Graphing Y = 3 - 2x: Find X Values And Intersections

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Hey guys! Today, we're diving into the world of linear functions and graphs. We're going to take a close look at the function y = 3 - 2x, learn how to plot its graph, and then use that graph to figure out some cool stuff. Specifically, we'll be finding the values of x where the graph sits above and below the x-axis, as well as above and below the lines y = 2 and y = 4. This is a super important skill in algebra, so let's get started!

Understanding the Function y = 3 - 2x

First off, let's break down what the function y = 3 - 2x actually means. This is a linear function, which basically means that when we graph it, we're going to get a straight line. The equation is in the slope-intercept form, which is generally written as y = mx + b. In our case, m (the slope) is -2, and b (the y-intercept) is 3. The slope tells us how steep the line is and in which direction it's going (since it's negative, it's going downwards as we move from left to right). The y-intercept is where the line crosses the y-axis, which in this case is at the point (0, 3).

To really grasp this, think about what the equation is telling us. For every increase of 1 in x, y decreases by 2 (that's the slope at work). The '+ 3' part simply shifts the entire line upwards by 3 units on the y-axis. Understanding the components of this equation is key to visualizing the graph and predicting its behavior. We can choose some values of x, plug them into the equation, and calculate the corresponding y values. For instance, if x is 0, y is 3; if x is 1, y is 1; and so on. These pairs of (x, y) values give us points that we can plot on a graph. When we connect these points, we'll get our line! This process of selecting x values and calculating y is fundamental to graphing any function, not just linear ones. It allows us to see how the function behaves and to identify key features, like intercepts and slopes. Furthermore, by understanding the slope and y-intercept, we can quickly sketch a rough graph of the line without plotting a ton of points. This is a useful skill for quickly visualizing the function and making estimations. So, let's move on to the next step, where we'll actually plot the graph and see this all in action!

Plotting the Graph of y = 3 - 2x

Okay, let's get down to business and plot the graph of our function, y = 3 - 2x. To do this, we need a few points. Remember, since it's a straight line, we technically only need two points, but it's always good to plot a third one just to make sure we haven't made any calculation errors. Let's choose some easy x values to work with. How about x = -1, x = 0, and x = 1?

  • When x = -1, we plug it into the equation: y = 3 - 2(-1) = 3 + 2 = 5. So, our first point is (-1, 5).
  • When x = 0, y = 3 - 2(0) = 3. Our second point is (0, 3). This is also our y-intercept, which we already knew from looking at the equation!
  • When x = 1, y = 3 - 2(1) = 3 - 2 = 1. Our third point is (1, 1).

Now, grab some graph paper (or use a graphing calculator or online tool) and plot these points. You should have three points: (-1, 5), (0, 3), and (1, 1). Once you've plotted them, take a ruler or straightedge and draw a line through them. You should get a straight line that slopes downwards from left to right. This line is the graph of our function, y = 3 - 2x. The visual representation of this function as a line makes it much easier to understand its behavior. We can now see how the y values change as x changes, and we can easily identify key features such as the y-intercept (where the line crosses the y-axis) and the x-intercept (where the line crosses the x-axis). Furthermore, the graph allows us to quickly estimate the y value for any given x value, and vice versa. This visual connection between the equation and its graph is a powerful tool in algebra and beyond. By understanding how to plot graphs, we can gain insights into a wide range of mathematical relationships. So, now that we have our graph plotted, let's use it to answer some specific questions about our function!

Finding x Values Above and Below the X-Axis

Alright, now that we've got the graph of y = 3 - 2x plotted, let's use it to answer some questions. First up, we want to find the values of x for which the graph lies above the x-axis and the values of x for which it lies below the x-axis. Remember, the x-axis is simply the line y = 0. So, we're essentially looking for where y > 0 and where y < 0.

  • Above the x-axis (y > 0): Look at your graph. The line is above the x-axis to the left of the point where it crosses the x-axis. To find the exact x value where it crosses, we need to find the x-intercept. This is where y = 0. So, let's set 3 - 2x = 0 and solve for x:
    • 3 - 2x = 0
    • -2x = -3
    • x = 3/2 = 1.5 So, the line crosses the x-axis at x = 1.5. Therefore, the graph is above the x-axis when x < 1.5.
  • Below the x-axis (y < 0): The line is below the x-axis to the right of the point where it crosses. So, the graph is below the x-axis when x > 1.5. These observations about where the graph lies relative to the x-axis are crucial for understanding the sign of the function's output. The x-intercept acts as a critical point, dividing the x-axis into regions where the function is either positive (above the x-axis) or negative (below the x-axis). This concept extends beyond linear functions and is fundamental in analyzing the behavior of various types of functions, including quadratic, exponential, and trigonometric functions. By determining the x-intercepts and observing the graph's behavior around these points, we can gain valuable insights into the function's properties, such as its intervals of increase and decrease, its maximum and minimum values, and its overall shape. This understanding is essential for solving equations, inequalities, and optimization problems in mathematics and real-world applications. Now, let's move on to comparing our function to other lines, specifically y = 2 and y = 4.

Finding x Values Above and Below the Lines y = 2 and y = 4

Now let's kick things up a notch! We've figured out when our graph, y = 3 - 2x, is above and below the x-axis. Let's see when it's above and below the horizontal lines y = 2 and y = 4. This is super useful because it helps us understand how our function compares to other constant values.

  • Above the line y = 2: We need to find the x values where 3 - 2x > 2. Let's solve this inequality:
    • 3 - 2x > 2
    • -2x > -1
    • x < 1/2 (Remember to flip the inequality sign when dividing by a negative number!) So, the graph is above the line y = 2 when x < 1/2. Look at your graph – does this make sense? It should!
  • Below the line y = 2: This is the opposite of above, so we're looking for x values where 3 - 2x < 2. We already did most of the work solving the inequality above. The solution is just the flip side: x > 1/2. The graph is below the line y = 2 when x > 1/2.
  • Above the line y = 4: Now let's tackle y = 4. We need to solve the inequality 3 - 2x > 4:
    • 3 - 2x > 4
    • -2x > 1
    • x < -1/2 So, the graph is above the line y = 4 when x < -1/2.
  • Below the line y = 4: And finally, for below the line y = 4, we want 3 - 2x < 4. Again, we've almost already done the work. The solution is x > -1/2. The graph is below the line y = 4 when x > -1/2. Comparing the graph of a function to horizontal lines like y = 2 and y = 4 helps us visualize the function's range, which is the set of all possible output values (y-values). By finding the x-values where the graph is above or below these lines, we're essentially determining the intervals where the function's output is greater or less than these specific values. This is a fundamental concept in understanding the function's overall behavior and its limitations. For instance, if we find that the graph is always below a certain line, it tells us that the function's output never exceeds the y-value of that line. This kind of analysis is crucial in various applications, such as optimization problems where we want to find the maximum or minimum value of a function within a given range.

Verifying Results Algebraically

Okay, we've done a bunch of work graphically, which is awesome for visualizing what's going on. But to be super sure of our answers, let's verify them algebraically. This means we'll use equations and inequalities to double-check our findings. This step is crucial because it reinforces our understanding and ensures the accuracy of our results. Graphical solutions can sometimes be approximate due to the limitations of visual interpretation, but algebraic solutions provide precise answers. By combining both approaches, we gain a more comprehensive understanding of the problem and build confidence in our solutions. Let's revisit our previous findings and verify them algebraically:

  • Graph above the x-axis (y > 0): We found that x < 1.5. Let's check this. If x = 1, then y = 3 - 2(1) = 1, which is greater than 0. If x = 2, then y = 3 - 2(2) = -1, which is less than 0. So, it checks out!
  • Graph below the x-axis (y < 0): We found that x > 1.5. We already tested x = 2 above, and it worked. Let's try x = 1: y = 3 - 2(1) = 1, which is not less than 0. Awesome, it checks out here too!
  • Graph above the line y = 2: We found that x < 1/2. Let's try x = 0: y = 3 - 2(0) = 3, which is greater than 2. If x = 1, y = 3 - 2(1) = 1, which is not greater than 2. Check!
  • Graph below the line y = 2: We found that x > 1/2. Let's try x = 1: y = 3 - 2(1) = 1, which is less than 2. If x = 0, y = 3 - 2(0) = 3, which is not less than 2. Perfect!
  • Graph above the line y = 4: We found that x < -1/2. Let's try x = -1: y = 3 - 2(-1) = 5, which is greater than 4. If x = 0, y = 3 - 2(0) = 3, which is not greater than 4. Excellent!
  • Graph below the line y = 4: We found that x > -1/2. Let's try x = 0: y = 3 - 2(0) = 3, which is less than 4. If x = -1, y = 3 - 2(-1) = 5, which is not less than 4. Nailed it! By substituting specific values of x into the equation and comparing the resulting y-values with the target lines, we can confirm the accuracy of our graphical solutions. This process not only reinforces our understanding of the relationship between the graph and the equation but also provides a practical method for verifying solutions in more complex problems. Furthermore, this algebraic verification step helps us develop a more critical approach to problem-solving, where we rely not only on visual intuition but also on rigorous mathematical reasoning.

Conclusion

Woohoo! You guys have just tackled a really important concept in algebra: graphing linear functions and using those graphs to solve problems. We took the function y = 3 - 2x, plotted its graph, and then figured out where it sits above and below the x-axis, as well as the lines y = 2 and y = 4. We even checked our answers algebraically to make sure we were spot on. Remember, graphing functions is a super powerful tool for visualizing mathematical relationships, and it's a skill that will come in handy again and again in your math journey. Keep practicing, and you'll become a graph-plotting pro in no time! Keep exploring different functions, experimenting with their graphs, and seeing how they behave. The more you practice, the more intuitive graphing will become, and the better you'll be at using it to solve a wide range of mathematical problems. And hey, if you ever get stuck, don't hesitate to ask for help. There are tons of resources available, from textbooks and online tutorials to teachers and classmates. The key is to keep learning, keep practicing, and keep having fun with math! You've got this!