1. Graph of Function EquationGradientY-intercept A B C D E F G H I J

2. Aims: To be able to read off the gradient and y-intercept of a straight line graph. To be able to use the graphical calculator to plot graphs and find straight line equations. To be able to reduce a relation to a linear law. Linear Laws Lesson 1 Starter – reading the grad and y-intercept w/s from straight line graphs

3. Linear Relations Introduction It is often necessary to find a relationship between two connected quantities. A table of experimental data, showing corresponding values of the two quantities, and a suggestion as to the form of the expected relationship are always given. The data should then be displayed graphically and, if the expected relationship is confirmed, the graph is used to obtain the unknown constants in this relationship. Linear Relations The simplest case is when the expected relation is a straight line of the form y = mx + c. In this case simply plot y against x. If the points plotted lie approximately on a straight line then the given linear relation is approximately true. The points rarely lie exactly on a straight line, so the line of ‘best fit’ is drawn ‘by eye’. Consequently everybody's answers are not identical. x012345 y5.88.712.51518.420.9 Example 1 Looks linear and points close to straight line so assume the relation y = mx + c By looking at the graph we can say c = and m =

4. Using the calculator to find m and C In Stats mode enter the x and y values into list 1 and list 2 respectively Now press F1 – GRPH F6 – SET Change Graph Type to Scatter and press EXE F1 – GPH1 see the points plotted! Press F1 – CALC Press F2 – X (a straight line relationship) Press F1 – aX+b Now you read the a value as m and the b value as C easy!! If you would like to see this line plotted and observe how accurate it fits the data, press F6 x012345 y5.88.712.51518.420.9

5. Non -Linear Relations Non -linear Relations If the expected relation is not in the form y = mx + c, then it must be transformed to a linear form before proceeding as before. Example 2 The data is thought to obey a law of the form. Test to find if the results do satisfy such a formula, and find suitable values for a and b. x0123 y-532467 Compare With Constructing a table for X = and Y = gives X Y-532467 We read from this that c = and m = That is a = and b = so Or use calc! Now plot the (X,Y) points and we see they form a linear relation, thus formula correct.

6. Non -Linear Relations Example 3 The data is thought to obey a law of the form. Test to find if the results do satisfy such a formula, and find suitable values for a and b. Compare With Constructing a table for X = and Y = gives Now plot the (X,Y) points and we see they form a linear relation, thus formula correct. We read from this that c = and m = That is a = and b = so u12345 t1.341.531.782.052.29 X = u 2 Y = t 3 Or use calc!

7. On w/b Test to find if the results do satisfy such a formula, and find suitable values for a and b.

8. Compare this with With Constructing a table for X = and Y = to give a straight line. Non -Linear Relations More transformations that you may see: Compare this with With So we’d need to plot Y = against X = to give a straight line. Re-write as Relationships of the form Gradient here would be m = and Y-intercept c = Relationships of the form Re-write as Gradient here would be m = and Y-intercept c = Do Worksheet for Monday and Moodle Numerical Methods for 2 nd lesson next week Linear form is: Y= m X + C Variable = constant  variable + constant

9. Can you do it? AQA Exam Question from 20089 marks 9 minutes!

12. Introduction x0123 y13919 Suspect this graph to be: Compare it with the straight line equation: Y=mX + c So Y = and X = also m = and c = Complete the new table and plot to see what’s happened X Y13919 We can read off the gradient and intercept of a line more easily so: Y = ____X + ____ Hence the original was y = ___x 2 + _____