1. 1: Straight Lines and Gradients © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules

2. Straight Lines and Gradients Module C1 "Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages"

3. Straight Lines and Gradients c is the point where the line meets the y -axis, the y -intercept and y -intercept, c = e.g. has gradient m = The equation of a straight line is m is the gradient of the line gradient = 2 x intercept on y -axis

4. Straight Lines and Gradients gradient = 2 x intercept on y -axis ( 4, 7 ) x The coordinates of any point lying on the line satisfy the equation of the line showing that the point ( 4,7 ) lies on the line. e.g. Substituting x = 4 in gives

5. Straight Lines and Gradients Notice that to find c, the equation has been solved from right to left. This takes a bit of practice but reduces the chance of errors.  Finding the equation of a straight line when we know e.g.Find the equation of the line with gradient passing through the point its gradient, m and the coordinates of a point on the line. Solution: So, Using, m is given, so we can find c by substituting for y, m and x. (-1, 3) x

6. Straight Lines and Gradients If we don’t know the gradient, we have to find it using two points on the line. We develop the formula by reminding ourselves about the meaning of a gradient. To do this, we can use a formula.

7. Straight Lines and Gradients 4 2 e.g.

8. Straight Lines and Gradients

10.  The gradient of the straight line joining the points and is e.g. Find the gradient of the straight line joining the points and To use this formula, we don’t need a diagram! Solution:

11. Straight Lines and Gradients  To find the equation of a straight line given 2 points on the line. Solution: First find the gradient: e.g. Find the equation of the line through the points Now on the line: Equation of line is

12. Straight Lines and Gradients SUMMARY  Equation of a straight line  Gradient of a straight line where and are points on the line where m is the gradient and c is the intercept on the y -axis

13. Straight Lines and Gradients 2. Find the equation of the line through the points Exercise 1. Find the equation of the line with gradient 2 which passes through the point. Solution: So, Solution: So,

14. Straight Lines and Gradients We sometimes rearrange the equation of a straight line so that zero is on the right-hand side ( r.h.s. ) We must take care with the equation in this form. e.g. can be written as e.g. Find the gradient of the line with equation Solution: Rearranging to the form : so the gradient is

15. Straight Lines and Gradients  Parallel and Perpendicular Lines  They are parallel if  They are perpendicular if If 2 lines have gradients and, then:

16. Straight Lines and Gradients e.g. 1Find the equation of the line parallel to which passes through the point Solution: The given line has gradient 2. Let For parallel lines, is the equation of any line parallel to Using on the line

17. Straight Lines and Gradients We don’t usually leave fractions ( or decimals ) in equations. So, multiplying by 2 : e.g.Find the equation of the line perpendicular to passing through the point. Solution: The given line has gradient 2. Let Perpendicular lines: Equation of a straight line: on the line

18. Straight Lines and Gradients If the gradient isn’t given, find the gradient using  Method of finding the equation of a straight line: Substitute for y, m and x in into to find c. either parallel lines: or 2 points on the line: or perpendicular lines: SUMMARY

19. Straight Lines and GradientsExercise Solution: So, Solution: So, 1.Find the equation of the line parallel to the line which passes through the point. Parallel line is 2.Find the equation of the line through the point (1, 2), perpendicular to the line So,

20. Straight Lines and Gradients A Second Formula for a Straight Line ( optional ) Let ( x, y ) be any point on the line Let be a fixed point on the line x x

21. Straight Lines and Gradients Solution: First find the gradient We could use the 2 nd point, (-1, 3) instead of (2, -3) To use the formula we need to be given either: one point on the line and the gradient or: two points on the line e.g. Find the equation of the line through the points Now use with

22. Straight Lines and Gradients

23. The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied. For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

24. Straight Lines and Gradients  They are parallel if  They are perpendicular if  If 2 lines have gradients and, then:  Equation of a straight line  Gradient of a straight line where and are points on the line where m is the gradient and c is the intercept on the y -axis SUMMARY

25. Straight Lines and Gradients Solution: First find the gradient: e.g. Find the equation of the line through the points Now on the line: Equation of line is

26. Straight Lines and Gradients We don’t usually leave fractions ( or decimals ) in equations. So, multiplying by 2 : e.g.Find the equation of the line perpendicular to passing through the point. Solution: The given line has gradient 2. Let Perpendicular lines: Equation of a straight line: on the line