Finding The Slope-Intercept Form: A Step-by-Step Guide
Hey math enthusiasts! Today, we're diving into a fundamental concept in algebra: finding the equation of a line in slope-intercept form. We'll be using a table of values to crack this code, so let's get started. Understanding slope-intercept form is super important because it provides a clear picture of a line's characteristics: its steepness (slope) and where it crosses the y-axis (y-intercept). This knowledge unlocks the ability to graph lines quickly, predict values, and solve a wide array of problems. Let's break down the process, step by step, to ensure we grasp the concept completely. You'll soon see how straightforward and powerful this method is.
Decoding the Slope-Intercept Form
First things first, let's refresh our memory about what the slope-intercept form actually is. The equation is typically written as: y = mx + b. In this equation, 'y' represents the dependent variable (the output), 'x' is the independent variable (the input), 'm' is the slope of the line, and 'b' is the y-intercept (where the line crosses the y-axis). Our goal is to figure out the values of 'm' and 'b' using the given table of x and y values. The table contains several (x, y) coordinates, representing specific points on the line. By analyzing these points, we can determine both the slope and the y-intercept. The slope tells us how much the y-value changes for every unit change in the x-value, and the y-intercept gives us the starting point of the line when x is zero. Knowing these two components is like having the blueprint of the line, allowing us to describe its behavior fully.
Calculating the Slope (m)
The slope is the measure of how steep a line is, and we calculate it using the formula: m = (y₂ - y₁) / (x₂ - x₁), where (x₁, y₁) and (x₂, y₂) are any two points on the line. Let's grab two points from our table. For instance, we can choose (-6, -18) and (-1, -8). Now, we'll plug these values into the formula. So, m = (-8 - (-18)) / (-1 - (-6)). Simplifying this gives us m = (-8 + 18) / (-1 + 6). Thus, m = 10 / 5, which simplifies to m = 2. Great job, guys! We've successfully calculated the slope. This means that for every one-unit increase in 'x', the 'y' value increases by 2 units. This also confirms that the line is increasing in the positive direction.
Now, let's take another pair of points just to confirm our calculations. Let's take (4, 2) and (9, 12). Using the same slope formula, m = (12 - 2) / (9 - 4) = 10 / 5 = 2. So, we're consistent, and our slope (m) is indeed 2. Both calculations should yield the same result. The fact that the slope is the same regardless of which points we use is what characterizes a linear function.
Finding the Y-Intercept (b)
Now that we've nailed down the slope, our next mission is to find the y-intercept (b). We know that the slope-intercept form is y = mx + b. We've calculated 'm' (the slope) to be 2. Now we can pick any point from the table and substitute the values of 'x' and 'y' into the equation. Let's pick (-1, -8) from the table. So, we have: -8 = 2 * (-1) + b. This simplifies to -8 = -2 + b. To solve for 'b', we need to isolate it, so we add 2 to both sides of the equation. This gives us b = -8 + 2, which simplifies to b = -6. Woohoo! We've found our y-intercept. The y-intercept represents the point where the line intersects the y-axis, and in our case, it's at the point (0, -6). It’s the starting point of the line on the graph.
Putting it All Together: The Equation
Alright, guys, we're at the finish line! We've calculated the slope (m = 2) and the y-intercept (b = -6). Now we can plug these values into the slope-intercept form equation: y = mx + b. Substituting our values, the equation of the line is y = 2x - 6. This equation fully represents the linear function described by the table. Every point (x, y) that satisfies this equation lies on the line. You can verify this by substituting the x-values from the table into the equation and confirming that the calculated y-values match the ones in the table. This is how you confirm your solution is accurate and that the equation is correctly derived.
Summary
In summary, finding the equation of a line in slope-intercept form from a table involves these key steps: first, calculate the slope (m) using the slope formula with two points from the table; second, pick any point from the table and substitute the x and y values, and the calculated slope (m), into the slope-intercept form equation (y = mx + b), then solve for the y-intercept (b); and finally, substitute the values of 'm' and 'b' into the slope-intercept form, resulting in your final equation, like y = 2x - 6. Remember, understanding this process helps you not only solve problems but also visualize and interpret linear relationships. It is a fundamental skill that will serve you well in various areas of mathematics and beyond. Keep practicing, and you'll become a pro at these problems in no time! Keep in mind that we're talking about linear functions. That means that a constant change in 'x' will always produce a constant change in 'y', and the graph of the function is a straight line.
Correct Answer
Looking back at the options you provided: A. y = -2x - 6, B. y = -2x - 2, C. y = 2x - 6, D. y = 2x - 2. Given our calculations, the correct answer is C. y = 2x - 6. This aligns with our derived equation. It's a great habit to work through the problem and then compare your result with the multiple-choice options, just to confirm everything is perfect. Also, always double-check your calculations, especially the arithmetic, to avoid any simple mistakes.
Keep practicing, and you'll become a master of linear equations in no time. If you have any questions, feel free to ask! Math can be fun and rewarding, and with practice, you can conquer any equation!