5.2. Point-slope form

b. Point-slope form:

𝑦−𝑦₁ = 𝑚(𝑥 −𝑥₁)

This form is useful when given a point and slope.

The point-slope form is a way to express a linear equation when you know a point on the line and the slope. The formula is written as y - y₁ = m(x - x₁), where (x₁, y₁) is a specific point on the line and m is the slope. This format allows you to easily write the equation of a line when you have these two pieces of information.

Imagine you're walking along a hill, and you notice a point where you're standing at coordinates (2, 3). If you know the hill rises at a certain angle (slope), let’s say it rises 1 meter for every 2 meters you move right, you can use point-slope form to describe the path of the hill starting from where you are.

To create an equation in point-slope form, follow these steps:

1. Identify a point (x₁, y₁) on the line.
2. Determine the slope (m) of the line.
3. Substitute the point and slope into the formula: y - y₁ = m(x - x₁).

To use the point-slope form, you need to start by identifying a point on the line, which is known as (x₁, y₁). Next, find the slope (m) of the line, which indicates how steep it is. Once you have these two details, simply plug them into the point-slope formula. This gives you a complete equation for the line that goes through that point with the specified slope.

Think of a bus that stops at a certain bus station, and you know how it travels from that station. If the bus station is located at (1, 2) and the bus goes up by 3 units for every 4 units it travels forward, you can write the equation for the bus's route using point-slope form to help predict where it will be after traveling a certain distance.

Once you have the point-slope equation, you can graph it by:

1. Plotting the point (x₁, y₁).
2. Using the slope m to find another point on the line.
3. Drawing the line through the plotted points.

When you have your equation in point-slope form, to graph it, start by plotting the point (x₁, y₁) on the coordinate plane. Then use the slope to find another point: if the slope is, say, 2, you would rise 2 units and run 1 unit to the right from your first point. Plot this second point and draw a straight line through both points. This visual representation will show you how the line behaves.

Imagine you are trying to map out a road between two towns. You've established one town's coordinates as (3, 4), and you've calculated that for every 5 km you go north, you should go 2 km east. You would use point-slope form to mark the direction of the road, helping you see the best route from one town to the other.

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