Problem 10 - Algebra and Trigonometry

Question

\(5-14\) . Suppose the graph of \(f\) is given. Describe how the graph of each function can be obtained from the graph of \(f .\) (a) \(y=f(x-4)+\frac{3}{4} \quad\) (b) \(y=f(x+4)-\frac{3}{4}\)

Step-by-Step Solution

Answer

(a) Shift right 4 units and up 0.75 units, (b) Shift left 4 units and down 0.75 units.

Step 1: Translate Right and Up

To transform the graph of the function from \(f(x)\) to \(y=f(x-4)+\frac{3}{4}\), start by translating the graph 4 units to the right. This is because the transformation \(f(x-4)\) shifts the graph horizontally by 4 units to the right. Next, move the entire graph vertically 0.75 units upwards due to the \(+\frac{3}{4}\) term, lifting every point on the graph evenly.

Step 2: Translate Left and Down

For the transformation from \(f(x)\) to \(y=f(x+4)-\frac{3}{4}\), begin by translating the graph 4 units to the left, as the \(f(x+4)\) indicates a shift in the opposite horizontal direction. Then lower the graph by 0.75 units to account for the \(-\frac{3}{4}\) term, moving each point down evenly.

Key Concepts

Graph Translations

In mathematics, graph translations refer to shifting a graph horizontally or vertically without changing its shape. This concept is essential for understanding how functions can be manipulated and visualized on a coordinate plane. Graph translations are crucial for analyzing real-world phenomena or altering functions to better fit a certain set of data points.

A translation involves two main transformations:

• Horizontal Shifts
• Vertical Shifts

These movements are performed by adjusting the domain or output of the function. When graphing transformations, the primary goal is to learn how these shifts influence the position of the graph relative to its original form. Understanding graph translations makes it easier to predict how the graph will look after applying certain transformations.

Horizontal Shifts

Horizontal shifts occur when a graph is moved left or right along the x-axis. This transformation is indicated by alterations within the parenthesis of the function. For instance, changing the variable from \(f(x)\) to \(f(x-4)\) or \(f(x+4)\) will cause the graph to shift horizontally.

• If the function changes to \(f(x-4)\), the graph moves 4 units to the **right**. This rightward movement happens because you are effectively delaying the input by 4 units.
• Conversely, if the function changes to \(f(x+4)\), the graph shifts 4 units to the **left**. This means you're advancing the input by 4 units, so the shift is leftward.

When executing horizontal shifts, it's important to note that only the x-values of the function change; the height or y-values of the function remain unchanged initially. This independent alteration of x-values results in a shift along the horizontal plane without modifying the function's shape.

Vertical Shifts

Vertical shifts involve moving a graph up or down along the y-axis. This kind of shift is achieved by adding or subtracting a constant from the function. This addition or subtraction affects the output of the function, changing its graph's position vertically.

• A positive constant added to the function, such as \(+\frac{3}{4}\), results in an upward shift of the graph by that exact amount. For example, in \(f(x-4) + \frac{3}{4}\), after moving horizontally, you'll raise the graph by 0.75 units.
• In contrast, a negative constant, like \(-\frac{3}{4}\), causes the graph to move downward by that value. In the function \(f(x+4) - \frac{3}{4}\), you first adjust horizontally and then move the graph down by 0.75 units.

In vertical shifts, the x-values remain unaffected, while the y-values are the ones being adjusted. This results in a transformation along the vertical axis, ensuring that all points of the graph rise or fall equally.