Exploring Functions: Tables, Ordered Pairs, And Graphs

Hey guys! Let's dive into the world of functions, specifically focusing on how to represent them using tables, ordered pairs, and graphs. We'll be working with a specific function, defined by the formula f(x) = 4x - 7, and a given domain (the set of input values). Don't worry, it's not as scary as it sounds! We'll break it down step-by-step so you can understand the process clearly. By the end of this, you'll be able to create tables, identify ordered pairs, and visualize functions through their graphs. This knowledge is fundamental in math and opens doors to understanding more complex concepts. So, let's get started and make math fun!

(a) Creating a Function Table: Your First Step

Creating a function table is like organizing your data in a clear and understandable format. It's the first step in understanding how a function works. For the function f(x) = 4x - 7, and the domain {-4, -3, -2, -1, 0, 1, 2, 3, 4}, we'll create a table that shows the input values (x) and their corresponding output values (f(x)). Each input value from our domain will be substituted into the function, and we'll calculate the result. This table is a great way to see the relationship between inputs and outputs at a glance.

Let's break it down further. We'll have two columns: one for x (the input values) and one for f(x) (the output values). The domain gives us our input values: -4, -3, -2, -1, 0, 1, 2, 3, and 4. Now, for each x, we substitute it into the formula f(x) = 4x - 7. For example, if x = -4, then f(-4) = 4(-4) - 7 = -16 - 7 = -23*. So, we place -23 in the f(x) column next to -4 in the x column. We repeat this process for all the values in the domain. This might seem tedious, but it gives you a very clear picture of how the function transforms each input.

Here’s what the table will look like:

As you can see, each x value from the domain has a corresponding f(x) value, and each pair represents a point on the graph of the function. Understanding how to create and read function tables is critical for not only this problem but for understanding function behavior in general. By doing this, you're building a strong foundation for more complex mathematical concepts like calculus and beyond. So keep up the great work, and you'll find that these seemingly simple steps unlock a deeper comprehension of how math really works!

(b) Determining Ordered Pairs: The Coordinates of Your Function

Alright, let's talk about ordered pairs. These are the coordinates that describe the location of points on a graph. Each ordered pair consists of an x-value (the input) and a y-value (the output), written as (x, y). In the context of functions, the y-value is the same as f(x). So, each row in our function table from part (a) can be directly translated into an ordered pair. These pairs are what we'll use to plot the graph of our function, revealing its visual shape and behavior.

Now, let's convert the function table into a set of ordered pairs. We simply take the x and f(x) values from each row of the table and write them as a coordinate pair. For example, from the first row of our table, we have x = -4 and f(x) = -23. This gives us the ordered pair (-4, -23). We continue this process for each row, obtaining a set of ordered pairs that completely defines our function for the given domain. This set of ordered pairs provides the precise information needed to understand where this function's graph will be located in relation to the x and y axes.

Here are the ordered pairs for our function:

• (-4, -23)
• (-3, -19)
• (-2, -15)
• (-1, -11)
• (0, -7)
• (1, -3)
• (2, 1)
• (3, 5)
• (4, 9)

Each of these ordered pairs represents a specific point on the graph of f(x) = 4x - 7. When you plot these points on a coordinate plane, you'll see a straight line. This is a characteristic of linear functions like ours. Recognizing these patterns and the connection between the formula, the table, the ordered pairs, and the graph is a key element of mastering functions. Keep in mind that ordered pairs are essential for creating the visual representation of functions, and also play an important role in defining many mathematical relationships.

(c) Creating a Function Graph: Visualizing the Function

Finally, let's get visual! Creating a function graph is all about plotting the ordered pairs we found in part (b) on a coordinate plane (also known as the Cartesian plane). This plane has two perpendicular axes: the horizontal x-axis and the vertical y-axis. Each point on the plane represents a unique pair of x and y values. The graph of a function is a visual representation of all the ordered pairs that satisfy the function's equation.

To graph our function f(x) = 4x - 7, follow these steps:

1. Draw the axes: Draw a horizontal x-axis and a vertical y-axis. Make sure to label them clearly.
2. Scale the axes: Decide on appropriate scales for the x and y axes. Since our y values range from -23 to 9, you'll need to make sure your y-axis covers that range. Your x-axis will need to accommodate the values from -4 to 4. Choose a scale that allows you to easily plot the points without the graph becoming too cramped or too spread out.
3. Plot the points: For each ordered pair, move along the x-axis to the x-value and then up or down along the y-axis to the y-value. Mark the point where the lines intersect. For example, for the ordered pair (-4, -23), go 4 units to the left on the x-axis and then 23 units down on the y-axis. Plot a point at that location. Do this for all of your ordered pairs.
4. Connect the points: Since this is a linear function, the points should all fall on a straight line. Use a ruler to draw a straight line that passes through all the plotted points. If the points do not form a straight line, double-check your calculations and plotting.

The graph of f(x) = 4x - 7 will be a straight line that slopes upwards from left to right. It will intersect the y-axis at the point (0, -7) - this is the y-intercept. The slope of the line represents the rate of change of the function, which in this case is 4. The graphical representation of a function gives you an immediate picture of its behavior, allowing you to quickly identify its trends and key characteristics. Being able to go from the equation to the table, to the ordered pairs, and finally to the graph is a powerful skill. It provides a visual understanding of the function, and it is a fundamental aspect of mathematics. Keep practicing, and you'll find that these concepts will become easier and more intuitive!