You can describe a line in a flat plane as having an X and Y axis. Both are points that contribute to that line. The relationship would be y = [a number with x]. What goes with x determines the line you have. An example of that would be the quadratic function – y = x2 + x, which is a parabola. The relationship of a straight line (with b and m as numbers) are y = mx + b.
With this slope intercept form calculator, you work with the straight line. You can use a parabola calculator to learn more about that side of things.
Straight line equations, or linear equations, have no terms with exponents. You will see an x and y, but you won’t see an x2 or y2. A linear equation describes a straight line, using a slope-intercept form to express it.
The slope-intercept form has an equation that looks like y = mx + b, as mentioned before. You learn the slope is m and the y-intercept is b. These values come in handy for linear interpolation.
The slope part refers to a gradient or incline. It determines any changes in y due to a fixed shift in x. If the difference in x is positive, y increases. If it’s negative, y decreases.
The y-intercept has a value of y. It crosses the straight line at the y-axis. In a linear equation, you find it by substituting x for 0 (x = 0). You will learn more about that below, including the importance of this rule in linear equations.
By now, you’ll want to cut to the chase and find the slope-intercept form of a linear equation. Two points go through the straight line. The first will be (x1, y1), and the second (x2, y2). What you don’t know is the slope (m) and the y-intercept (b).
1. Substitute the two points’ coordinates in the equation.
y1 = mx₁ + b
y₂ = mx₂ + b
2. Subtract the first equation from the second.
y₂ - y₁ = m (x₂ - x₁)
3. Divide the sides of the equation by (x₂ - x₁) to identify the slope.
m = (y₂ - y₁) / (x₂ - x₁)
Now that you know the slope, get the y-intercept by substituting it into the equation (first or second.)
y₁ = x₁ (y₂ - y₁) / (x₂ - x₁) + b
then:
b = y₁ - x₁ (y₂ - y₁) / (x₂ - x₁)
In the slope intercept form, find the line’s equation. You will need the two points that go through your line. Follow these steps below.
1. Write down the first point’s coordinates. For the example, it’s:
x₁ = 1 and y₁ = 1
2. Find the slope with the slope intercept formula
m = (y₂ - y₁) / (x₂ - x₁) = (3-1) / (2-1) = 2/1 = 2
3. Calculate your y intercept. You can use x₂ and y₂ instead of x₁ and y₂
b = y₁ - m x x₁ = 1 – 2 x 1 = -1
4. Construct the slope intercept form with the values
y = 2x – 1
You find the x-intercept value when the x crosses through the straight line. The value of x is when Y = 0. You can calculate it like this:
0 = mx + b
x = -b/m
If you use the intercept form calculator, you will see it displays both values.
After reading the formulas, you might wonder how this applies to real life. Let’s run through a couple of examples involving y-intercepts and x-intercepts.
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Imagine you see a skateboarder moving at full speed. You can plot the movement based on the time it will take to reach you as well as his distance from you. The x-axis is the time that passes, and the y-axis is the distance from you to the skateboarder.
Look at the y-intercept, which shows x = 0. It shows t = 0 when you first noticed the skateboarder. The y’s value will be the starting position of the skateboarder and its starting distance to you. It’s the same as b’s value in the straight line equation of a slope-intercept form.
Now, look at the x-intercept with y = 0. It’s when the distance from you to the skateboarder will be 0. The value of x will be the time you and the skateboarder end up in the same place.
Using a skateboarder in an example will show you why the slope-intercept form is critical. It also shows you the meaning of x-intercept and y-intercept. Even though we’ve mainly explained a straight line, you can use intercept points in a curve.
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The example involving the skateboarder didn’t need a linear equation. It would be the same for other shapes unless they had labeled names. An equation that always has a y-intercept but not always having an x-intercept is a parabola. Depending on orientation, it has a maximum and minimum. If the x-axis is above the maximum, or below the minimum, there won’t be an x-intercept point.
That won’t always the case, as not all equations are equal. Some formulas won’t have to intercept the x-axis or y-axis. Read more about these situations below.
You can identify equations in three ways. They have a y-intercept, x-intercept, or none at all. With only a y-intercept, you can have linear and other equation types. For example, y = 3. This line will be parallel to the x-axis and can therefore never touch it.
With only an x-intercept or none at all, it gets challenging. That’s where asymptote comes in. A curve or function can get close to an asymptote, but it won’t touch it. For example, you have an equation of y = 1/x. When you substitute x = 0 to find the y-intercept, you will get nothing but an error. In mathematical terms, it’s an undefined expression. This error happens when it makes no sense to divide by zero.
When you have numbers that are almost zero, such as 0.001, your y value increases. When x = 0, y would have a significant value. You will not be able to find the exact value. A typical equation would be 1/0 = ∞ but ∞ is a concept or symbol, not a number.
For our example, x = 0 is the linear equation that represents y = 1/x = the asymptote of the function. Y = 1/x won’t intercept the line, and the equation won’t have a y-intercept. When a function has an asymptote on an axis, it won’t have all the intercept points.
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In the previous example, we used y = 1/x. The example is also an asymptote for y = 0 (the x-axis). You can’t use y = 0 in the formula because it has an infinite answer. As you now know, infinity is not a number.
It can be difficult to think you’ll ever use such information in daily life. They can seem too simple. Linear equations are more common than you think. They appear in optimization and minimization problems often.
You will use linear equations if you are working out how to make variables smaller. An example would be differences between reality and predictions. You would also use them for minimum values of equations or formulas in the Newton Method. Such a method uses x-intercepts, linear equations, and derivatives.
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When you use this method, you have to choose x’s value for the equation. You then calculate the derivative of it at that point. With the derivative, it becomes the slope of a linear equation. It passes through the (x, y) point, then you calculate the x-intercept. With such an equation, the slope-intercept form is useful.
When you calculate the x-intercept, you use the value to repeat the process. With repetition, you get the y value, with a derivative of zero. In real life, you can’t get that exact minimal point. Most people will settle for an approximate value.
