Graph the line with slope 2 passing through the point (-4, -1).
Understand the Problem
The question asks to graph a line given its slope and a point it passes through. We can start by plotting the given point (-4, -1) on the coordinate plane. Then, using the slope of 2, we can find other points on the line by moving 1 unit to the right and 2 units up, and so on. Finally, we can draw a line through these points.
Answer
The line with slope $2$ passing through the point $(-4, -1)$ is $y = 2x + 7$ ``` graph Copy code ---------------------------------------------------------------------------------------------- | | | * | 10 | | | | 9 | | | | 8 | | | | 7 | | | | 6 | | | | 5 | | | | 4 | | | * | 3 | / | | / | 2 | / | | / | 1 | / | |--------------/---------------------------------------------------------------------------------| 0 | / | | / | -1 | / | | / | -2 | / | | / | -3 | / | | / | -4 | / | | / | -5 | / | | / | -6 | / | |*----------------------------------------------------------------------------------------------| -7 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 ```
Answer for screen readers
The line which has a slope of 2 and passes through point (-4, -1) is graphed below:
graph
Copy code
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| * | 10
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| | 9
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| | 8
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| | 7
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| | 6
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| | 5
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| | 4
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| * | 3
| / |
| / | 2
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| / | 1
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|--------------/---------------------------------------------------------------------------------| 0
| / |
| / | -1
| / |
| / | -2
| / |
| / | -3
| / |
| / | -4
| / |
| / | -5
| / |
| / | -6
| / |
|*----------------------------------------------------------------------------------------------| -7
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
Steps to Solve
- Plot the given point
The line passes through the point $(-4, -1)$. Plot this point on the graph.
- Use the slope to find another point
The slope is 2, which can be written as $\frac{2}{1}$. This means for every 1 unit we move to the right (increase in $x$), we move 2 units up (increase in $y$). Starting from the point $(-4, -1)$, move 1 unit to the right to $x = -3$ and 2 units up to $y = 1$. This gives us the point $(-3, 1)$.
- Draw the line
Draw a line that passes through the points $(-4, -1)$ and $(-3, 1)$.
The line which has a slope of 2 and passes through point (-4, -1) is graphed below:
graph
Copy code
----------------------------------------------------------------------------------------------
| |
| * | 10
| |
| | 9
| |
| | 8
| |
| | 7
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| | 6
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| | 5
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| | 4
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| * | 3
| / |
| / | 2
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| / | 1
| / |
|--------------/---------------------------------------------------------------------------------| 0
| / |
| / | -1
| / |
| / | -2
| / |
| / | -3
| / |
| / | -4
| / |
| / | -5
| / |
| / | -6
| / |
|*----------------------------------------------------------------------------------------------| -7
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
More Information
To check if the line is correct, we can use the point-slope form of a linear equation, which is: $y - y_1 = m(x - x_1)$, where $m$ is the slope and $(x_1, y_1)$ is a point on the line. In our case, $m = 2$ and $(x_1, y_1) = (-4, -1)$. So, the equation of the line is: $y - (-1) = 2(x - (-4))$ $y + 1 = 2(x + 4)$ $y + 1 = 2x + 8$ $y = 2x + 7$
We can verify that the point $(-3, 1)$ lies on this line: $1 = 2(-3) + 7$ $1 = -6 + 7$ $1 = 1$ The equation is correct.
Tips
A common mistake is to miscalculate the new point using the slope. For example, adding the slope incorrectly or moving in the wrong direction. Another common mistake is to plot the initial point incorrectly. It's important to carefully count the units when plotting points and applying the slope.
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