Problem 41 - Algebra and Trigonometry

Question

\(21-44\) . Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations. $$ y=|x+2|+2 $$

Step-by-Step Solution

Answer

Shift the graph of \(y=|x|\) 2 units left and 2 units up.

Step 1: Identify the Base Graph

The function given is based on the absolute value function. The base graph is the graph of \(y = |x|.\)

Step 2: Horizontal Shift

In the function \(y = |x + 2| + 2\), the expression "+2" inside the absolute value indicates a horizontal shift. Specifically, the graph of \(y = |x + 2|\) is obtained from the graph of \(y = |x|\) by shifting it 2 units to the left.

Step 3: Vertical Shift

The "+2" outside the absolute value in \(y = |x + 2| + 2\) indicates a vertical shift. Take the previously shifted graph and move it 2 units up. This results in the final graph of the function.

Key Concepts

Absolute Value Function

The absolute value function is a fundamental piece in algebra, represented as \( y = |x| \). This function forms a 'V'-shaped graph centered at the origin, with the vertex located at \( (0, 0) \). It features two symmetrical arms: one rising to the right and the other rising to the left. Understanding this standard function helps in recognizing how transformations affect its shape and position. Key properties include:

• Domain: All real numbers \( (-\infty, \infty) \)
• Range: All non-negative real numbers \( [0, \infty) \)
• Key characteristic: The output is always non-negative, reflecting any negative input across the x-axis.

Horizontal Shift

A horizontal shift in a graph occurs when you pick up the entire graph and shift it left or right across the x-axis. For our function \( y = |x+2| \), the "+2" inside the absolute value signifies a shift. But here's the interesting part — the graph of \( y = |x| \) moves in the opposite direction of the sign! Therefore, with \( x+2 \), the graph actually shifts 2 units to the left. Think of the horizontal shift as changing the point where the "V" of the absolute value graph touches the x-axis:

• Shift left: Add a number inside the absolute value (\( x+c \))
• Shift right: Subtract a number inside the absolute value (\( x-c \))

Vertical Shift

Vertical shifts move a graph up or down along the y-axis. In the function \( y = |x+2| + 2 \), the "+2" outside the absolute value term signals a vertical shift.This time, it's straightforward — you move the graph up by 2 units. Here's a breakdown:

• Positive value ("+c") outside moves the graph up
• Negative value ("-c") outside moves the graph down

These shifts alter the vertical position of the graph but leave the shape unchanged. All points on the graph are raised by the same amount, effectively changing the range of the graph without affecting the domain.