Graph Y=70x: A Step-by-Step Guide

Hey guys! Today, let's dive into graphing the equation y = 70x. This is a fundamental concept in algebra, and understanding how to visualize such equations is super important for grasping more complex mathematical ideas. We'll break it down step by step, ensuring you get a solid handle on it. So, let’s get started!

Understanding the Equation y = 70x

At its heart, y = 70x is a linear equation. Linear equations always graph as straight lines, which makes them relatively straightforward to understand and plot. The general form of a linear equation is y = mx + b, where 'm' represents the slope of the line and 'b' is the y-intercept (the point where the line crosses the y-axis). In our case, y = 70x, we can see that 'm' is 70 and 'b' is 0. This tells us a couple of things right off the bat. First, the slope of the line is quite steep since 70 is a large number. Second, because 'b' is 0, the line passes through the origin (0,0) on the coordinate plane. Understanding these basic components is crucial before we start plotting points and drawing the graph. The slope, which is 70 in this scenario, indicates how much 'y' changes for every unit change in 'x.' A slope of 70 means that for every increase of 1 in 'x,' 'y' increases by 70. This steep slope will be quite evident when we draw the graph. To truly understand this equation, imagine different values for 'x' and how they affect 'y.' For instance, if x = 1, then y = 70 * 1 = 70. If x = 2, then y = 70 * 2 = 140. As 'x' increases, 'y' increases rapidly, which is why the line is so steep. Grasping this relationship between 'x' and 'y' is fundamental to understanding linear equations and their graphs. By recognizing that this equation is a straight line that passes through the origin with a steep positive slope, we can predict its general appearance on a graph even before plotting specific points. This understanding forms the foundation for accurately and confidently graphing the equation y = 70x.

Setting Up the Coordinate Plane

Before we start plotting, we need to set up our coordinate plane. A coordinate plane, also known as the Cartesian plane, consists of two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical). The point where these two axes intersect is called the origin, and it's represented by the coordinates (0,0). To graph y = 70x effectively, we need to choose appropriate scales for both axes. Since the slope is quite steep, the y-values will increase rapidly as 'x' increases. Therefore, we need to ensure our y-axis can accommodate these larger values. A good approach is to start by selecting a few x-values and calculating the corresponding y-values. For example, if we choose x = 1, y = 70; if x = 2, y = 140; and if x = 3, y = 210. This gives us an idea of the range of y-values we need to plot. Based on these calculations, we can decide on an appropriate scale for the y-axis. We might choose to have each increment on the y-axis represent 10, 20, or even 50 units, depending on how large we want our graph to be. For the x-axis, we can use a smaller scale since the x-values are relatively small. Each increment on the x-axis could represent 1 unit. Once we have determined the scales for both axes, we need to label them clearly. This includes marking the increments along each axis and indicating the units. For example, we might label the x-axis with numbers like 1, 2, 3, and the y-axis with numbers like 50, 100, 150. Setting up the coordinate plane correctly is essential for creating an accurate and readable graph. Without proper scaling and labeling, the graph may be misleading or difficult to interpret. By carefully planning the setup of the coordinate plane, we ensure that our graph effectively represents the relationship between 'x' and 'y' as defined by the equation y = 70x.

Calculating Points for the Graph

To accurately draw the graph of y = 70x, we need to calculate a few points that lie on the line. Since it's a linear equation, only two points are technically needed to define the line, but plotting a third point is a good way to check for accuracy. Let's choose a few easy values for 'x' and calculate the corresponding 'y' values. We'll start with x = 0. When x = 0, y = 70 * 0 = 0. So, our first point is (0,0), which we already knew because the line passes through the origin. Next, let's choose x = 1. When x = 1, y = 70 * 1 = 70. This gives us the point (1,70). Now, let's choose x = 2. When x = 2, y = 70 * 2 = 140. This gives us the point (2,140). So, we have three points: (0,0), (1,70), and (2,140). These points should be sufficient to draw an accurate graph of the line. If we wanted to, we could choose a negative value for 'x' to see what the graph looks like on the other side of the y-axis. For example, if x = -1, y = 70 * -1 = -70. This gives us the point (-1,-70). However, for most purposes, the positive values are enough to understand the behavior of the line. Calculating these points is a straightforward process of substituting different values of 'x' into the equation and solving for 'y.' By choosing simple values for 'x,' we can easily calculate the corresponding 'y' values and plot these points on the coordinate plane. This step is crucial for visualizing the relationship between 'x' and 'y' and accurately drawing the graph of the equation y = 70x.

Plotting the Points

Now that we have our points—(0,0), (1,70), and (2,140)—it's time to plot them on the coordinate plane. Plotting points involves locating each point on the grid based on its x and y coordinates. For the point (0,0), we start at the origin, which is the intersection of the x and y axes. Since both the x and y coordinates are zero, the point is located right at the origin. Next, let's plot the point (1,70). To do this, we start at the origin and move 1 unit to the right along the x-axis. Then, we move 70 units up along the y-axis. Mark this location with a dot or a small cross. Finally, let's plot the point (2,140). We start at the origin again and move 2 units to the right along the x-axis. Then, we move 140 units up along the y-axis. Mark this location as well. As you plot these points, you'll notice that they form a straight line. This is expected since we are graphing a linear equation. If the points do not appear to be in a straight line, it could indicate a mistake in our calculations or plotting. Double-checking our work is always a good idea to ensure accuracy. Plotting the points accurately is a critical step in creating a correct graph. It allows us to visualize the relationship between 'x' and 'y' and provides the foundation for drawing the line that represents the equation y = 70x. By carefully locating each point on the coordinate plane, we ensure that our graph is a faithful representation of the equation.

Drawing the Line

With our points plotted, the next step is to draw the line that connects them. Since y = 70x is a linear equation, we know that the graph will be a straight line. To draw the line, simply take a ruler or straightedge and align it with the points we've plotted. Make sure the ruler extends beyond the points to create a continuous line. Draw the line carefully, ensuring it passes through all the points. If the line doesn't pass through all the points, double-check your calculations and plotting to see if there were any errors. Once the line is drawn, extend it in both directions to indicate that it continues infinitely. Add arrows at both ends of the line to show that it goes on forever. This is an important convention in graphing linear equations. The line represents all possible solutions to the equation y = 70x. Any point on the line corresponds to a pair of x and y values that satisfy the equation. Drawing the line accurately is the final step in creating the graph. It provides a visual representation of the relationship between 'x' and 'y' and allows us to quickly see how 'y' changes as 'x' changes. By carefully drawing the line and extending it in both directions, we create a complete and accurate graph of the equation y = 70x.

Final Touches and Observations

After drawing the line, there are a few final touches we can add to make the graph more complete and informative. First, label the line with its equation, y = 70x. This helps to clearly identify the graph and its corresponding equation. Place the label near the line, but not directly on top of it, so it doesn't obscure the graph. Next, take a moment to observe the graph and make some observations about its characteristics. Notice that the line passes through the origin (0,0), which we already knew from the equation. Also, observe that the line has a steep positive slope. This means that as 'x' increases, 'y' increases rapidly. The slope of the line is 70, which indicates that for every increase of 1 in 'x,' 'y' increases by 70. This steep slope is visually evident in the graph. Another observation we can make is that the line extends infinitely in both directions. This means that there are infinitely many solutions to the equation y = 70x. Any point on the line represents a solution to the equation. Finally, we can use the graph to estimate the value of 'y' for any given value of 'x,' or vice versa. For example, if we want to find the value of 'y' when x = 1.5, we can locate x = 1.5 on the x-axis, move vertically up to the line, and then move horizontally to the y-axis to read the corresponding value of 'y.' Adding these final touches and making observations about the graph helps to solidify our understanding of the equation y = 70x and its visual representation. It also allows us to use the graph as a tool for solving problems and making predictions.

Conclusion

So, there you have it! Graphing the equation y = 70x involves understanding the equation, setting up the coordinate plane, calculating points, plotting those points, and drawing the line. It's a fundamental skill in algebra and provides a visual way to understand the relationship between variables. Keep practicing, and you'll become a pro in no time!