Problem 151 - College Algebra

Question

Assume that \((a, b)\) is a point on the graph of \(f .\) What is the corresponding point on the graph of each of the following functions? $$ y-f(x-3) $$

Step-by-Step Solution

Answer

The corresponding point on the graph of \(y=f(x-3)\) is \((a+3, b)\).

Step 1: Understand the Transformation

A function of the form \(y=f(x-h)\) represents a horizontal shift of the graph of function \(f\). If \(h\) is positive, the shift is \(h\) units to the right. Here, \(h=3\), so this a right shift by 3 units.

Step 2: Apply the Transformation

Taking a point \((a, b)\) on the graph of \(f\), the corresponding point on the graph of \(y=f(x-3)\) is found by adding 3 (the amount of the shift) to the x-coordinate. The y-coordinate remains the same.

Step 3: Obtain the Result

The corresponding point on the graph of \(y=f(x-3)\) is \((a+3, b)\).

Key Concepts

Horizontal Shift

A horizontal shift in a function transformation is akin to moving the entire graph left or right along the x-axis.
For a function in the form of \( y = f(x-h) \), the horizontal shift occurs by \( h \) units. If \( h \) is positive, the graph shifts \( h \) units to the right, and if \( h \) is negative, it moves \( |h| \) units to the left.
In our exercise, the function given is \( y = f(x-3) \), which means the graph is shifted 3 units to the right.

• Add 3 to every x-coordinate.
• The y-coordinate remains unchanged.

A key point about horizontal shifts is that they do not affect the vertical position of points on the graph, leaving the y-values intact.

Graph of a Function

The graph of a function visually represents the relationship between the input \( x \) and the output \( y \).
Each point \( (x, y) \) on this graph illustrates how \( y \) changes according to \( x \).
The graph can take various forms: lines, curves, shapes, depending on the function type.

• A linear function results in a straight line.
• Quadratic functions lead to a parabolic curve.
• More complex functions can produce various shapes.

Understanding the graph provides valuable insights into the behavior and characteristics of a function, such as continuity, slope, and intercepts. In transformations like horizontal shifts, the shape of the graph remains identical though the position changes.

Coordinate Transformation

Coordinate transformation involves changing the position of points based on a rule or a series of rules.
In math, specifically with function transformations, this often means altering the x or y-coordinates or both, based on the kind of transformation applied.

• For horizontal shifts, adjust the x-coordinates.
• For vertical shifts, modify the y-coordinates.
• Scaling involves multiplying coordinates to stretch or compress the graph.

These transformations allow for repositioning the graph without altering its fundamental shape and properties. By mastering coordinate transformations, one can predict how a given function’s graph will look after adjustments. In our specific example with \((a, b)\), the graph of \(y = f(x-3)\) results in the coordinate \((a+3, b)\), showing a simple transformation without modifying the y-value.