Problem 4 - Precalculus

Question

Suppose (2,-3) is on the graph of \(y=f(x) .\) In Exercises \(1-18,\) use Theorem 1.7 to find a point on the graph of the given transformed function. $$ y=f(x-1) $$

Step-by-Step Solution

Answer

The transformed point is (3, -3).

Step 1: Understanding the Given Function

We start by noting that the original point (2, -3) lies on the graph of \(y=f(x)\). This means that when \(x=2\), the value of \(f(x)\) is -3, i.e., \(f(2)=-3\).

Step 2: Identify the Transformation

The transformation given is \(y=f(x-1)\). This is a horizontal shift to the right by 1 unit. According to Theorem 1.7, this shift means we replace \(x\) with \(x-1\) in the function.

Step 3: Apply Transformation to the Point

To find the new x-coordinate, add 1 to the original x-coordinate of the point (2, -3), because shifting right by 1 unit means we need to find \(x-1 = 2\). Solving gives \(x = 3\).

Step 4: Determine the New Point

Since the transformation only affects the x-coordinate of the point, the new point will be \((3, -3)\). The y-coordinate remains unchanged as we are only shifting the point horizontally.

Key Concepts

Horizontal Shift

When we talk about a horizontal shift in the context of function transformations, it refers to moving the entire graph of a function left or right along the x-axis. This happens without modifying the shape of the graph.
A horizontal shift can be recognized in a function as a change in the inside part of the function's argument. For example, in the function given in the problem, we have a horizontal shift:

• In the function \(y = f(x-1)\), the graph shifts 1 unit to the right. This is because subtracting 1 from \(x\) means every x-value in the function has been increased by 1 to maintain the original output of the function.
• Conversely, if we had \(y = f(x+1)\), it would represent a shift to the left by 1 unit, essentially decreasing each x-value by 1.

Horizontal shifts are useful for aligning functions or changing their position without altering their appearance on the graph. This concept is crucial for understanding more complex transformations where multiple shifts and stretches or compressions may occur in tandem.

Graphing Functions

Graphing functions is a fundamental skill in mathematics that allows us to visualize how a function behaves. By graphing, we transform a set of rules or equations into a visual plot that we can analyze.
When you graph a basic function like \(y = f(x)\), you plot points based on inputs \(x\) and their corresponding outputs \(y\).
To track changes in the function's graph, such as a horizontal shift:

• Start with the original function's points, like (2, -3) in the exercise.
• Apply any given transformations and calculate the new positions of the key points.
• Plot the new points using these adjusted coordinates. In our example, after shifting, the point becomes (3, -3).

Visualizing transformations in this way makes it easier to understand the effects of different algebraic changes to a function, aiding in problem-solving and deeper comprehension of the relationship between algebraic expressions and their graphical counterparts.

Theorem 1.7

Theorem 1.7 provides a mathematical framework for applying transformations to functions, simplifying the understanding of how various components of a function change its graph.
In the context of the exercise problem, Theorem 1.7 helps us understand how to modify the point according to the transformation described by \(y = f(x-1)\). This theorem teaches us that:

• When a transformation replaces \(x\) with \(x-c\), the graph of the function shifts horizontally by \(c\) units.
• Specifically, if the penalty in the equation is subtraction, as with \(x-1\), the graph moves to the right.
• If it were \(x+c\), the shift would be to the left.

Understanding this theorem allows students to predict and articulate the effect of transformations on graphs accurately, without having to visualize each step mentally. It offers a concise way to grasp how algebraic manipulations relate directly to physical shifts in the graph. This makes graphing more intuitive and less daunting, particularly for beginners.