Problem 74 - 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 \(y = f(x) - 3\) to the point \((a, b)\) on the graph of \(f(x)\) is \((a, b - 3)\)

Step 1: Understanding the transformation

A function of the form \(y = f(x) - k\) represents a vertical shift of the graph of \(f(x)\) downwards by `k` units. Here \(k = 3\), so the entire graph of \(f(x)\) is shifted 3 units downwards.

Step 2: Apply the transformation to the point

So a point \((a, b)\) on the graph of `f(x)` transforms to \((a, b - 3)\) on the graph of \(y = f(x) - 3\). Hence the corresponding point to \((a, b)\) on the function \(y = f(x) - 3\) is \((a, b - 3)\)

Key Concepts

Vertical Shift

In the world of function transformations, a **vertical shift** is one of the simplest types of transformations to understand. It involves moving the entire graph of a function up or down without changing its shape. Imagine you have a graph of a function, and you lift it straight up or push it down vertically.

When you see an equation in the form of \(y = f(x) - k\), it indicates that the graph has been vertically shifted downward by \(k\) units. Conversely, \(y = f(x) + k\) would shift it upward by \(k\) units.

This shift affects all points on the graph. For example, if you start with a point \((a, b)\) on the graph of \(f(x)\), a downward shift of 3 units, such as in \(y = f(x) - 3\), will move this point to \((a, b - 3)\).

• Vertical shift doesn't alter the x-value of points, only the y-value.
• It keeps the original graph's proportions intact.
• Useful for creating variations of a function by simple adjustments.

Graph Translation

Graph translation generally refers to moving a graph either vertically, horizontally, or both, without changing its size or orientation. It’s like sliding a piece of transparent paper with the graph drawn on it across your page. For vertical and horizontal shifts, the shape of the graph remains the same.

In our specific example, we dealt with a vertical shift, but translations can also occur horizontally. If we have \(y = f(x - h)\), the entire graph shifts horizontally to the right by \(h\) units. Conversely, \(y = f(x + h)\) will shift it to the left by \(h\) units.

• Vertical translation: Shifting up or down.
• Horizontal translation: Shifting left or right.
• Does not affect the nature of the graph, only its position in the coordinate plane.

Understanding graph translation allows us to predict how functions will behave under various transformations, providing insights into their properties and symmetries.

Algebraic Functions

Algebraic functions are functions defined by polynomials or rational expressions involving mathematical operations like addition, subtraction, multiplication, division, and raising to a power. They are fundamental in understanding transformations since any modification to these functions can significantly affect their graphical representation.

In our exercise, the function involved is an algebraic function, and altering it with an operation like subtraction directly influenced the graph's positioning. Consider \(f(x)\) as a base polynomial function; its alterations such as adding or subtracting a constant leads to vertical shifts.

• Algebraic functions can be as simple as linear functions \(f(x) = x\) or as complex as rational functions \(f(x) = \frac{p(x)}{q(x)}\).
• Graphical transformations involve direct manipulation of these algebraic expressions.
• Identifying these functions allows us to predict and manipulate transformations more effectively.

Understanding algebraic functions is key to mastering function transformations, as they form the groundwork for exploring more intricate relationships in algebra.