Problem 2 - Algebra and Trigonometry

Question

\(1-2\) Fill in the blank with the appropriate direction (left, right.up, or down). (a) The graph of \(y=f(x)-3\) is obtained from the graph of \(y=f(x)\) by shifting_________3 units. (b) The graph of \(y=f(x-3)\) is obtained from the graph of \(y=f(x)\) by shifting_________3 units.

Step-by-Step Solution

Answer

(a) Down; (b) Right.

Step 1: Understand Vertical Shifts

Given the equation \(y = f(x) - 3\), this implies a vertical shift in the graph of the function \(y = f(x)\). Subtracting a constant value from the function shifts the graph **down** by that amount. Therefore, \(y = f(x) - 3\) results in shifting the graph of \(y = f(x)\) down by 3 units.

Step 2: Understand Horizontal Shifts

For the equation \(y = f(x - 3)\), this involves a horizontal shift. A function of the form \(y = f(x - a)\) shifts the graph of \(y = f(x)\) to the **right** by \(a\) units (since \(a = 3\) in this case). Therefore, \(y = f(x - 3)\) shifts the function \(y = f(x)\) right by 3 units.

Step 3: Conclusion

Based on the analysis, the graph of \(y=f(x)-3\) is shifted **down** 3 units, and the graph of \(y=f(x-3)\) is shifted **right** 3 units. These are the directions needed to fill in the blanks.

Key Concepts

Vertical Shifts in Graph Transformations

Vertical shifts happen when we move the entire graph of a function up or down along the y-axis. Think of it as lifting or lowering the graph without altering its shape. If you have the function \( y = f(x) \) and modify it to \( y = f(x) + c \), then you shift the graph upward by \( c \) units. Conversely, if your equation becomes \( y = f(x) - c \), you move it down by \( c \) units.

For instance, consider the transformation \( y = f(x) - 3 \). Here, the graph of \( y = f(x) \) shifts down 3 units because of the subtraction. Vertical shifts do not affect the x-values of your function; they only alter the y-coordinates, moving the graph up or down as needed.

Horizontal Shifts in Graph Transformations

Horizontal shifts occur when you slide the graph left or right without altering its appearance. These shifts affect the graph along the x-axis. To implement a horizontal shift, you can modify your function \( y = f(x) \) to \( y = f(x - a) \) or \( y = f(x + a) \).

For \( y = f(x - a) \), the graph shifts to the right by \( a \) units. Conversely, the equation \( y = f(x + a) \) will shift the graph to the left by \( a \) units. The function \( y = f(x - 3) \) exemplifies a right horizontal shift of 3 units, where each point on the graph moves 3 units to the right on the x-axis.

Horizontal shifts are a bit counterintuitive; despite what the minus sign may suggest, subtracting moves the graph to the right, while adding moves it to the left. This shift changes only the x-coordinates of the points on the graph.

Understanding Graph Transformations

Graph transformations involve various methods of modifying a function's graph to achieve a desired visual outcome. These transformations include shifts (both vertical and horizontal), reflections, stretches, and compressions.

Shifts are a straightforward type of transformation that don't change the graph's shape, only its position. They provide a simple way to adjust where the graph lies on the coordinate plane:

• Vertical Shifts: Add or subtract from the entire function to move the graph up or down.
• Horizontal Shifts: Adjust the input (x-value) to move the graph left or right.

When dealing with graph transformations, it's crucial to understand how these changes affect the coordinates and the visual placement of the graph on the plane. Recognizing and applying vertical and horizontal shifts can greatly help in graphing functions and interpreting their behavior.