Graphing Y=-(3/2)x+3: A Step-by-Step Guide

Hey guys! Today, we're diving into the fascinating world of graphing lines. Specifically, we'll be tackling the equation $y=-\frac{3}{2}x+3$. Graphing lines is a fundamental skill in mathematics, and it's super useful in many real-world situations, from calculating distances to predicting trends. In this article, we'll break down the process step-by-step, making it easy to understand and apply. So, grab your graph paper (or your favorite digital graphing tool), and let's get started!

Understanding Linear Equations

Before we jump into graphing, let's quickly recap what a linear equation actually is. Linear equations are equations that, when graphed, form a straight line. The general form of a linear equation is $y = mx + b$, where:

• $y$ is the dependent variable (usually plotted on the vertical axis).
• $x$ is the independent variable (usually plotted on the horizontal axis).
• $m$ is the slope of the line, indicating its steepness and direction.
• $b$ is the y-intercept, the point where the line crosses the y-axis.

In our equation, $y = -\frac{3}{2}x + 3$, we can easily identify the slope ($m$) as $-\frac{3}{2}$ and the y-intercept ($b$) as 3. These two values are our key ingredients for graphing this line. Understanding the role of slope and y-intercept is crucial. The slope tells us how much the line rises or falls for every unit we move to the right. A negative slope, like ours, indicates that the line slopes downwards from left to right. The y-intercept, on the other hand, gives us a fixed point on the line – the point where the line intersects the y-axis. This point serves as our starting point for drawing the entire line. By knowing these two parameters, we can accurately plot any linear equation.

Furthermore, let's delve deeper into why linear equations are so important. They model a wide range of real-world phenomena, from simple scenarios like the cost of buying a certain number of items to more complex situations like the trajectory of a projectile. Understanding how to graph and interpret linear equations is therefore a valuable skill in various fields, including physics, economics, and engineering. For instance, businesses use linear equations to model their revenue and expenses, while scientists use them to describe relationships between variables in experiments. In essence, mastering linear equations opens up a world of analytical possibilities and practical applications.

Method 1: Using Slope-Intercept Form

Our first method utilizes the slope-intercept form directly. Remember, the slope-intercept form is $y = mx + b$. We already know that the slope ($m$) is $-\frac{3}{2}$ and the y-intercept ($b$) is 3.

Step 1: Plot the Y-Intercept

Start by plotting the y-intercept, which is the point (0, 3) on the graph. This is where the line crosses the y-axis. Think of it as your starting point for drawing the line. It's a fixed point that helps anchor the line in the coordinate plane. Make a clear dot at (0, 3) on your graph. This will be our reference point for the next step.

Step 2: Use the Slope to Find Another Point

The slope, $-\frac{3}{2}$, tells us how to move from the y-intercept to another point on the line. The slope can be interpreted as "rise over run." In this case, a slope of $-\frac{3}{2}$ means we go down 3 units (the rise) and right 2 units (the run). Starting from the y-intercept (0, 3), move down 3 units and right 2 units. This brings us to the point (2, 0). Plot this point as well. Understanding how to interpret the slope as rise over run is crucial. A positive rise means moving upwards, while a negative rise means moving downwards. Similarly, a positive run means moving to the right, while a negative run means moving to the left. By applying this concept, we can easily find additional points on the line and ensure our graph is accurate.

Step 3: Draw the Line

Now that you have two points, (0, 3) and (2, 0), simply draw a straight line through them. Extend the line in both directions to cover the entire graph. This line represents all the possible solutions to the equation $y = -\frac{3}{2}x + 3$. Make sure your line is straight and passes precisely through the two points you've plotted. Use a ruler or a straight edge to ensure accuracy. The line should extend beyond the plotted points to indicate that the solutions continue infinitely in both directions.

Method 2: Using Two Points

Another way to graph a line is by finding any two points that satisfy the equation. This method is particularly useful if you prefer plugging in values for $x$ and solving for $y$.

Step 1: Choose Two Values for X

Pick any two values for $x$. Let's choose $x = 0$ and $x = 2$ for simplicity. These values are easy to work with and will give us two distinct points on the line. The key is to choose values that are relatively small and easy to substitute into the equation. This will minimize the chances of making calculation errors and make the process smoother. You could choose any two values, but selecting smaller numbers often makes the arithmetic simpler.

Step 2: Calculate the Corresponding Y Values

Substitute each $x$ value into the equation $y = -\frac{3}{2}x + 3$ to find the corresponding $y$ values.

• For $x = 0$: $y = -\frac{3}{2}(0) + 3 = 3$. So, the first point is (0, 3).
• For $x = 2$: $y = -\frac{3}{2}(2) + 3 = -3 + 3 = 0$. So, the second point is (2, 0).

These calculations give us the coordinates of two points that lie on the line. Substituting the $x$ values into the equation and solving for $y$ is a direct way to find points on the line. Remember to perform the multiplication before the addition to follow the order of operations correctly. The resulting $y$ values, paired with their corresponding $x$ values, give us the coordinates we need for plotting.

Step 3: Plot the Points and Draw the Line

Plot the points (0, 3) and (2, 0) on the graph. Draw a straight line through these points, extending it in both directions. This line is the graph of the equation $y = -\frac{3}{2}x + 3$. Just like in the previous method, ensure your line is straight and accurate. Use a ruler to connect the plotted points, and extend the line beyond the points to show that the line continues indefinitely. This visual representation accurately depicts the relationship described by the linear equation.

Method 3: Using X and Y Intercepts

Our third method involves finding the x and y-intercepts. These intercepts are the points where the line crosses the x-axis and the y-axis, respectively. They provide two key points that define the line.

Step 1: Find the Y-Intercept

To find the y-intercept, set $x = 0$ in the equation and solve for $y$. This is the same as what we did in Method 2, and we already know the y-intercept is (0, 3).

Setting $x$ to zero essentially isolates the constant term in the equation, which directly gives us the y-coordinate of the y-intercept. The y-intercept is a crucial point because it's where the line intersects the vertical axis. It provides a fixed reference point for drawing the line. Understanding that setting $x$ to zero helps find the y-intercept is a fundamental concept in graphing linear equations.

Step 2: Find the X-Intercept

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

$0 = -\frac{3}{2}x + 3$

Subtract 3 from both sides:

$-3 = -\frac{3}{2}x$

Multiply both sides by $-\frac{2}{3}$:

$x = -3 * -\frac{2}{3} = 2$

So, the x-intercept is (2, 0).

Setting $y$ to zero allows us to isolate the $x$ term and solve for the value of $x$ where the line crosses the x-axis. This point, the x-intercept, is another key reference point for graphing the line. The process involves a few algebraic steps, such as subtracting the constant term and then multiplying by the reciprocal of the coefficient of $x$. Understanding these steps is essential for accurately finding the x-intercept.

Step 3: Plot the Intercepts and Draw the Line

Plot the points (0, 3) and (2, 0) on the graph. Draw a straight line through these points, extending it in both directions. This line is the graph of the equation $y = -\frac{3}{2}x + 3$. This step is similar to the previous methods, where we use the plotted points to draw a straight line. The line should accurately pass through both the x and y-intercepts, visually representing the equation. Using a ruler ensures that the line is straight and extends beyond the plotted points, indicating the infinite nature of the line.

Tips for Accurate Graphing

To make sure your graphs are accurate and easy to read, here are a few helpful tips:

• Use graph paper: Graph paper provides a grid that helps you plot points accurately and draw straight lines. The grid lines serve as a visual aid, making it easier to maintain consistent spacing and alignment.
• Use a ruler or straightedge: This will ensure that your lines are straight and precise. Freehand lines can often be wobbly, which can affect the accuracy of your graph.
• Label your axes: Always label the x and y-axes to indicate what they represent. This helps others understand your graph and avoids confusion. You might also include units of measurement if applicable.
• Label key points: Label the intercepts and any other key points you've plotted. This makes it easier to read and interpret the graph. Labeling helps highlight important features of the line and makes the graph more informative.
• Double-check your work: Before finalizing your graph, double-check your calculations and plotted points to ensure accuracy. A small mistake in the calculations can lead to a significant error in the graph.

Conclusion

Graphing the line $y = -\frac{3}{2}x + 3$ can be done in several ways, each with its own advantages. Whether you prefer using the slope-intercept form, finding two points, or utilizing the intercepts, the key is to understand the relationship between the equation and its visual representation. With practice, you'll become a pro at graphing lines! So, go ahead and try graphing other linear equations. Experiment with different slopes and intercepts, and see how they affect the position and direction of the line. The more you practice, the more confident you'll become in your graphing skills. Remember, graphing lines is not just a mathematical exercise; it's a powerful tool for visualizing and understanding relationships between variables. Keep exploring, and you'll discover even more applications of this fundamental skill!