Problem 70 - 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.$$y=2 f(x)$$.

Step-by-Step Solution

Answer

The x-intercepts of the function \(y = 2f(x)\) are \(x = 2\) and \(x = -3\).

Step 1: Understand the relationship between the function and its x-intercepts

For any function, the x-intercepts are the values of x which make the function value 0. That is, \(f(x) = 0\). If \(x = a\) is an x-intercept of \(f(x)\), then \(f(a) = 0\)

Step 2: Apply the properties of the x-intercepts

Applying this knowledge, since \(x = 2\) and \(x = -3\) are x-intercepts of \(f(x)\), we have \(f(2) = 0\) and \(f(-3) = 0\). i.e., when \(x = 2\) and \(x = -3\), our function \(f(x)\) becomes 0.

Step 3: Consider the new function

Now, we have a new function \(y = 2f(x)\). Other than the function being scaled vertically by a factor of 2, the function \(2f(x) = 2 * 0 = 0\). This means that the function \(2f(x)\) is also 0 when \(x = 2\) and \(x = -3\). Hence, \(x = 2\) and \(x = -3\) will also be x-intercepts of the function \(y = 2f(x)\).

Key Concepts

Understanding Functions

A function is a fundamental concept in algebra where each input is assigned a unique output. Imagine it as a machine where you put something in, it does its magic, and gives you the result. For example, if we have a function \( f(x) = x^2 \), putting an input of \( x = 3 \) would give us an output of \( f(3) = 9 \).Functions can be represented as equations, tables, or graphs, and they can have different types of behavior or shapes, such as linear, quadratic, or exponential. In the context of the exercise, the function \( f(x) \) determines the shape and position of the graph. Understanding the behavior of the function allows you to find important points like x-intercepts, which are the points where the graph crosses the x-axis.These x-intercepts are crucial since they represent the inputs where the function's output is zero. In simpler terms, they are the roots or solutions to the equation \( f(x) = 0 \). For the function \( y = f(x) \), if \( f(x) \) crosses the x-axis at \( x = 2 \) and \( x = -3 \), then those are its x-intercepts.

Exploring Graph Transformations

Graph transformations involve altering the position or shape of a graph, based on the mathematical operations applied to the function's equation. These transformations can include shifting, reflecting, stretching, or compressing the graph.In the exercise where the function changes from \( y = f(x) \) to \( y = 2f(x) \), this is an example of vertical stretching. Multiplying the function by a coefficient greater than 1 stretches it vertically, making it taller and narrower. However, this transformation does not affect the x-intercepts.The reason is that the x-intercepts depend on where the value of the function is zero. Since multiplying by two still results in zero when the original function's value is zero, the x-intercepts remain unchanged. Thus, the new function \( y = 2f(x) \) maintains the same x-intercepts at \( x = 2 \) and \( x = -3 \). Understanding this concept is essential as it allows you to predict how changes in functions affect their graphs.

Diving into Algebra Concepts

Algebra provides a framework for dealing with variables and understanding their relationships. One of the key concepts in algebra is solving equations to find specific values for variables that make the equation true.Consider the function's role in determining the x-intercepts. If you are given \( f(x) \) and asked to find where \( f(x) = 0 \), you're solving an algebraic equation. This involves setting the equation equal to zero and finding the x-values that satisfy this condition. In the given exercise, knowing that \( f(2) = 0 \) and \( f(-3) = 0 \) means these values of \( x \) make the function zero, marking them as the x-intercepts.In algebra, transformations like multiplying a function by a number lead to learning about the properties that remain unchanged. By recognizing these properties, such as unchanged x-intercepts when scaling a function, you gain a deeper understanding of algebraic manipulation and how transformations work.