Problem 84 - Precalculus : Building Concepts and Connections

Question

The point (-2,2) on the graph of \(f(x)=|x|\) has been shifted horizontally and vertically to the point (3,4) Identify the shifts and write a new function \(g(x)\) in terms of \(f(x)\).

Step-by-Step Solution

Answer

The point (-2,2) has been shifted 5 units to the right and 2 units up to get to the point (3,4). The new function \(g(x)\) that represents these shifts is \(g(x) = |x - 5| + 2.

Step 1: Identify the Original Point

The original point on the function \(f(x) = |x|\) is given as (-2,2).

Step 2: Identify the Shifted Point

The point has been moved to position (3,4).

Step 3: Calculate Shifts

The change in the x-values from -2 to 3 represents a horizontal shift. This is a shift of 5 units to the right. The change in the y-values from 2 to 4 shows a vertical shift of 2 units upwards.

Step 4: Formulate New Function

A horizontal shift of \(h\) units to the right corresponds to the replacement of \(x\) by \((x - h)\) in the function. A vertical shift of \(v\) units upwards corresponds to the addition of \(v\) to the function. So by replacing \(x\) with \((x - 5)\) in \(f(x) = |x|\) and adding 2, we get the new function \(g(x) = |x - 5| + 2\).

Key Concepts

Understanding Horizontal Shifts

A horizontal shift occurs when a function moves left or right along the x-axis. This happens without altering the shape of the graph itself. In mathematical terms, if you shift a function horizontally by \( h \) units to the right, you substitute \( x \) in the function with \( x-h \). Conversely, for a left shift by \( h \) units, \( x \) is replaced with \( x+h \).

• To identify a horizontal shift, observe the change in x-coordinates between the original and new positions.
• If the x-value increases, the function has undergone a shift to the right.
• If it decreases, the shift is to the left.

In our exercise, the original point is (-2,2) and the shifted point is (3,4). The change in x is from -2 to 3, indicating a 5 units shift to the right. This is why \(f(x) = |x|\) is changed to \(g(x) = |x - 5| + 2\), reflecting the horizontal transition.

Exploring Vertical Shifts

Vertical shifts refer to moving a graph up or down the y-axis. Similar to horizontal shifts, the shape of the function remains intact, but its position changes. When a function is shifted \( v \) units up, you simply add \( v \) to the whole function. Conversely, subtracting \( v \) shifts the graph down.

• To find a vertical shift, look at how the y-coordinate changes between points.
• A positive change means an upward shift, whereas a negative one means down.

In the given problem, the original y-coordinate 2 is changed to 4, indicating a vertical shift of 2 units upward. Therefore, after finding the horizontal shift, we add 2, modifying the function to \(g(x) = |x - 5| + 2\). This adjustment shows the function's new vertical placement.

The Role of Absolute Value Function

The absolute value function, represented as \( f(x) = |x| \), is a V-shaped graph centered at the origin. It reflects the distance of x from zero, rendering it always positive or zero. Understanding the properties of absolute value is essential for applying transformations correctly.

• Absence of horizontal shifts often positions the vertex at the origin.
• Vertical transformations displace the vertex by directly altering its y-coordinate.

In the exercise, we start with \( f(x) = |x| \), and apply a horizontal shift rightwards and a vertical shift upwards with \( g(x) = |x - 5| + 2 \). Here, the vertex shifts from (0,0) in the original graph to (5,2), aligning with the calculated transformations. Absolute value functions are uniquely influenced by shifts due to their distinctive V-shape, making them a useful practice for understanding graph transformations.