Problem 72 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the fact that the graph of \(y=f(x)\) has \(x\) -intercepts at \(x=2\) and \(x=-3\) to find the \(x\) -intercepts of the given graph. If not possible, state the reason.$$y=f(x-3)$$.

Step-by-Step Solution

Answer

The x-intercepts of the function \(y = f(x - 3)\) are at \(x = 5\) and \(x = 0\).

Step 1: Identify the Original X-Intercepts

The original function \(y = f(x)\) has x-intercepts at \(x = 2\) and \(x = -3\). These are the points where the function crosses the x-axis.

Step 2: Understand Function Transformations

A function of the form \(f(x - h)\) represents a translation (or shift) of the graph of function \(f(x)\) h units to the right. Here, \(h = 3\), so the function \(y = f(x - 3)\) is the graph of \(y = f(x)\) shifted three units to the right.

Step 3: Find New X-Intercepts

The x-intercepts of \(y = f(x - 3)\) will be at \(x = 2 + 3 = 5\) and \(x = -3 + 3 = 0\), because we shift each of the original x-intercepts 3 units to the right.

Key Concepts

Understanding X-Intercepts

X-intercepts are the points where a graph crosses the x-axis. At these points, the value of the function is zero. In mathematical terms, this means that if you have a function \(y = f(x)\), then at the x-intercepts, \(f(x) = 0\).
To locate x-intercepts:

• Set the function equal to zero.
• Solve for \(x\).

In our example, the function \(y = f(x)\) has original x-intercepts at \(x = 2\) and \(x = -3\). These values tell us where the function was originally zero. Think of these as the roots or solutions of the equation \(f(x) = 0\).
Understanding x-intercepts is essential for graphically analyzing functions since these points often give crucial information about the behavior and characteristics of a graph.

Exploring Function Translation

Function translation involves shifting the entire graph of a function without altering its shape. The graph of \(f(x - h)\) is a horizontal translation of the function \(f(x)\) by \(h\) units.

• If \(h\) is positive, shift the graph right.
• If \(h\) is negative, shift it left.

For our exercise, the function given is \(y = f(x - 3)\). This represents a translation of the original function \(f(x)\) by 3 units to the right. The horizontal shift of the graph leaves the shape unchanged, just relocated along the x-axis.
This type of transformation helps us predict where the new x-intercepts will be. Every point on the graph moves equally, maintaining the relative position of each point to others.

Concept of Graph Shifting

Graph shifting is a crucial concept in understanding how graphs of functions move across the coordinate plane. When a graph is shifted, each point moves a specific distance without changing the function's other properties, such as its shape or orientation.
There are two primary types of shifts:

• Horizontal Shifts: These occur when the graph moves left or right, as seen in the form \(y = f(x - h)\).
• Vertical Shifts: These occur when the graph moves up or down, expressed by \(y = f(x) + k\).

For the problem \(y = f(x - 3)\), the graph of the function is shifted 3 units to the right. Thus, each x-intercept of the original function \(f(x)\) is moved 3 units right, resulting in new x-intercepts at \(x = 5\) and \(x = 0\). This consistent shift behavior is why function transformations are a powerful tool in graphing and analysis.