Graph The Line With Slope $\frac{1}{2}$ Passing Through The Point $(1, -4$\].

Introduction

Graphing a line with a given slope and point is a fundamental concept in mathematics, particularly in algebra and geometry. In this article, we will discuss how to graph a line with a slope of $\frac{1}{2}$ passing through the point $(1, -4)$. We will use the point-slope form of a linear equation, which is a powerful tool for graphing lines.

What is the Point-Slope Form?

The point-slope form of a linear equation is given by:

$$y - y_1 = m(x - x_1)$$

where $(x_1, y_1)$ is a point on the line, and $m$ is the slope of the line. This form is useful because it allows us to easily graph a line given a point and the slope.

Graphing the Line

To graph the line with a slope of $\frac{1}{2}$ passing through the point $(1, -4)$, we can use the point-slope form. We will substitute the given point and slope into the equation:

$$y - (-4) = \frac{1}{2}(x - 1)$$

Simplifying the equation, we get:

$$y + 4 = \frac{1}{2}x - \frac{1}{2}$$

To graph the line, we can use the following steps:

1. Plot the given point: We will plot the point $(1, -4)$ on the coordinate plane.
2. Use the slope to find another point: We will use the slope of $\frac{1}{2}$ to find another point on the line. We can do this by moving up or down from the given point by a distance equal to the slope.
3. Draw the line: We will draw a line through the two points we have found.

Finding Another Point on the Line

To find another point on the line, we can use the slope of $\frac{1}{2}$. We will move up from the given point $(1, -4)$ by a distance equal to the slope. This means we will move up by $\frac{1}{2}$ unit.

To find the new x-coordinate, we can add the slope to the x-coordinate of the given point:

$$x_2 = x_1 + m$$

$$x_2 = 1 + \frac{1}{2}$$

$$x_2 = \frac{3}{2}$$

To find the new y-coordinate, we can add the slope to the y-coordinate of the given point:

$$y_2 = y_1 + m$$

$$y_2 = -4 + \frac{1}{2}$$

$$y_2 = -\frac{7}{2}$$

So, the new point is $(\frac{3}{2}, -\frac{7}{2})$.

Drawing the Line

Now that we have two points on the line, we can draw the line through them. We will draw a line through the points $(1, -4)$ and $(\frac{3}{2}, -\frac{7}{2})$.

Conclusion

Graphing a line with a given slope and point is a fundamental concept in mathematics. We have used the point-slope form of a linear equation to graph a line with a slope of $\frac{1}{2}$ passing through the point $(1, -4)$. We have found another point on the line using the slope and have drawn the line through the two points.

Example Problems

1. Graph the line with a slope of $\frac{3}{4}$ passing through the point $(2, 1)$.
2. Graph the line with a slope of $-\frac{2}{3}$ passing through the point $(3, -2)$.

Solutions

1. To graph the line with a slope of $\frac{3}{4}$ passing through the point $(2, 1)$, we can use the point-slope form:

$$y - 1 = \frac{3}{4}(x - 2)$$

Simplifying the equation, we get:

$$y - 1 = \frac{3}{4}x - \frac{3}{2}$$

To graph the line, we can use the following steps:

• Plot the given point $(2, 1)$ on the coordinate plane.
• Use the slope of $\frac{3}{4}$ to find another point on the line.
• Draw the line through the two points.

2. To graph the line with a slope of $-\frac{2}{3}$ passing through the point $(3, -2)$, we can use the point-slope form:

$$y - (-2) = -\frac{2}{3}(x - 3)$$

Simplifying the equation, we get:

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

To graph the line, we can use the following steps:

• Plot the given point $(3, -2)$ on the coordinate plane.
• Use the slope of $-\frac{2}{3}$ to find another point on the line.
• Draw the line through the two points.

Tips and Tricks

• When graphing a line with a given slope and point, make sure to use the point-slope form of a linear equation.
• When finding another point on the line, use the slope to move up or down from the given point by a distance equal to the slope.
• When drawing the line, make sure to draw a line through the two points you have found.

Conclusion

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Introduction

Graphing a line with a given slope and point is a fundamental concept in mathematics, particularly in algebra and geometry. In this article, we will provide a Q&A section to help you better understand how to graph a line with a given slope and point.

Q: What is the point-slope form of a linear equation?

A: The point-slope form of a linear equation is given by:

$$y - y_1 = m(x - x_1)$$

where $(x_1, y_1)$ is a point on the line, and $m$ is the slope of the line.

Q: How do I use the point-slope form to graph a line?

A: To graph a line using the point-slope form, follow these steps:

1. Plot the given point: Plot the point $(x_1, y_1)$ on the coordinate plane.
2. Use the slope to find another point: Use the slope $m$ to find another point on the line. You can do this by moving up or down from the given point by a distance equal to the slope.
3. Draw the line: Draw a line through the two points you have found.

Q: How do I find another point on the line using the slope?

A: To find another point on the line using the slope, follow these steps:

1. Add the slope to the x-coordinate: Add the slope $m$ to the x-coordinate of the given point to find the new x-coordinate.
2. Add the slope to the y-coordinate: Add the slope $m$ to the y-coordinate of the given point to find the new y-coordinate.

Q: What if I don't have a point on the line? Can I still graph the line?

A: Yes, you can still graph the line even if you don't have a point on the line. You can use the slope-intercept form of a linear equation, which is given by:

$$y = mx + b$$

where $m$ is the slope of the line, and $b$ is the y-intercept.

Q: How do I find the y-intercept of a line?

A: To find the y-intercept of a line, follow these steps:

1. Set the x-coordinate to 0: Set the x-coordinate to 0 in the equation $y = mx + b$.
2. Solve for y: Solve for y to find the y-intercept.

Q: What if I have a negative slope? How do I graph the line?

A: If you have a negative slope, you can still graph the line. To do this, follow these steps:

1. Plot the given point: Plot the point $(x_1, y_1)$ on the coordinate plane.
2. Use the slope to find another point: Use the negative slope to find another point on the line. You can do this by moving down from the given point by a distance equal to the slope.
3. Draw the line: Draw a line through the two points you have found.

Q: Can I graph a line with a slope of 0?

A: Yes, you can graph a line with a slope of 0. To do this, follow these steps:

1. Plot the given point: Plot the point $(x_1, y_1)$ on the coordinate plane.
2. Draw a horizontal line: Draw a horizontal line through the given point.

Conclusion

Graphing a line with a given slope and point is a fundamental concept in mathematics. We have provided a Q&A section to help you better understand how to graph a line with a given slope and point. We have covered topics such as the point-slope form, finding another point on the line, and graphing lines with negative slopes and slopes of 0.