Graphing Linear Functions A Step By Step Guide To F(x) = -(1/3)x - 6

Hey guys! Today, we're diving deep into the world of linear functions and tackling a super common question in algebra: completing a table and graphing a function. Specifically, we're going to focus on the function f(x) = -(1/3)x - 6. Don't worry if that looks a little intimidating – we'll break it down step-by-step, making it super easy to understand. This is a crucial skill for anyone learning algebra, so let's jump right in and master it together!

Completing the Table for f(x) = -(1/3)x - 6

First, let's tackle the table. This is the foundation for graphing our function. A table helps us organize our thoughts and see the relationship between x and f(x). Remember, f(x) is just another way of writing y, representing the output of our function for a given input x. We've got a table with specific x values (-9, -3, 0, 3, and 6), and our mission is to find the corresponding f(x) values. Think of it like a math puzzle – we're going to plug in each x value into our function and solve for f(x). This process, known as function evaluation, is the key to unlocking the table. So, grab your pencils, and let's calculate these values one by one!

Step-by-Step Calculation

Okay, let's get down to the nitty-gritty and walk through the calculations. We'll take each x value and substitute it into the function f(x) = -(1/3)x - 6. Remember the order of operations (PEMDAS/BODMAS)? We'll be using that to make sure we get the correct answers. So, first up, we have x = -9. Substituting this into our function, we get f(-9) = -(1/3)(-9) - 6. A negative times a negative is a positive, so -(1/3)(-9) becomes 3. Then, 3 - 6 equals -3. So, f(-9) = -3. Next, let's tackle x = -3. Plugging this in, we have f(-3) = -(1/3)(-3) - 6. Again, a negative times a negative gives us a positive, so -(1/3)(-3) is 1. Then, 1 - 6 equals -5. So, f(-3) = -5. Now for x = 0. This one's pretty straightforward! f(0) = -(1/3)(0) - 6. Anything times zero is zero, so -(1/3)(0) is just 0. That leaves us with 0 - 6, which is -6. So, f(0) = -6. Moving on to x = 3, we have f(3) = -(1/3)(3) - 6. -(1/3) times 3 is -1. Then, -1 - 6 equals -7. So, f(3) = -7. Finally, let's do x = 6. Plugging this in, we get f(6) = -(1/3)(6) - 6. -(1/3) times 6 is -2. Then, -2 - 6 equals -8. So, f(6) = -8. We've successfully evaluated the function for all our given x values! This methodical approach ensures we get the right results every time. Remember, practice makes perfect, so the more you do this, the easier it'll become.

The Completed Table

Now that we've crunched the numbers, let's fill in our table. We've found the corresponding f(x) values for each x value. This table is going to be our roadmap for graphing the function. It gives us the specific points we need to plot on our coordinate plane. It's like having a treasure map – the x and f(x) values are our coordinates, guiding us to the correct location on the graph. By organizing our data in this way, we make the graphing process much smoother and more accurate. So, let's take a look at our completed table and get ready to transform this data into a visual representation!

Here's what our completed table looks like:

Graphing the Function f(x) = -(1/3)x - 6

Alright, guys, we've conquered the table, and now it's time for the fun part: graphing! Graphing a linear function is like connecting the dots – literally! We'll use the points we found in our table to draw a straight line that represents our function, f(x) = -(1/3)x - 6. Remember, each row in our table gives us a coordinate pair (x, f(x)), which corresponds to a specific point on the graph. The x value tells us how far to move horizontally on the x-axis, and the f(x) value (or y value) tells us how far to move vertically on the y-axis. The coordinate plane is our canvas, and these points are our guide. So, let's get our graph paper ready and start plotting these points!

Plotting the Points

Now, let's talk about how to actually plot these points. Grab your graph paper (or a digital graphing tool) and let's get started. The coordinate plane has two axes: the horizontal x-axis and the vertical y-axis. The point where they intersect is called the origin, and it's represented by the coordinates (0, 0). Each point on the plane is defined by its x and y coordinates. Let's take our first point from the table: (-9, -3). The x-coordinate is -9, so we move 9 units to the left along the x-axis (since it's negative). Then, the y-coordinate is -3, so we move 3 units down along the y-axis (again, since it's negative). Mark that spot – that's our first point! Now, let's do the next point: (-3, -5). We move 3 units to the left on the x-axis and 5 units down on the y-axis. Mark that spot too. Next up is (0, -6). This one's easy! Since the x-coordinate is 0, we stay on the y-axis. We just move 6 units down from the origin. Mark that point. Moving on to (3, -7), we go 3 units to the right on the x-axis and 7 units down on the y-axis. Mark it. And finally, (6, -8). We move 6 units to the right on the x-axis and 8 units down on the y-axis. Mark that last point. We've successfully plotted all the points from our table! Each point represents a solution to our equation, and together, they form the foundation for our line. This careful plotting ensures that our graph accurately represents the function.

Drawing the Line

We've plotted our points, and now the magic happens – we connect the dots! Since we're dealing with a linear function, we know that these points should form a straight line. Grab a ruler (or use a digital line tool) and carefully draw a line through all the points we've plotted. Make sure the line extends beyond the points, as a line technically goes on infinitely in both directions. If your points don't quite line up perfectly, don't panic! This can sometimes happen due to slight inaccuracies in plotting. Just try to draw the line that best represents the overall trend of the points. The line you've drawn is the visual representation of our function, f(x) = -(1/3)x - 6. It shows us all the possible solutions to the equation. Every point on that line corresponds to a pair of x and f(x) values that satisfy the function. Isn't that cool? We've taken a mathematical equation and turned it into a visual masterpiece! Remember, the key to a good graph is accuracy and neatness. A well-drawn graph makes it much easier to understand the behavior of the function.

Understanding the Graph

Now that we have our beautiful line graph, let's take a moment to really understand what it's telling us. The graph of f(x) = -(1/3)x - 6 is a straight line, and straight lines have some key characteristics. First, they have a slope, which tells us how steep the line is and in what direction it's going. In our function, the slope is -(1/3). The negative sign tells us that the line is decreasing – as x increases, f(x) decreases. The fraction (1/3) tells us that for every 3 units we move to the right on the x-axis, the line goes down 1 unit on the y-axis. Think of it like walking down a gentle slope. Next, our line has a y-intercept, which is the point where the line crosses the y-axis. This is the point where x is 0. Looking back at our table, we see that when x is 0, f(x) is -6. So, our y-intercept is the point (0, -6). This point is super important because it gives us a starting point for graphing the line. It's like the anchor that holds our line in place. The slope and y-intercept are the two fundamental properties that define a linear function. By understanding these properties, we can quickly sketch a graph of any linear equation. Moreover, by looking at the graph, we can easily read off solutions to the equation. For example, if we want to find f(x) when x is 3, we just find the point on the line where x is 3 and read off the corresponding f(x) value. This visual representation makes solving equations and understanding functions much more intuitive.

Key Takeaways

Alright, guys, we've covered a lot today! Let's recap the key takeaways so you can confidently tackle similar problems in the future. First, we learned how to complete a table for a linear function by substituting different x values into the function and solving for f(x). This is a fundamental skill in algebra and is the foundation for graphing. Next, we mastered the art of graphing a linear function. We learned how to plot points from the table on a coordinate plane and then draw a straight line through those points. Remember, each point on the line represents a solution to the equation. Finally, we discussed the importance of understanding the graph. We talked about the slope and y-intercept and how they define the behavior of the line. By understanding these key characteristics, we can quickly sketch graphs and read off solutions. Graphing linear functions might seem a little daunting at first, but with practice, it becomes second nature. The key is to break it down into smaller steps, like we did today. Completing the table, plotting the points, and drawing the line – it's all about taking it one step at a time. And remember, math is like building with Lego bricks. Each skill builds on the previous one. Mastering linear functions is a crucial step towards understanding more advanced concepts in algebra and beyond. So, keep practicing, keep exploring, and keep having fun with math!

Practice Makes Perfect

Okay, guys, now that we've walked through this example together, it's time for you to put your newfound skills to the test! Practice is the secret ingredient to mastering any math concept, and linear functions are no exception. Try working through some similar problems on your own. You can find plenty of examples in your textbook, online, or even create your own! The more you practice, the more confident you'll become. Start by choosing a different linear function, like f(x) = 2x + 1 or f(x) = -x - 4. Create a table of values by choosing a few x values and calculating the corresponding f(x) values. Then, plot those points on a coordinate plane and draw a line. As you practice, pay attention to the slope and y-intercept of the line. How does the equation of the function affect the way the line looks on the graph? This is where the real learning happens – when you start to see the connections between the equation and the graph. Don't be afraid to make mistakes! Mistakes are a natural part of the learning process. When you make a mistake, take the time to understand why you made it and what you can do differently next time. And if you get stuck, don't hesitate to ask for help. Your teacher, your classmates, or online resources can all provide valuable assistance. The journey of learning math is a marathon, not a sprint. So, pace yourself, celebrate your successes, and keep practicing! You've got this!