Graphing Y−6=−4(x+2): A Step-by-Step Guide

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Graphing y−6=−4(x+2): A Step-by-Step Guide

Hey guys! Today, we're diving into graphing the equation y−6=−4(x+2) on a coordinate plane. This might seem tricky at first glance, but don't worry! We'll break it down into simple, manageable steps. By the end of this guide, you'll be graphing like a pro.

Understanding the Equation

Before we start graphing, let's understand what the equation y−6=−4(x+2) represents. This is a linear equation, meaning it will form a straight line when graphed. The equation is given in point-slope form, which is super handy for identifying key information. The point-slope form of a linear equation is y−y1=m(x−x1), where (x1,y1) is a point on the line and m is the slope.

In our equation, y−6=−4(x+2), we can identify that the slope m=−4 and a point on the line is (−2,6). This is because the equation can be read as "the slope of the line is -4 and it passes through the point (-2, 6)". Knowing this, we can easily plot the line on a coordinate plane.

Why is Point-Slope Form Useful?

The point-slope form is incredibly useful because it directly gives you a point on the line and the slope. This makes it easy to visualize and graph the line without having to convert it to slope-intercept form (y=mx+b) first, although we will do that later to double-check our work. Understanding point-slope form allows you to quickly interpret linear equations and sketch their graphs efficiently. For example, if you see an equation like y+3=2(x−1), you immediately know the line passes through (1,−3) and has a slope of 2. This form is particularly helpful when you're given a point and a slope and need to find the equation of the line.

Converting to Slope-Intercept Form

To further understand our equation, let's convert it to slope-intercept form (y=mx+b), where m is the slope and b is the y-intercept. Starting with y−6=−4(x+2), we can distribute the −4 on the right side:

y−6=−4x−8

Now, add 6 to both sides to isolate y:

y=−4x−8+6

y=−4x−2

Now we have the equation in slope-intercept form. From this, we can see that the slope m=−4 and the y-intercept is −2. This means the line crosses the y-axis at the point (0,−2). Knowing the slope and y-intercept gives us another way to graph the line.

Step-by-Step Graphing Guide

Now that we understand the equation, let's get to the fun part: graphing! Follow these steps to graph y−6=−4(x+2) on a coordinate plane.

Step 1: Identify a Point on the Line

From the point-slope form y−6=−4(x+2), we know that the line passes through the point (−2,6). Locate this point on the coordinate plane and mark it.

Step 2: Use the Slope to Find Another Point

The slope m=−4 tells us how the line rises or falls. A slope of −4 can be written as −4/1, which means for every 1 unit we move to the right on the x-axis, we move 4 units down on the y-axis. Starting from the point (−2,6), move 1 unit to the right and 4 units down. This brings us to the point (−1,2). Mark this point on the coordinate plane.

Step 3: Draw the Line

Now that we have two points, (−2,6) and (−1,2), we can draw a straight line through them. Use a ruler or straightedge to ensure the line is accurate. Extend the line beyond the two points to fill the coordinate plane.

Step 4: Verify with the y-intercept

From the slope-intercept form y=−4x−2, we know the y-intercept is −2. Check if the line you drew crosses the y-axis at (0,−2). If it does, you've graphed the line correctly. If not, double-check your points and slope.

Alternative Method: Using Slope-Intercept Form Directly

If you prefer working with the slope-intercept form, you can graph the line using the equation y=−4x−2.

Step 1: Plot the y-intercept

Start by plotting the y-intercept, which is (0,−2). Mark this point on the coordinate plane.

Step 2: Use the Slope to Find Another Point

Again, the slope is −4, or −4/1. From the y-intercept (0,−2), move 1 unit to the right and 4 units down. This brings us to the point (1,−6). Mark this point on the coordinate plane.

Step 3: Draw the Line

Draw a straight line through the points (0,−2) and (1,−6). Extend the line to fill the coordinate plane.

Common Mistakes to Avoid

Graphing lines can be straightforward, but here are some common mistakes to watch out for:

  • Incorrectly Identifying the Point: Make sure you correctly identify the point from the point-slope form. Remember that y−6=−4(x+2) gives you the point (−2,6), not (2,−6).
  • Misinterpreting the Slope: A negative slope means the line goes downwards as you move from left to right. Make sure you move down when the slope is negative.
  • Inaccurate Plotting: Always use a ruler or straightedge to draw the line. Inaccurate plotting can lead to a wrong graph.
  • Forgetting to Extend the Line: The line should extend beyond the points you've plotted to show that it continues infinitely in both directions.

Practice Problems

Want to test your skills? Try graphing these equations:

  1. y−2=3(x−1)
  2. y+1=−2(x+3)
  3. y=1/2x+4

Conclusion

Graphing the equation y−6=−4(x+2) is a breeze once you understand the point-slope and slope-intercept forms. By following the steps outlined in this guide, you can accurately graph any linear equation. Remember to double-check your work and avoid common mistakes. Keep practicing, and you'll become a graphing master in no time!

So there you have it, guys! Graphing linear equations is not as intimidating as it seems. With a little practice, you'll be able to tackle any linear equation that comes your way. Keep up the great work, and happy graphing!