Gradient and intercepts

Gradient and Intercepts

The gradient of a straight line measures how steep the line is. It is defined as the change in the vertical direction (y) divided by the change in the horizontal direction (x). The formula for the gradient (m) between two points (x_1, y_1) and (x_2, y_2) is:

m = (y_2 - y_1) / (x_2 - x_1)

Calculating the Gradient

To find the gradient, you need two points on the line. Let's consider the points A(2, 3) and B(5, 11).

1. Identify the coordinates of the points:

• A(2, 3) where x_1 = 2 and y_1 = 3
• B(5, 11) where x_2 = 5 and y_2 = 11

2. Substitute the values into the gradient formula:

m = (11 - 3) / (5 - 2)

m = 8 / 3

The gradient of the line passing through points A and B is 8/3.

Finding the Y-Intercept

The y-intercept of a line is the point where the line crosses the y-axis. It can be found by setting x = 0 in the equation of the line. To find the equation of the line, you can use the point-slope form:

y - y_1 = m(x - x_1)

Using the gradient we found (8/3) and point A(2, 3), the equation becomes:

y - 3 = (8/3)(x - 2)

Now, simplify this equation:

y - 3 = (8/3)x - (16/3)

y = (8/3)x - (16/3) + 3

y = (8/3)x - (16/3) + (9/3)

y = (8/3)x - (7/3)

The y-intercept occurs when x = 0:

y = (8/3)(0) - (7/3)

y = -7/3

The y-intercept is -7/3. This means the line crosses the y-axis at the point (0, -7/3).

Finding the X-Intercept

The x-intercept is the point where the line crosses the x-axis. It can be found by setting y = 0 in the equation of the line:

0 = (8/3)x - (7/3)

Now, solve for x:

(8/3)x = (7/3)

x = (7/3) / (8/3)

x = 7/8

The x-intercept is 7/8. This means the line crosses the x-axis at the point (7/8, 0).

Graphing the Line

To graph the line, plot the y-intercept (0, -7/3) and the x-intercept (7/8, 0). Draw a straight line through these points. The gradient indicates how steep the line is.

Parallel and Perpendicular Lines

Lines can be parallel or perpendicular based on their gradients. Parallel lines have the same gradient, while perpendicular lines have gradients that multiply to -1. If a line has a gradient m, the gradient of a perpendicular line is -1/m.

For example, if a line has a gradient of 2, a perpendicular line will have a gradient of -1/2.

Summary

• The gradient measures the steepness of a line.
• The y-intercept is where the line crosses the y-axis.
• The x-intercept is where the line crosses the x-axis.
• Parallel lines have equal gradients; perpendicular lines have gradients that multiply to -1.

Check your understanding

1. Calculate the gradient of the line that passes through the points (1, 2) and (4, 8).
2. Find the y-intercept of the line with the equation y = 2x + 5.
3. Determine the x-intercept of the line y = -3x + 9.
4. Are the lines with gradients of 1/2 and -2 parallel or perpendicular? Explain your answer.