Problem 6 - 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 \(f .\) \(\begin{array}{ll}{\text { (a) } y=f(x+7)} & {\text { (b) } y=f(x)+7}\end{array}\)

Step-by-Step Solution

Answer

(a) Shift the graph of \(f\) 7 units left; (b) Shift the graph of \(f\) 7 units up.

Step 1: Understand Function Transformations

Before tackling the specific problems, it's useful to understand the basic concept of function transformations with respect to translation. A function is transformed by shifting it either horizontally or vertically without altering its shape.

Step 2: Analyze Horizontal Shift for y = f(x+7)

Consider the function \( y = f(x+7) \). This expression represents a horizontal shift. Specifically, replacing \( x \) with \( x+7 \) shifts the graph 7 units to the left. This is because adding inside the function's argument causes a translation in the opposite direction along the x-axis.

Step 3: Analyze Vertical Shift for y = f(x) + 7

Now, consider the function \( y = f(x) + 7 \). Adding 7 outside of the function corresponds to a vertical shift. This translates the graph 7 units upwards since each output value of the function is increased by 7.

Key Concepts

Horizontal Shift

A horizontal shift in a function graph occurs when you adjust the input values, or "x-values," affecting the graph along the x-axis. It can be a bit tricky because the direction of the shift might be counterintuitive. For example, in the function \( y = f(x+7) \), the graph of the function \( f \) will shift 7 units to the left. You might expect that adding a positive number would move the graph to the right, but it actually shifts the graph to the opposite direction of the sign inside the function's argument. Here's why:

• Replacing \( x \) with \( x + a \) translates the graph \( a \) units to the **left**.
• Conversely, replacing \( x \) with \( x - a \) shifts the graph \( a \) units to the **right**.

Think of it like adjusting where you start on the x-axis: adding moves you left, subtracting moves you right. This concept is essential when exploring function transformations, so let's keep this logic in our toolbox as we explore more shifts.

Vertical Shift

Vertical shifts move the graph of a function up or down and are generally more intuitive than horizontal shifts. When you change the output values, or "y-values," by adding or subtracting a number to the entire function, the whole graph simply slides up or down. Considering the function \( y = f(x) + 7 \), this demonstrates a vertical shift upwards of 7 units.

• Adding \( a \) to the function, \( y = f(x) + a \), shifts the graph **up** \( a \) units.
• Subtracting \( a \), \( y = f(x) - a \), shifts the graph **down** \( a \) units.

This is like lifting the graph off the page and moving it up or down while maintaining its shape and features. Every point on the graph moves consistently by the same amount vertically. This concept is especially straightforward and acts as a foundation for understanding more complex transformations.

Graph Translation

Graph translation is a term used to describe any shift of a graph on a coordinate plane. This includes both horizontal and vertical shifts. It's like picking up the entire graph and relocating it to a new position while keeping its overall shape intact.

• Horizontal and vertical translations don't change the size or orientation of the graph, only its position.
• Combining both types of shifts, you can move the graph in any direction desired.

Graph translation is useful in real-life scenarios where you need to model situations by simply repositioning existing functions. In the problem's context, it's important to distinguish between shifting inputs (horizontal) and outputs (vertical). Graph translations enable you to see relationships and modifications of functions as simple positional adjustments on a graph. By understanding this, you can navigate through different transformations seamlessly, gaining deeper insights into the behavior and characteristics of functions.