Problem 2 - College Algebra with Modeling and Visualization

Question

Evaluating and Representing Functions If \(f(3)=-9.7,\) identity a point on the graph of \(f .\)

Step-by-Step Solution

Answer

The point on the graph is (3, -9.7).

Step 1: Understanding Function Notation

In function notation, if we see something like \(f(x) = y\), it signifies that when the input \(x\) is put into the function \(f\), the output is \(y\). Therefore, the given \(f(3) = -9.7\) means when the input \(x\) is 3, the output \(f(x)\) is -9.7.

Step 2: Identifying the Point on the Graph

From the function notation \(f(3) = -9.7\), we can translate this into a point on the graph. This means the point where \(x = 3\) and \(y = -9.7\) is on the graph of \(f\).

Step 3: Constructing the Point

Using the ordered pair form \((x, y)\), we construct the point on the graph from our earlier step. The point on the graph of \(f\) is \((3, -9.7)\).

Key Concepts

Evaluating Functions

Evaluating functions is all about finding the output value when an input is plugged into a function. This can be understood through function notation. When you see something like \(f(x) = y\), it simply indicates that \(y\) is the result we get when \(x\) is used as an input. An example is \(f(3) = -9.7\), which tells us that by substituting \(3\) into the function \(f\), the output we receive is \(-9.7\). This process is helpful when you want to determine the value of a function at specific points on its graph.
Here are some key points to consider while evaluating functions:

• Identify the function and understand its notation.
• Substitute the input value into the function.
• Calculate or identify the output value.

Evaluating a function at a specific input helps us understand its behavior and can be further used to plot or analyze its graph.

Graphing Functions

Graphing functions involves plotting the relationship between inputs and their corresponding outputs on a coordinate plane. Each pair of input and output values represents a point on the graph. Understanding how to evaluate a function helps in identifying these points.
When graphing a function, follow these steps:

• Evaluate the function at various input values to find their corresponding outputs.
• Plot these ordered pairs \((x, y)\) on the coordinate plane.
• Connect the dots to visualize the behavior of the function.

Graphing provides a visual representation of the function's behavior, like its rise, fall, or constancy. For example, if you evaluate the function and find the point \((3, -9.7)\), this point will be part of the graph. Understanding the graph of a function helps in interpreting the changes in its values over the domain.

Ordered Pairs

Ordered pairs are fundamental in representing points on a graph. They are written in the form \((x, y)\) and depict the relationship between inputs \(x\) and their outputs \(y\) in a function. With an ordered pair like \((3, -9.7)\), \(3\) corresponds to the x-coordinate, and \(-9.7\) corresponds to the y-coordinate.
Some important aspects of ordered pairs include:

• The \(x\)-value, or input, is always written first.
• The \(y\)-value, or output, follows the \(x\)-value.
• Ordered pairs are used to pinpoint an exact location on the graph.

Understanding ordered pairs is crucial for transitioning numeric function evaluations into graphical representations. These pairs allow us to see how a function behaves visually and help us find specific points on its graph.