Finding Point Q: Slope Of PQ With P(-1, 2)

by Alex Johnson 43 views

Let's dive into the fascinating world of coordinate geometry! In this article, we're tackling a classic problem: finding the possible locations of a point Q, given the coordinates of another point P and the desired slope of the line connecting them. We'll break down the concept of slope, explore the formula, and then work through several examples to solidify your understanding. So, grab your graph paper (or your favorite digital drawing tool) and let's get started!

Understanding the Slope

At its heart, the slope of a line is a measure of its steepness and direction. Think of it as the 'rise over run' – how much the line goes up (or down) for every unit it moves to the right. A positive slope indicates an upward trend, a negative slope signifies a downward trend, a zero slope represents a horizontal line, and an undefined slope corresponds to a vertical line. Understanding the slope is important for coordinate geometry problems, allowing us to calculate the equation of the line and how one line relates to another. Specifically, if we understand the slope, we can find the position of one point given another point on the line.

The slope formula is the cornerstone of our calculations. Given two points, P(x₁, y₁) and Q(x₂, y₂), the slope (often denoted by m) is calculated as:

m = (y₂ - y₁) / (x₂ - x₁)

This formula essentially quantifies the 'rise' (the difference in y-coordinates) divided by the 'run' (the difference in x-coordinates). It's a simple yet powerful tool that we'll use extensively in this article. The application of the slope formula is one of the crucial concepts in coordinate geometry. Different slopes provide different information about the lines we are looking at. For example, if two lines are parallel, they have the same slope. If two lines are perpendicular, the slope of one line is the negative reciprocal of the slope of the other. By using this formula, we can perform sophisticated analysis of linear relationships in a coordinate system. Understanding the slope and its formula gives us the power to accurately describe and compare lines, laying the groundwork for more advanced geometric and algebraic studies.

Point P and the Quest for Q

Our starting point is P(-1, 2). We want to find two different locations for point Q such that the line PQ has various specified slopes. To do this, we will use the slope formula, plugging in the coordinates of P and the desired slope, and then solving for the coordinates of Q. We will see how the relationship between P and Q changes as the slope varies. This will give us a deeper understanding of how slopes dictate the orientation and direction of lines in the coordinate plane.

Case a) Slope = 2

Let's find two possible locations for point Q such that the slope of line PQ is 2. Let Q have coordinates (x, y). Using the slope formula:

2 = (y - 2) / (x - (-1))
2 = (y - 2) / (x + 1)

To find possible locations for Q, we need to solve this equation. We can rewrite it as:

y - 2 = 2(x + 1)
y = 2x + 4

Now, we can choose two different values for x and calculate the corresponding y values. This will give us two points Q that satisfy the condition.

  1. If x = 0:

    y = 2(0) + 4 = 4

    So, one possible location for Q is (0, 4).

  2. If x = 1:

    y = 2(1) + 4 = 6

    Another possible location for Q is (1, 6).

Therefore, two positions for point Q with a slope of 2 are (0, 4) and (1, 6). Let's see how this translates graphically. If you were to plot these points and draw the line, you would see a line that rises steeply upwards from left to right, consistent with a positive slope of 2.

Case b) Slope = -3

Now, let’s find two possible locations for point Q such that the slope of line PQ is -3. Again, let Q have coordinates (x, y). Using the slope formula:

-3 = (y - 2) / (x - (-1))
-3 = (y - 2) / (x + 1)

Solving for y:

y - 2 = -3(x + 1)
y = -3x - 1

Let's choose two different values for x:

  1. If x = 0:

    y = -3(0) - 1 = -1

    One possible location for Q is (0, -1).

  2. If x = -1:

    y = -3(-1) - 1 = 2

    Another possible location for Q is (-1, 2). However, notice that this is the same as point P! So, we need to choose a different value for x. Let's try x = 1:

    y = -3(1) - 1 = -4

    Another possible location for Q is (1, -4).

So, two positions for point Q with a slope of -3 are (0, -1) and (1, -4). In this case, the negative slope indicates that the line will trend downwards from left to right. The steeper the absolute value of the slope (in this case, -3), the steeper the descent of the line.

Case c) Slope = 1/3

Now, let's find two possible locations for point Q with a slope of 1/3. Using the slope formula:

1/3 = (y - 2) / (x - (-1))
1/3 = (y - 2) / (x + 1)

Solving for y:

y - 2 = (1/3)(x + 1)
y = (1/3)x + 7/3

Choosing two different values for x:

  1. If x = 2:

    y = (1/3)(2) + 7/3 = 9/3 = 3

    One possible location for Q is (2, 3).

  2. If x = -1:

    y = (1/3)(-1) + 7/3 = 6/3 = 2

    This gives us point P again, so let's try x = 5:

    y = (1/3)(5) + 7/3 = 12/3 = 4

    Another possible location for Q is (5, 4).

Therefore, two positions for point Q with a slope of 1/3 are (2, 3) and (5, 4). This flatter, positive slope tells us that the line connecting P and Q will rise gently as we move from left to right.

Case d) Slope = -2/5

Next, we'll find two possible locations for point Q with a slope of -2/5:

-2/5 = (y - 2) / (x - (-1))
-2/5 = (y - 2) / (x + 1)

Solving for y:

y - 2 = (-2/5)(x + 1)
y = (-2/5)x + 8/5

Choosing two different values for x:

  1. If x = 0:

    y = (-2/5)(0) + 8/5 = 8/5

    One possible location for Q is (0, 8/5).

  2. If x = 4:

    y = (-2/5)(4) + 8/5 = 0

    Another possible location for Q is (4, 0).

So, two positions for point Q with a slope of -2/5 are (0, 8/5) and (4, 0). The fractional negative slope indicates a relatively gentle downward slope from left to right.

Case e) Slope = 0

Now, let's consider the case where the slope is 0. A line with a slope of 0 is a horizontal line. This means that the y-coordinate of point Q must be the same as the y-coordinate of point P (which is 2), but the x-coordinate can be anything.

So, the slope formula becomes:

0 = (y - 2) / (x + 1)

This implies y - 2 = 0, so y = 2.

Choosing two different values for x:

  1. If x = 0, then Q is (0, 2).
  2. If x = 1, then Q is (1, 2).

Therefore, two positions for point Q with a slope of 0 are (0, 2) and (1, 2). As expected, both points have the same y-coordinate, creating a horizontal line.

Case f) Slope = Undefined

Finally, let's tackle the case where the slope is undefined. An undefined slope corresponds to a vertical line. This occurs when the denominator of the slope formula is zero, i.e., x₂ - x₁ = 0. This means that the x-coordinate of point Q must be the same as the x-coordinate of point P (which is -1), but the y-coordinate can be anything.

So, the slope formula would have a zero in the denominator:

m = (y - 2) / (x - (-1))

For the slope to be undefined, x + 1 = 0, so x = -1.

Choosing two different values for y:

  1. If y = 0, then Q is (-1, 0).
  2. If y = 1, then Q is (-1, 1).

Thus, two positions for point Q with an undefined slope are (-1, 0) and (-1, 1). Both points share the same x-coordinate, forming a vertical line.

Conclusion

In this article, we've explored how to find different locations for a point Q, given the coordinates of point P and a specified slope for the line PQ. We've used the slope formula to solve for the coordinates of Q in various cases, covering positive, negative, zero, and undefined slopes. This exercise has highlighted how the slope dictates the direction and steepness of a line and how different slopes yield vastly different results. Understanding these concepts is fundamental to mastering coordinate geometry.

For further exploration of slope and linear equations, you might find the resources at Khan Academy helpful. They offer a wealth of practice problems and video explanations.