Equations of a Straight Line (DP IB Analysis & Approaches (AA)): Revision Note

Equations of a Straight Line

How do I find the gradient of a straight line?

• Find two points that the line passes through with coordinates (x₁, y₁) and (x₂, y₂)

• The gradient between these two points is calculated by

   

  • This is given in the formula booklet

• The gradient of a straight line measures its slope

  • A line with gradient 1 will go up 1 unit for every unit it goes to the right

  • A line with gradient -2 will go down two units for every unit it goes to the right

What are the equations of a straight line?

• 

  • This is the gradient-intercept form

  • It clearly shows the gradient m and the y-intercept (0, c)

• 

  • This is the point-gradient form

  • It clearly shows the gradient m and a point on the line (x₁, y₁)

• 

  • This is the general form

  • You can quickly get the x-intercept and y-intercept

How do I find an equation of a straight line?

• You will need the gradient

  • If you are given two points then first find the gradient

• It is easiest to start with the point-gradient form

  • then rearrange into whatever form is required

    • multiplying both sides by any denominators will get rid of fractions

• You can check your answer by using your GDC

  • Graph your answer and check it goes through the point(s)

  • If you have two points then you can enter these in the statistics mode and find the regression line 

Parallel Lines

How are the equations of parallel lines connected?

• Parallel lines are always equidistant meaning they never intersect

• Parallel lines have the same gradient

  • If the gradient of line l₁ is m₁ and gradient of line l₂ is m₂ then...

    • 

    • 

• To determine if two lines are parallel:

  • Rearrange into the gradient-intercept form 

  • Compare the coefficients of 

  • If they are equal then the lines are parallel

Perpendicular Lines

How are the equations of perpendicular lines connected?

• Perpendicular lines intersect at right angles

• The gradients of two perpendicular lines are negative reciprocals

  • If the gradient of line l₁ is m₁ and gradient of line l₂ is m₂ then...

    • 

    •  

• To determine if two lines are perpendicular:

  • Rearrange into the gradient-intercept form 

  • Compare the coefficients of 

  • If their product is -1 then they are perpendicular

• Be careful with horizontal and vertical lines

  • and  are perpendicular where p and q are constants