Point Slope Form Calculator
How to use the ‘Point Slope Form Calculator’?
Follow these steps to use the point slope form calculator:
Step 1: Enter the coordinate of a point and the slope of a line(m) into the respective input boxes.
Step 2: Click on the ‘Solve’ button to obtain the point slope form and the graph of the line.
Step 3: Click on the ‘Show Steps’ button to understand the stepwise method to graph and write in the standard form.
Step 4: Click on the refresh button to enter new inputs and start again.
Step 5: Click on the ‘Example’ button to find results with random input values.
Step 6: Click on the ‘Explore’ button to observe the variation in the graph and point slope form with the use of sliders.
Step 7: When on the ‘Explore’ page, click the ‘Calculate’ button to go back to the calculator.
What is ‘Point Slope Form’?
A straight line is represented using its slope and a point on the line using ‘point slope form’. In other words, point slope form represents the equation of a line passing through a point \(\left( x_{1},\text{ }y_{1} \right)\) and has a slope ‘m’.
The equation of a straight line can be expressed in various ways. ‘Point slope shape’ is one of them. The point slope form is written as:
\(y \text{ }-\text{ } y_{1}\text{ }=\text{ }m \left( x\text{ }-\text{ }x_{1} \right)\)
Where \(\left( x_{1},\text{ }y_{1} \right)\) is a randomly selected point on the line and m is the slope of the line.
Derivation of Point Slope Form:
Let us assume that \(\left( x_{1},\text{ }y_{1} \right)\) is a known point on a line. Also \(\left( x,\text{ }y \right)\) is the point whose coordinates are not known.
As we know that the slope of a line is calculated by:
Slope, \(m = \frac{Difference\text{ }of\text{ }y\text{ }coordinates}{Difference\text{ }of\text{ }x\text{ }coordinates}\)
\(m = \frac{y\text{ }-\text{ }y_{1}}{x\text{ }-\text{ }x_{1}}\)
This equation can also be written as,
\(y \text{ }- \text{ }y_{1} = m \left( x \text{ }-\text{ } x_{1} \right)\)
This is called the point slope form.
Solved Examples
Example 1: Write the point slope form of a line that passes through (9, 10) and its slope is 5.
Solution:
Given point = (9, 10)
\(m=5\)
The point slope form of a line is written as:
\(y \text{ }- \text{ }y_{1} = m \left( x \text{ }- \text{ }x_{1} \right)\)
\(y \text{ }-\text{ } 10 = m \left( x-9 \right)\)
\(y \text{ }-\text{ } 10 = 5 \left( x-9 \right)\)
So, the point slope form of the line is \( y\text{ }-\text{ }10 = 5\left( x-9 \right)\)
Example 2: Express the point slope form of a line that passes through (3, 4) and its slope is 4.
Solution:
Given point = (3, 4)
\(m=4\)
The point slope form of a line is written as:
\(y \text{ }- \text{ }y_{1} = m \left( x\text{ }-\text{ }x_{1} \right)\)
\(y \text{ }-\text{ } 4 = m\left( x-3 \right)\)
\(y \text{ }-\text{ } 4 = 4\left( x-3 \right)\)
So, the point slope form of the line is y – 4 = 4 (x – 3)
Example 3: Write the point slope form of a line that passes through (-5, 4) and with a slope of -2.
Solution:
Given point = (-5, 4)
\(m=-2\)
The point slope form of a line is written as:
\(y-y_{1} = m\left( x-x_{1} \right)\)
\(y -4=m \left( x-\left( -5 \right) \right)\)
\(y -4=m \left( x+5 \right)\)
\(y -4=(-2) \left( x+5 \right)\)
So, the point slope form of the line is \(y -4=(-2) \left( x+5 \right)\)
A point slope formula is one of the formulas used to find the equation of a line. The equation of a line with slope ‘m’ and a point with coordinates(x1, y1) , is the representation of the point slope formula.
The slope of a line is the ratio of the difference of y coordinates and the x coordinates. It is denoted by ‘m’.
Using the point slope formula, students can find:
- A point on a line with the specified slope and the equation of a line
- The slope and a single point on a line are used to graph the line
- The slope of a line