.

Vertical Shift

Vertical shifts adjust a function's graph along the y-axis. This adjustment is achieved by adding or subtracting a constant to the output of the function. For the function \(f(x) = |x|\), if we apply a vertical shift upward by \(k\) units, we modify the function to \(f(x) = |x| + k\). This moves the graph upward by \(k\) units. A vertical shift downward by \(k\) units changes the function to \(f(x) = |x| - k\), contracting the graph downward by \(k\) units.

• Example of shifting down: If moving \(f(x) = |x| - 3\) three units further down, it becomes \(f(x) = |x| - 6\).
• Example of shifting up: To move \(f(x) = |x| - 3\) 2 units up, it becomes \(f(x) = |x| - 1\).

These vertical shifts are essential for altering the graph's elevation, allowing us to move shapes vertically while keeping the horizontal axis unchanged.

Absolute Value Function

The absolute value function is foundational in mathematics, often denoted as \(f(x) = |x|\). It represents the distance of a number \(x\) from zero on the number line, always yielding a non-negative result. This characteristic shapes its distinctive "V" form on a graph. The point at the vertex of this "V," where the slope changes direction, occurs at the origin \((0,0)\) of the standard absolute value function\(f(x) = |x|\).
Understanding how transformations apply to the absolute value function is crucial. When transformations like horizontal and vertical shifts are applied, they modify the position of this "V" shape on the graph. Despite these shifts, the shape and orientation of the graph remain the same; only its position changes.
The combination of horizontal and vertical shifts allows for a versatile manipulation of the absolute value function, aiding in solving problems where adjustments to the position matter but the fundamental "V" shape must be preserved.