Graphing Y = (3/4)x - 4: What To Expect?

Hey guys! Today, we're diving into the world of linear equations and graphs, and we're going to specifically look at the equation y = (3/4)x - 4. Imagine you're Priya, and you're about to graph this line, but you want to get a sneak peek of what it's going to look like. No worries, we've got you covered! We'll break down this equation, figure out its key features, and predict which quadrants it'll pass through. Think of this as your guide to understanding linear equations – let's jump right in!

Understanding the Equation y = (3/4)x - 4

Let's break down the equation y = (3/4)x - 4 piece by piece. This equation is in slope-intercept form, which is a super helpful way to represent linear equations. The general form of slope-intercept form is y = mx + b, where 'm' is the slope and 'b' is the y-intercept. Understanding these two components, the slope and the y-intercept, is crucial for visualizing the line. They are the key to unlocking the secrets of our line's behavior on the graph. The beauty of the slope-intercept form is that it immediately tells you two very important things about the line: its steepness (slope) and where it crosses the vertical axis (y-intercept). So, let’s decode what these values tell us about Priya's line.

Slope (m)

In our equation, y = (3/4)x - 4, the slope (m) is 3/4. What does this mean? Well, the slope tells us how steep the line is and in what direction it's going. A positive slope, like ours, means the line is going uphill from left to right. If you were walking along the line from left to right, you would be climbing. The slope of 3/4 means that for every 4 units we move to the right on the graph (the 'run'), we move 3 units up (the 'rise'). Think of it as a set of stairs – for every 4 steps forward, you climb 3 steps up. This ratio of rise over run gives the line its characteristic tilt. A larger slope (like 2 or 3) would mean a steeper climb, while a smaller slope (like 1/2 or 1/4) would mean a gentler climb. So, a slope of 3/4 indicates a moderately steep line that rises as we move from left to right.

Y-intercept (b)

The y-intercept (b) is the point where the line crosses the y-axis (the vertical axis). In our equation, y = (3/4)x - 4, the y-intercept is -4. This means the line intersects the y-axis at the point (0, -4). Imagine the coordinate plane – the y-axis is the vertical line that runs straight up and down. The y-intercept is the specific point where our line will cross this axis. In this case, it crosses the y-axis at -4, which is four units below the origin (0, 0). The y-intercept is a fixed point that anchors the line on the graph. It's like the starting point for our line. From this point, the slope dictates how the line will extend across the coordinate plane. So, knowing that our line crosses the y-axis at -4 gives us a crucial reference point for sketching or graphing the line.

Visualizing the Line

Now that we know the slope and the y-intercept, we can start to visualize the line. Imagine a coordinate plane with the x-axis (horizontal) and the y-axis (vertical) intersecting at the origin (0, 0). We know our line crosses the y-axis at -4, so let's put a point there. This is our anchor, our starting point. From this point, we use the slope to guide us. The slope is 3/4, which means for every 4 units we move to the right, we move 3 units up. So, starting from (0, -4), we move 4 units to the right and 3 units up, placing another point. We could repeat this process to find more points, but two points are enough to define a line. Now, imagine drawing a straight line through these two points – that's the line represented by the equation y = (3/4)x - 4.

This line slopes upwards from left to right, reflecting the positive slope. It's not a very steep line, as the slope is less than 1, but it's definitely ascending. The y-intercept of -4 places the line below the x-axis at that point. Mentally extending this line in both directions, we can start to anticipate which quadrants it will traverse. Remember, the quadrants are the four regions of the coordinate plane, numbered I through IV in a counter-clockwise direction, starting from the upper right.

Determining the Quadrants the Line Passes Through

Okay, let's figure out which quadrants our line will visit. Remember, the coordinate plane has four quadrants:

• Quadrant I: Top right (x and y are positive)
• Quadrant II: Top left (x is negative, y is positive)
• Quadrant III: Bottom left (x and y are negative)
• Quadrant IV: Bottom right (x is positive, y is negative)

Our line, y = (3/4)x - 4, has a y-intercept of -4. This means it definitely passes through the y-axis below the origin, which is in Quadrant IV. As the line slopes upwards from left to right, it will continue through Quadrant I (where both x and y are positive). To determine if it passes through Quadrant III and Quadrant II, we need to think about where the line crosses the x-axis. The x-axis represents the points where y = 0. So, let's find the x-intercept.

To find the x-intercept, we set y = 0 in our equation and solve for x:

0 = (3/4)x - 4
4 = (3/4)x
x = 4 * (4/3)
x = 16/3

The x-intercept is 16/3, which is approximately 5.33. This means the line crosses the x-axis at a positive value, so it passes through Quadrant I as expected. Since the line crosses the y-axis at -4 (Quadrant IV) and the x-axis at approximately 5.33 (Quadrant I), it must also pass through Quadrant IV. It will not pass through Quadrant II because to do so, it would have to reverse direction, which is impossible for a straight line. So, Priya's line will pass through Quadrants I, and IV.

Conclusion

So, there you have it! By understanding the slope and y-intercept of the equation y = (3/4)x - 4, we were able to describe the line Priya will draw. It's a line that slopes upwards from left to right, crosses the y-axis at -4, and passes through Quadrants I, and IV. Armed with this knowledge, Priya can confidently graph the line, knowing what to expect. Understanding the components of a linear equation like this is a powerful tool in mathematics, guys! It allows us to visualize and predict the behavior of lines, which is super useful in many real-world applications. Keep practicing, and you'll become a pro at graphing lines in no time!