2. Introduction This Chapter focuses on sketching Graphs We will also be looking at using them to solve Equations There will also be some work on Graph transformations

3. Sketching Curves Sketching Cubics You need to be able to sketch equations of the form: This involves finding the places where the graph crosses the axes, in the same way you do when sketching a Quadratic. 4A or A cubic equation will take one of the following shapes For any x 3 For any -x 3

4. Sketching Curves Sketching Cubics You need to be able to sketch equations of the form: This involves finding the places where the graph crosses the axes, in the same way you do when sketching a Quadratic. 4A or Example Sketch the graph of the function: If y = 0 So x = 2, 1 or -1 (-1,0) (1,0) and (2,0) If x = 0 So y = 2 (0,2)

5. Sketching Curves Sketching Cubics You need to be able to sketch equations of the form: This involves finding the places where the graph crosses the axes, in the same way you do when sketching a Quadratic. 4A or Example Sketch the graph of the function: (-1,0) (1,0) (2,0) (0,2) x y 2 2 1 If we substitute in x = 3, we get a value of y = 8. The curve must be increasing after this point…

6. Sketching Curves Sketching Cubics You need to be able to sketch equations of the form: This involves finding the places where the graph crosses the axes, in the same way you do when sketching a Quadratic. 4A or Example Sketch the graph of the function: If y = 0 So x = 2, 1 or -1 (-1,0) (1,0) and (2,0) If x = 0 So y = -2 (0,-2)

7. Sketching Curves Sketching Cubics You need to be able to sketch equations of the form: This involves finding the places where the graph crosses the axes, in the same way you do when sketching a Quadratic. 4A or Example Sketch the graph of the function: (-1,0) (1,0) (2,0) (0,-2) x y 2 -2 1 If we substitute in x = 3, we get a value of y = -8. The curve must be decreasing after this point…

8. Sketching Curves Sketching Cubics You need to be able to sketch equations of the form: This involves finding the places where the graph crosses the axes, in the same way you do when sketching a Quadratic. 4A or Example Sketch the graph of the function: If y = 0 So x = 1 or -1 (-1,0) and (1,0) If x = 0 So y = 1 (0,1)

9. Sketching Curves Sketching Cubics You need to be able to sketch equations of the form: This involves finding the places where the graph crosses the axes, in the same way you do when sketching a Quadratic. 4A or Example Sketch the graph of the function: (-1,0) (1,0) (0,1) x y 1 1 If we substitute in x = 2, we get a value of y = 3. The curve must be increasing after this point… ‘repeated root’

10. Sketching Curves Sketching Cubics You need to be able to sketch equations of the form: This involves finding the places where the graph crosses the axes, in the same way you do when sketching a Quadratic. 4A or Example Sketch the graph of the function: If y = 0 So x = 0, 3 or -1 (0,0) (3,0) and (-1,0) If x = 0 So y = 0 (0,0) Factorise Factorise fully

11. Sketching Curves Sketching Cubics You need to be able to sketch equations of the form: This involves finding the places where the graph crosses the axes, in the same way you do when sketching a Quadratic. 4A or Example Sketch the graph of the function: (0,0) (3,0) (-1,0) x y 03 If we substitute in x = 4, we get a value of y = 20. The curve must be increasing after this point…

12. Sketching Curves Sketching Cubics You need to be able to sketch and interpret cubics that are variations of y = x 3 This will be covered in more detail in C2. You can still plot the graphs in the same way we have seen before. This topic is offering a ‘shortcut’ if you can understand it. 4B Example Sketch the graph of the function: x y y = x 3

13. Sketching Curves Sketching Cubics You need to be able to sketch and interpret cubics that are variations of y = x 3 This will be covered in more detail in C2. You can still plot the graphs in the same way we have seen before. This topic is offering a ‘shortcut’ if you can understand it. 4B Example Sketch the graph of the function: x y y = x 3 y = -x 3 A cubic with a negative ‘x 3 ’ will be reflected in the x-axis ‘Whatever you get for x 3, you now have the negative of that..’ 5 -5

14. Sketching Curves Sketching Cubics You need to be able to sketch and interpret cubics that are variations of y = x 3 This will be covered in more detail in C2. You can still plot the graphs in the same way we have seen before. This topic is offering a ‘shortcut’ if you can understand it. 4B Example Sketch the graph of the function: x y y = x 3 When a value ‘a’ is added to a cubic, inside a bracket, it is a horizontal shift of ‘-a’ ‘I will now get the same values for y, but with values of x that are 1 less than before’ y = (x + 1) 3 1 When x = 0: y-intercept

15. Sketching Curves Sketching Cubics You need to be able to sketch and interpret cubics that are variations of y = x 3 This will be covered in more detail in C2. You can still plot the graphs in the same way we have seen before. This topic is offering a ‘shortcut’ if you can understand it. 4B Example Sketch the graph of the function: x y y = x 3 y = (3 - x) 3 27 When x = 0: y-intercept Reflected in the x-axis Horizontal shift, 3 to the right 3

16. Sketching Curves The Reciprocal Function You need to be able to sketch the ‘reciprocal’ function. This takes the form: Where ‘k’ is a constant. Example Sketch the graph of the function and its asymptotes. 124-4-2y 10.50.25-0.25-0.5x x y y = 1 / x 4C You cannot divide by 0, so you get no value at this point These are where the graph ‘never reaches’, in this case the axes…

17. Sketching Curves The Reciprocal Function You need to be able to sketch the ‘reciprocal’ function. This takes the form: Where ‘k’ is a constant. Example Sketch the graph of the function and its asymptotes. x y y = 1 / x 4C y = 3 / x The curve will be the same, but further out…

18. Sketching Curves The Reciprocal Function You need to be able to sketch the ‘reciprocal’ function. This takes the form: Where ‘k’ is a constant. Example Sketch the graph of the function and its asymptotes. x y y = 1 / x 4C y = - 1 / x The curve will be the same, but reflected in the x-axis

19. Sketching Curves Solving Equations and Sketching You need to be able to sketch 2 equations on a set of axes, as well as solve equations based on graphs. Example On the same diagram, sketch the following curves: 4D and x y Quadratic ‘U’ shape Crosses through 0 and 3 03 Cubic ‘negative’ shape Crosses through 0 and 1. The ‘0’ is repeated so just ‘touched’ 1

20. Sketching Curves Solving Equations and Sketching You need to be able to sketch 2 equations on a set of axes, as well as solve equations based on graphs. Example On the same diagram, sketch the following curves: 4D and Find the co-ordinates of the points of intersection  These will be where the graphs are equal… x y 031 Expand brackets Group together Factorise

21. Sketching Curves Solving Equations and Sketching You need to be able to sketch 2 equations on a set of axes, as well as solve equations based on graphs. Example On the same diagram, sketch the following curves: 4D and Find the co-ordinates of the points of intersection  These will be where the graphs are equal… Expand brackets Group together Factorise x=-√3x=0x=√3 (0,0)(-√3, 3+3√3)(√3, 3-3√3)

22. Sketching Curves Solving Equations and Sketching You need to be able to sketch 2 equations on a set of axes, as well as solve equations based on graphs. Example On the same diagram, sketch the following curves: 4D and x y Cubic ‘positive’ shape Crosses through 0 and 1. The ‘0’ is repeated. 0 Reciprocal ‘positive’ shape Does not cross any axes 1 y = 2 / x

23. Sketching Curves Solving Equations and Sketching You need to be able to sketch 2 equations on a set of axes, as well as solve equations based on graphs. How does the graph show there are 2 solutions to the equation.. Example On the same diagram, sketch the following curves: 4D and x y 0 1 y = 2 / x Set equations equal, and re- arrange And they cross in 2 places…