GCSE Maths - How to Find the Gradient Using the Coordinates of Two Points #66

Summary

This video explains how to calculate the gradient between two points when a graph is not provided. It introduces a simple formula using the differences in y and x values (y2 - y1) / (x2 - x1), emphasizing the importance of correctly labeling the points. The video walks through examples to clarify the concept, including a scenario where the coordinates must be chosen carefully, and shows how to apply the formula for both given coordinates and graph-based points. Viewers are encouraged to practice and apply the method for calculating gradients effectively.

Takeaways

• 😀 The gradient between two points can be found using the equation (y2 - y1) / (x2 - x1).
• 😀 If only the coordinates are given, the gradient formula still applies by substituting values for x1, y1, x2, and y2.
• 😀 x1 and y1 represent the coordinates of the first point, while x2 and y2 represent the coordinates of the second point.
• 😀 When working with coordinates, it is easier to assign the larger numbers (for x and y) to x2 and y2, respectively.
• 😀 The order of coordinates doesn't affect the result as long as you keep the formula consistent, but choosing the larger values for x2 and y2 simplifies calculations.
• 😀 After plugging values into the formula, simplify the difference in y-values and x-values to find the gradient.
• 😀 The gradient of two points represents the slope of the line connecting them and is calculated by dividing the vertical change (y2 - y1) by the horizontal change (x2 - x1).
• 😀 If given a graph, pick any two points on the line, then apply the same gradient formula to find the slope.
• 😀 The gradient can be positive, negative, or zero, depending on the relative position of the points on the graph.
• 😀 The method for calculating gradients works both with raw coordinates or graph points, providing flexibility in solving gradient problems.

Q & A

What is the formula used to find the gradient between two points?

The formula to find the gradient between two points is (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the coordinates of the two points.

Why is it easier to use the formula (y2 - y1) / (x2 - x1) when we are only given the coordinates of the points?

The formula (y2 - y1) / (x2 - x1) allows us to directly calculate the gradient without needing the graph, using just the differences in the y and x coordinates of the two points.

What do x1, y1, x2, and y2 represent in the gradient formula?

x1 and y1 represent the coordinates of the first point, while x2 and y2 represent the coordinates of the second point.

Is it important to choose a specific order for the coordinates when applying the formula?

It’s not strictly necessary which set of coordinates you choose as the first or second point, but it’s typically easier if you use the larger values for x2 and y2.

In the example with coordinates (2, 3) and (8, 12), how is the gradient calculated?

The gradient is calculated as (12 - 3) / (8 - 2), which simplifies to 9 / 6, resulting in a gradient of 1.5.

In the second example with coordinates (11, 7) and (8, -2), how do we calculate the gradient?

First, choose the coordinates with the larger x-value as the second point: (11, 7) as (x2, y2) and (8, -2) as (x1, y1). Then, the gradient is calculated as (7 - (-2)) / (11 - 8), which simplifies to 9 / 3, giving a gradient of 3.

Can we use the gradient formula if we are given a graph instead of coordinates?

Yes, you can use the gradient formula by choosing any two points on the line from the graph, labeling the coordinates, and applying the same formula to calculate the gradient.

How do we handle negative values when calculating the gradient?

When subtracting negative values, remember that subtracting a negative number is the same as adding the positive equivalent. For example, (7 - (-2)) becomes (7 + 2), which equals 9.

In the graph example, how do you calculate the gradient between the points (0, -1) and (4, -7)?

For the points (0, -1) and (4, -7), we calculate the gradient as (-7 - (-1)) / (4 - 0), which simplifies to (-6) / 4, resulting in a gradient of -1.5.

Why is the gradient for the line between (0, -1) and (4, -7) negative?

The gradient is negative because the line slants downwards as we move from left to right. A negative gradient indicates a downward slope.