Problem 10 - College Algebra with Corequisite Support

Question

For the following exercises, describe how the graph of the function is a transformation of the graph of the original function \(f\). \(y=f(x-49)\)

Step-by-Step Solution

Answer

The graph of \( y = f(x-49) \) is the graph of \( f(x) \) shifted 49 units to the right.

Step 1: Identify the Original Function

The original function is given as \( f(x) \). This function can represent any general curve or graph in the coordinate plane. For this exercise, we focus on how transformations affect this base graph.

Step 2: Understand the Transformation

The given function is \( y = f(x-49) \). This transformation is specifically a horizontal shift. To interpret \( f(x-49) \), recognize that it indicates a horizontal shift to the right by 49 units.

Step 3: Explain the Effect of Transformation

A horizontal shift does not change the shape or the size of the graph, only its position. Specifically, \( y = f(x-49) \) means for every point \((x_0, y_0)\) on the graph of \( f(x) \), there is a corresponding point \((x_0+49, y_0)\) on the graph of \( f(x-49) \). This effectively moves every point 49 units to the right.

Key Concepts

horizontal shift

A horizontal shift is one of the simplest ways to transform the graph of a function on the coordinate plane. When you see a function expressed as \(y = f(x - h)\), it indicates a horizontal shift. The "\(-h\)" inside the function's argument tells us we are shifting to the right by \(h\) units. In this case, with \(f(x-49)\), you shift the graph 49 units to the right.

Why does this happen? Because you're essentially delaying the input value that affects the output. Think of it like delaying a train that has a strict schedule. If you push the departure time by 49 minutes, everything at each station also shifts by 49 minutes.

Remember, this horizontal shift moves all points of the graph in the same direction by the same amount. The shape and orientation of the graph remain unchanged, but its position on the plane differs.

coordinate plane

The coordinate plane is a fundamental concept in algebra and calculus. It's a two-dimensional plane formed by the intersection of the vertical y-axis and the horizontal x-axis. This plane is a backdrop on which functions like \(y = f(x)\) or \(y = f(x-49)\) are graphically represented.

Think of it like a map grid system, where every point is defined by coordinates \((x, y)\). The x-coordinate measures horizontal distance, whereas the y-coordinate measures vertical distance.

When graphing transformations such as horizontal shifts, the coordinate plane helps us visualize how the entire graph moves without altering its geometry. Each shift translates to modifying the x-coordinates of all points, making the coordinate plane an essential tool for analyzing function transformations.

graphing functions

Graphing functions is a crucial technique to visually understand and analyze mathematical expressions. By plotting the function's input-output pairs on a coordinate plane, you can "see" the relationship it describes. When you have a function like \(f(x)\), its graph represents all the points \((x, f(x))\).

Function transformations allow you to modify these graphs to understand their characteristics better. With horizontal shifts like \(y = f(x-49)\), each graphing point moves horizontally but retains its vertical position. This lets you track how changes affect the entire function's "picture" on a graph, aiding in visual learning.

To become proficient at graphing functions, follow these tips:

• Identify key points such as intercepts and peaks that are easy to track after a transformation.
• Pay close attention to scale, so that shifts are accurately represented.
• Use graphing tools or graph paper to maintain precision and clarity.

Mastering these techniques aids significantly in learning and applying mathematical transformations.