Problem 2 - Algebra and Trigonometry

Question

If \(f(2)=3,\) then the point (2,________) is on the graph of \(f\).

Step-by-Step Solution

Answer

The point is (2, 3).

Step 1: Understanding the Problem

We are given that the function value of \( f \) at \( x = 2 \) is \( f(2) = 3 \). The problem asks us to find the corresponding \( y \)-coordinate for the point on the graph.

Step 2: Applying the Given Information

Since \( f(2) = 3 \), the function \( f \) assigns the value \( 3 \) to the input \( 2 \). This means that when \( x = 2 \), the corresponding \( y \)-coordinate on the graph of \( f \) is \( 3 \).

Step 3: Writing the Point

Using the information from the previous steps, the point on the graph of \( f \) that corresponds to \( x = 2 \) is \( (2, 3) \), where \( 3 \) is the \( y \)-coordinate determined by \( f(2) = 3 \).

Key Concepts

Understanding Function Values

When we talk about the function value, we're referring to the output we get when we input a certain value into a function. This is often denoted as \( f(x)\), where \( x \) is the input, and \( f(x) \) is the function value.
For example, in the exercise we see \( f(2) = 3 \). Here, \( 3 \) is the function value when \( x \) equals \( 2 \). Why is this important? Knowing the function value helps us understand how the function behaves at that particular point. It tells us what output the function will produce for an input.
In a broader sense, function values are crucial for plotting graphs, solving equations, and analyzing the behavior of different types of functions.

The Art of Graphing Functions

Graphing functions is a visual way to represent the relationship between inputs and outputs of a function. When graphing, the function's inputs become the \( x \)-coordinates, while the outputs become the \( y \)-coordinates.
The graph is a collection of all points \((x, y)\) such that \( y = f(x) \).To graph a function, follow these steps:

• Identify a range of \( x \)-values. Plug each of these into the function to find the corresponding \( y \)-values.
• Plot these points on a coordinate plane.
• Connect the points to visualize the function's behavior.

Graphing allows you to see at a glance how your function behaves. It reveals key features such as intercepts, slopes, and any symmetry. This visual representation can help solve problems more quickly and intuitively.

Decoding Coordinates on a Graph

Coordinates are like address labels for points on a graph. An ordered pair \((x, y)\) specifies the exact location of a point on the coordinate plane.
The first number, \( x \), tells how far left or right to move from the origin (the center of the graph). The second number, \( y \), tells how far to move up or down.For instance, in our earlier example with \( f(2) = 3 \), the corresponding point is \((2, 3)\).
This means, from the origin, move 2 units to the right and 3 units up.Working with coordinates is essential when graphing functions because:

• They help plot exact points generated by the function.
• They provide a means to verify the function behavior at specific inputs.
• They aid in tracing the growth or decay patterns of a function across its domain.

Understanding coordinates is a foundational skill in learning math and solving real-world problems using graphs.