Line Analysis: Slope, Intercepts, And Graph
Understanding the Basics of Linear Equations
Before we dive into the specifics of our line, let's refresh our understanding of linear equations. Remember the slope-intercept form? It's the key to unlocking this problem! The slope-intercept form is y = mx + b, where m represents the slope and b represents the y-intercept. Think of the slope as the 'steepness' of the line – how much it rises or falls for every step you take to the right. The y-intercept, on the other hand, is where the line crosses the vertical y-axis. It's the point where x is equal to zero.
Now, let’s analyze our equation: y = 2 + 712x. Notice anything familiar? It might look slightly different from the classic y = mx + b, but we can easily rearrange it to fit the mold. Simply swapping the terms around, we get y = 712x + 2. Ah, much better! Now it's crystal clear that our slope (m) is 712 and our y-intercept (b) is 2. See? We're already making progress!
Finding the Slope: The Steepness of Our Line
Okay, so we've already identified the slope from our rearranged equation y = 712x + 2. The slope, m, is the number that multiplies x, which in our case is 712. That's a pretty big number! What does this actually mean? Well, a slope of 712 means that for every one unit you move to the right along the x-axis, the line goes up by a whopping 712 units on the y-axis. That's a steep climb! Imagine scaling a mountain with that kind of incline – you'd need some serious hiking gear. So, the slope tells us just how rapidly our line is ascending. The larger the slope (in absolute value), the steeper the line. A positive slope means the line goes upwards from left to right, while a negative slope would indicate a downward trend. In our case, we have a positive slope, so our line is definitely heading uphill.
Unveiling the Intercepts: Where the Line Crosses the Axes
Intercepts are crucial points that help us visualize and understand where our line interacts with the coordinate axes. We have two main intercepts to consider: the y-intercept and the x-intercept.
The Y-Intercept: Crossing the Vertical
The y-intercept is the point where our line crosses the y-axis. Remember, this happens when x is equal to zero. We already figured out the y-intercept when we rearranged our equation into slope-intercept form! In the equation y = 712x + 2, the y-intercept, b, is 2. This means our line crosses the y-axis at the point (0, 2). Easy peasy!
The X-Intercept: Crossing the Horizontal
The x-intercept is where our line crosses the x-axis. This occurs when y is equal to zero. To find the x-intercept, we need to substitute y = 0 into our equation and solve for x. Let's do it:
0 = 712x + 2
Now, we need to isolate x. First, subtract 2 from both sides:
-2 = 712x
Next, divide both sides by 712:
x = -2 / 712
Simplifying this fraction, we get:
x = -1 / 356
So, the x-intercept is the point (-1/356, 0). This is a tiny negative number, which means our line crosses the x-axis very close to the origin on the negative side.
Images and Pre-images: Mapping the Line
Now, let's explore the concepts of images and pre-images. These concepts help us understand how the line maps specific x values to y values (images) and vice versa (pre-images).
Finding the Image of 10: What Y Value Corresponds to X = 10?
The image of a value is simply the y value that corresponds to a given x value. In our case, we want to find the image of 10. This means we need to find the y value when x = 10. We can do this by substituting x = 10 into our equation:
y = 712(10) + 2
y = 7120 + 2
y = 7122
So, the image of 10 is 7122. That's a huge y value! This makes sense considering our steep slope. When x is 10, the line is way up high on the y-axis.
Finding the Pre-image of 8: What X Value Corresponds to Y = 8?
The pre-image is the reverse of the image. We're given a y value (8 in this case) and we need to find the corresponding x value. To do this, we substitute y = 8 into our equation and solve for x:
8 = 712x + 2
First, subtract 2 from both sides:
6 = 712x
Now, divide both sides by 712:
x = 6 / 712
Simplifying this fraction, we get:
x = 3 / 356
Therefore, the pre-image of 8 is 3/356. This is a small positive number, indicating that the point on the line where y is 8 is relatively close to the y-axis on the positive side.
Graphing the Line: Visualizing the Equation
Finally, let's bring it all together and graph our line! We have all the information we need: the slope (712), the y-intercept (0, 2), and the x-intercept (-1/356, 0). We can also use the image and pre-image we calculated as additional points on the line.
To graph the line, we can start by plotting the intercepts. Plot the point (0, 2) on the y-axis and the point (-1/356, 0) on the x-axis. Now, we can use the slope to find another point. Remember, the slope is the rise over the run. A slope of 712 means that for every 1 unit we move to the right, we go up 712 units. This isn't very practical to plot on a standard graph, so let's use the image we found earlier: when x = 10, y = 7122. This point (10, 7122) would be far off our graph, but it confirms the steepness of the line.
To get a more manageable point, we can use the pre-image we found. When y = 8, x = 3/356. This point (3/356, 8) will be closer to the origin. Now, we have a few points and a good understanding of the line's steepness. We can draw a straight line through these points, extending it in both directions. And there you have it – the graph of y = 712x + 2!
Conclusion: Mastering Linear Equations
So, we've successfully decoded the line represented by the equation y = 2 + 712x. We found its slope (712), y-intercept (0, 2), x-intercept (-1/356, 0), calculated the image of 10 (7122), and found the pre-image of 8 (3/356). We even graphed the line! This exercise demonstrates how understanding the fundamental concepts of linear equations – slope, intercepts, images, and pre-images – can help us analyze and visualize these lines. Keep practicing, guys, and you'll become linear equation masters in no time!
