Problem 152 - Precalculus Essentials

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

Step 1: Understand the given point

A point on a graph of a function is generally given in the form \((a, b)\) where \(a\) is the value on the x-axis (input) and \(b\) is the value on the y-axis (output of the function for the input \(a\)). Here, we are given a point on the graph of the function \(f\), which is \((a,b)\). This means that when the function \(f\) is evaluated at \(a\), the output value is \(b\).

Step 2: Understand the transformation

The function \(f(x) - 3\) represents a vertical shift of \(f(x)\) downwards by 3 units. This means that for every point on the graph of the function \(f\), it's image on the graph of \(f(x) - 3\) is a point that is 3 units lower.

Step 3: Apply the transformation to the given point

So, the corresponding point on the graph of \(f(x) - 3\) to the point \((a,b)\) on the graph of \(f\) would be \((a, b - 3)\) since it would be 3 units lower than \((a,b)\). This is because the original output \(b\) is decreased by 3 units to become \(b - 3\)