Simple Definition:

Slope = How much we go up or down ÷ How much we go forward

Real-Life Examples:

• The steepness of a slide at the playground
• A ramp for wheelchairs or bicycles
• The roof of a house
• Hills and mountains

Basic Slope Formula:

Slope = \(\frac{\text{Rise}}{\text{Run}}\) = \(\frac{\text{Vertical Change}}{\text{Horizontal Change}}\)

The slope of a line is the ratio of how much it rises (vertical change) to how much it runs (horizontal change).

Finding Slope from Two Points:

Slope = \(\frac{y_2 - y_1}{x_2 - x_1}\)

Types of Slopes:

Slope-Intercept Form:

y = mx + b

• m = slope
• b = y-intercept (where the line crosses the y-axis)

The coefficient m in the equation represents the slope of the line.

Point-Slope Form:

y - y₁ = m(x - x₁)

• m = slope
• (x₁, y₁) = coordinates of a point on the line

Standard Form:

Ax + By + C = 0

Slope = -\(\frac{A}{B}\)

Parallel Lines:

Parallel lines have the same slope.

If line 1 has slope m₁ and line 2 has slope m₂,
then the lines are parallel if m₁ = m₂

Perpendicular Lines:

Perpendicular lines have slopes that are negative reciprocals of each other.

If line 1 has slope m₁ and line 2 has slope m₂,
then the lines are perpendicular if m₁ × m₂ = -1

m₂ = -\(\frac{1}{m_1}\)

Average Rate of Change:

The slope between two points on a curve represents the average rate of change of the function.

Average Rate of Change = \(\frac{f(x_2) - f(x_1)}{x_2 - x_1}\)

Instantaneous Rate of Change (Derivative):

The slope of the tangent line at a point on a curve represents the instantaneous rate of change.

Instantaneous Rate of Change = \(\lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\) = f'(x)

Linear Regression (Line of Best Fit):

For a set of data points, the slope of the line of best fit can be calculated using:

m = \(\frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2}\)

• n = number of data points
• Σxy = sum of the products of x and y values
• Σx = sum of all x values
• Σy = sum of all y values
• Σx² = sum of squares of all x values