Problem 56 - Algebra 2

Question

The graph of \(y=|x-1|\) is translated 3 units left and 2 units down. What is the equation of the new graph? A. \(y=|x+2|-2\) B. \(y=|x-4|-2\) C. \(y=|x+4|+2\) D. \(y=|x-4|+2\)

Step-by-Step Solution

Answer

The equation of the new graph after translation is A. \(y=|x+2|-2\).

Step 1: Identify the Transformation

A translation of the graph 3 units left means we add 3 to the x-value inside the absolute value because going left is in the negative x-direction. A translation of 2 units down means we subtract 2 from the entire function, as moving down translates to decreasing the y-value.

Step 2: Apply Horizontal Translation

Adjust the original function by incorporating the left translation inside the absolute value: from \(y=|x-1|\) to \(y=|x-(1-3)|\), which simplifies to \(y=|x+2|\).

Step 3: Apply Vertical Translation

Adjust the translated function from the previous step by subtracting 2 to reflect the downwards shift: from \(y=|x+2|\) to \(y=|x+2|-2\).

Key Concepts

Absolute Value Transformations

Understanding absolute value transformations is a crucial part of algebra that deals with how the graph of an absolute value function changes in response to modifications made in its equation. The absolute function is represented as \(y = |x|\), and its graph is a V-shaped figure that opens upwards with its vertex at the origin (0,0).

When dealing with transformations, imagine that the V-shape can move around the coordinate plane without changing its 'openness' or 'width'. The movements can be up, down, left, or right, and each direction is associated with a specific algebraic manipulation on the function's equation. To translate this graph horizontally, we alter the x-component, while vertical translations affect the y-component. By understanding these transformations, you'll be able to predict and graph the behavior of absolute value functions quickly.

Horizontal Translation

A horizontal translation shifts the graph of a function left or right. In our example, translating the graph 3 units to the left modifies the input values. Algebraically, this means we add 3 to the x-values within the absolute value to compensate for the shift left since it is in the negative direction on the x-axis.

For our function \(y = |x - 1|\), adding 3 to the x-component yields \(y = |x - (1 - 3)|\). Simplifying the expression inside the absolute value results in \(y = |x + 2|\), displaying our horizontal translation. It's important to note that adding within the absolute value shifts left, while subtracting shifts the graph to the right.

Vertical Translation

Vertical translation involves moving the graph of a function up or down the y-axis. This movement is reflected directly in the equation of the function outside the absolute value component. To shift the graph downward by 2 units as in our example, we subtract 2 from the entire function's value.

Now, we adjust our horizontally translated function \(y = |x + 2|\) with a vertical translation, resulting in \(y = |x + 2| - 2\). This equation represents a downward shift of the original graph by 2 units. Remember, subtracting from the function moves it down the y-axis, while adding raises it up.

Algebraic Manipulation

Algebraic manipulation is the process of reworking equations to make transformations like translations easier to identify and apply. This step often involves simplifying expressions and ensuring that the transformations correspond correctly to the type of shift you need on the graph.

In the context of the exercise, we first deal with the horizontal shift by manipulating the stucture within the absolute value: \(y = |x - 1 + 3|\). Then, for the vertical movement, we use simple subtraction to adjust the function's entire value: \(y = |x + 2| - 2\). It's vital for students to practice these algebraic skills to correctly apply graph transformations.