How to find slope with one point

The slope of the function f at the point (a, f(a)) is the limit of the slope of the secant line between x = a and x = a + h as h gets closer to 0. Whew, that's a mouthful.

From the formula for the slope of this secant line, we find another formula:

How to find slope with one point

"The slope of f at a" is also called the derivative of f at a and is written f '(a). This nice formula is known as the limit definition of the derivative, and we'll write it again here, with correct notation:

How to find slope with one point

Visually, we move h closer and closer to 0 (equivalently, move a + h closer and closer to a) and watch what happens to the secant line, as in this animation.

If we could keep going until h reached 0 (equivalently, until a + h reached a) we would find a line that, instead of passing through the graph twice, would hit the graph at one spot and bounce off.

The slope of the function f at the single point x = a is the slope of this line, also called the tangent line.

Here are two important things to remember:

  • The slope of f at a and the derivative of f at a are the same thing.
  • Since the derivative of f at a is a limit, the derivative won't always exist.

Another phrase for f '(a) is the instantaneous rate of change of f with respect to x when x = a.

It's important to remember that the derivative is a limit. Later on, we'll find nifty ways to compute derivatives without having to take a limit every time, but fundamentally, every time we take a derivative we're finding a limit. In fact, if we look hard enough, every in calculus can be reduced to a limit.

The point-slope form calculator will show you how to find the equation of a line from a point on that line and the line's slope. Soon, you will know what is point-slope form equation, and learn how is it different from the slope-intercept form equation. We also came up with two exercises, and we'll explain how to solve them in the last paragraph.

Let's start with the basics. What is the slope? The slope, also known as the gradient, is the marker of a line's steepness. If it's positive, it means the line rises. If it's negative - the line decreases. If it's equal to zero, the line is horizontal.

You can find the slope between two points by estimating rise over run - the difference in height over a distance between two points.

How to find slope with one point

So, slope formula is:

m = change in y / change in x = (y - y₁) / (x - x₁)

The point-slope form equation is a rearranged slope equation.

To find the gradient of non-linear functions, you can use the average rate of change calculator.

🙋 For more information go to the slope calculator.

There is more than one way to form an equation of a straight line. Point-slope form is a form of a linear equation, where there are three characteristic numbers - two coordinates of a point on the line, and the slope of the line. The point slope form equation is:

y−y1=m⋅(x−x1)\small y - y_1 = m \cdot (x - x_1)yy1=m(xx1)

,where:

  • x1,x2\small x_1, x_2x1,x2 are the coordinates of a point, and
  • m\small mm is the slope.

Do you see the similarity to the slope formula? What you might not know is that it's not the only way to form a line equation. The more popular is the slope intercept form:

y=m⋅x+b\small y = m \cdot x + b y=mx+b

,where:

  • m\small mm is the slope; and
  • b\small bb is the intercept of the y-axis.

The truth is that this is nothing else than a more precise point-slope form. A straight line intercepts the y-axis in a point (0, b). If you choose this point - (0, b), as a point that you want to use in the point-slope form of the equation, you will get:

y−b=m⋅(x−0)\small y - b = m \cdot (x - 0)yb=m(x0), which is the same as y=m⋅x+b\small y = m \cdot x + by=mx+b.

In the two graphs below, you can see the same function, only described with two different forms of a linear equation:

How to find slope with one point

To learn how to find the x-intercept and y-intercept of a line, visit our x- and y-intercept calculator.

Let's have a look at two exercises, to understand the topic more clearly.

The slope of a line is 2. It passes through point A(2, -3). What is the general equation of the line?

  1. Identify the point coordinates:
  2. Identify the slope:
  3. Input the values into the point slope form formula:
    • y−y1=m(x−x1)\small y - y_1 = m (x - x_1)yy1=m(xx1)
    • y−(−3)=2(x−2)\small y - (-3) = 2(x - 2)y(3)=2(x2)
  4. Simplify to get the general equation:
    • y=2x−4−3\small y = 2x - 4 -3y=2x43
    • 0=2x−y−7\small 0 = 2x - y - 70=2xy7
    And you have the answer. Now, you can check your result with our point-slope form calculator.

Let's solve an exercise with a more relatable subject.

Let's say you got a puppy. When you got him he was 14 pounds. It grew 0.2 pounds every day, and after 30 days, he was 20 pounds. Find the general equation of the puppy's growth.

  1. The slope is the change of weight per day: m = 0.2
  2. The characteristic point is 20 pounds on 30th day: (x1, y1) = (30, 20)
  3. Now, input the values into the point-slope formula:

y−20=0.2∗(x−30)\small y - 20 = 0.2 * (x - 30)y20=0.2(x30) 4. Simplify the equation to get the general equation:

0=0.2x−y+14\small 0 = 0.2x - y + 140=0.2xy+14

💡 If you need to find a different point on your line click on the advanced mode button. Then, input one coordinate, and get the other.

And here you have it! We hope you enjoyed our point-slope form calculator! Before you go, check out more of our geometry calculators!