Point slope shape

The point slope form or point slope form is a special form of a straight line equation in mathematics . In the point slope form, a straight line is represented in the Euclidean plane with the help of a point on the straight line and the slope of the straight line.

presentation

Point slope form of a straight line equation

In the point slope form, a straight line in the plane passing through the point and having the slope is described as the set of those points whose coordinates satisfy the equation ${\displaystyle (x_{1},y_{1})}$ ${\displaystyle m}$ ${\displaystyle (x,y)}$

${\displaystyle y-y_{1}=m\cdot (x-x_{1})}$

fulfill. If the straight line equation is solved, the explicit representation is obtained ${\displaystyle y}$

${\displaystyle y=m\cdot (x-x_{1})+y_{1}}$.

The straight line is then the graph of the function with the function equation ${\displaystyle f}$

${\displaystyle f(x)=m\cdot (x-x_{1})+y_{1}}$.

example

In the picture above, for example, the given straight line point and the slope are obtained , and the straight line equation is obtained ${\displaystyle (2,2)}$ ${\displaystyle m=-{\tfrac {1}{2}}}$

${\displaystyle y-2=-{\tfrac {1}{2}}\cdot (x-2)}$

respectively

${\displaystyle y=-{\tfrac {1}{2}}\cdot (x-2)+2}$.

Derivation

Assuming the general shape of a straight line

${\displaystyle y=m\cdot x+n}$

off, then especially since the point lies on the straight line, ${\displaystyle (x_{1},y_{1})}$

${\displaystyle y_{1}=m\cdot x_{1}+n}$.

If this equation is solved for and inserted into the general form, it follows ${\displaystyle n}$

${\displaystyle y=m\cdot x+(y_{1}-m\cdot x_{1})}$.

By excluding from one gets the point slope form ${\displaystyle m}$

${\displaystyle y=m\cdot (x-x_{1})+y_{1}}$.

conversion

Is using the slope triangle through the point and another straight line point by ${\displaystyle m}$ ${\displaystyle (x_{1},y_{1})}$ ${\displaystyle (x_{2},y_{2})}$

${\displaystyle m={\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}}$

calculated, one obtains the two-point form of a straight line equation.

literature

• Frank Paech: Mathematics - clear and entertaining . Carl Hanser Verlag, 2012, ISBN 978-3-446-42874-4 .
• Karl-Heinz Pfeffer: Analysis for technical high schools . Springer, 2010, ISBN 978-3-8348-9646-9 .