Graph The Line.${ Y = -4x + 4 }$

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Understanding the Equation of a Line

In mathematics, the equation of a line is a fundamental concept that is used to describe the relationship between two variables, typically x and y. The equation of a line can be written in various forms, including the slope-intercept form, which is the most commonly used form. The slope-intercept form of a line is given by the equation y = mx + b, where m is the slope of the line and b is the y-intercept.

The Given Equation

In this article, we will be graphing the line given by the equation y = -4x + 4. This equation is in the slope-intercept form, where the slope (m) is -4 and the y-intercept (b) is 4.

What is the Slope?

The slope of a line is a measure of how steep the line is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. In the given equation, the slope is -4, which means that for every unit increase in x, the value of y decreases by 4 units.

What is the Y-Intercept?

The y-intercept of a line is the point where the line intersects the y-axis. In the given equation, the y-intercept is 4, which means that the line intersects the y-axis at the point (0, 4).

Graphing the Line

To graph the line, we need to plot two points on the coordinate plane. We can use the slope and y-intercept to find these points.

Finding the First Point

To find the first point, we can use the y-intercept, which is (0, 4). This point is already on the y-axis, so we don't need to do anything else.

Finding the Second Point

To find the second point, we can use the slope and the fact that the line passes through the point (0, 4). We can use the slope formula to find the x-coordinate of the second point:

y - y1 = m(x - x1)

where (x1, y1) is the first point (0, 4) and m is the slope (-4).

Plugging in the values, we get:

y - 4 = -4(x - 0)

Simplifying the equation, we get:

y - 4 = -4x

Adding 4 to both sides, we get:

y = -4x + 4

This is the same equation we started with, so we know that the line passes through the point (1, 0).

Plotting the Points

Now that we have the two points, we can plot them on the coordinate plane. The first point is (0, 4) and the second point is (1, 0).

Drawing the Line

To draw the line, we can connect the two points with a straight line. The line will pass through the points (0, 4) and (1, 0).

The Graph

Here is the graph of the line y = -4x + 4:

+---------------+
|              |
|  (0, 4)     |
|              |
+---------------+
|              |
|  (1, 0)     |
|              |
+---------------+

Key Takeaways

  • The equation of a line can be written in various forms, including the slope-intercept form.
  • The slope-intercept form of a line is given by the equation y = mx + b, where m is the slope and b is the y-intercept.
  • The slope of a line is a measure of how steep the line is.
  • The y-intercept of a line is the point where the line intersects the y-axis.
  • To graph a line, we need to plot two points on the coordinate plane and connect them with a straight line.

Conclusion

Q&A: Graphing the Line

Q: What is the slope of the line y = -4x + 4?

A: The slope of the line y = -4x + 4 is -4. This means that for every unit increase in x, the value of y decreases by 4 units.

Q: What is the y-intercept of the line y = -4x + 4?

A: The y-intercept of the line y = -4x + 4 is 4. This means that the line intersects the y-axis at the point (0, 4).

Q: How do I graph the line y = -4x + 4?

A: To graph the line y = -4x + 4, you need to plot two points on the coordinate plane and connect them with a straight line. You can use the slope and y-intercept to find these points.

Q: What are the two points that I need to plot on the coordinate plane?

A: The two points that you need to plot on the coordinate plane are (0, 4) and (1, 0). You can find these points by using the slope and y-intercept of the line.

Q: How do I find the x-coordinate of the second point?

A: To find the x-coordinate of the second point, you can use the slope formula:

y - y1 = m(x - x1)

where (x1, y1) is the first point (0, 4) and m is the slope (-4).

Q: What is the equation of the line in the slope-intercept form?

A: The equation of the line in the slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept.

Q: What is the significance of the slope-intercept form of a line?

A: The slope-intercept form of a line is significant because it allows us to easily identify the slope and y-intercept of the line. This makes it easier to graph the line and understand its behavior.

Q: Can I graph a line using other forms of equations?

A: Yes, you can graph a line using other forms of equations, such as the standard form (Ax + By = C) or the point-slope form (y - y1 = m(x - x1)). However, the slope-intercept form is the most commonly used form and is often the easiest to work with.

Q: How do I determine the slope and y-intercept of a line?

A: To determine the slope and y-intercept of a line, you need to examine the equation of the line. The slope is the coefficient of the x-term, and the y-intercept is the constant term.

Q: What are some common mistakes to avoid when graphing a line?

A: Some common mistakes to avoid when graphing a line include:

  • Not plotting the correct points on the coordinate plane
  • Not using the correct slope and y-intercept
  • Not drawing the line correctly
  • Not labeling the axes correctly

Conclusion

Graphing a line is a fundamental concept in mathematics that is used to describe the relationship between two variables. In this article, we provided a comprehensive guide to graphing the line y = -4x + 4, including the slope and y-intercept, and answered some common questions about graphing lines.