1. Graphs of Equations Mathematics Chapter 8 and 9

2. Linear Equations Graphs of Equations y x

3. Axes of x and y The x-axis is the horizontal line of a graph. Positive values of x lie after the y-axis while negative values of x lie before the y-axis. The y-axis is the vertical line of a graph. Positive values of y lie after the x-axis while negative values of y lie before the x-axis.

4. Graphs of y = c The lines of y are horizontal and x = 0. y y = 7 y = 4 x

5. Graphs of x = a The lines of x are vertical and y = 0. y x = 3 x = 5 x

6. Graphs of y = mx Example : y = 3x y y = 6 y = 3 x x = 2 x = 1

7. Graphs of y = mx + c Example : y = 2x + 3 y y = 9 y = 7 y = 5 y = 0 x x = 1 / 2 / 3 x = -1.5

8. Using graphs in Simultaneous Linear Equations Example : y = 2x + 3, y = -x + 4 y 5 3 x 0 1 1/3 , 32/3 Answer : x = 1/3and y = 3 2/3

9. Simultaneous Linear Equations with no solution Lines are parallel 2x - 2y = 4 y x - y = 1 x

10. Simultaneous Linear Equations with infinite solutions Lines are identical y x – y = 2 2x – 2y = 4 x

11. Length of Line Segments x2 y2– x1 y1 = (x2 – x1) 2 + (y2 – y1) 2 y (By Pythagoras Theorem) y 2 y 2 - y 1 y1 x 2 - x 1 x x2 x1

13. Equations by intercept of y = mx Gradient = m y – c x – 0 y = m B (x, y) y – c = mx y = mx + c A (0, c) x

14. Equations by intercept of y = mx Find the gradient of A (4, 5) and B (5, 8) Complete the equation with the gradient y = 3x + c Fill in x and y of one of the two points 5 = 3(4) + c Work out the value of c c = -7 Complete the equation with c y = 3x – 7 8 – 5 5 – 4 = 3

15. Quadratic Equations Graphs of Equations y x

16. Basic Concepts (y = ax2+ bx – c) When a increases, the line bends nearer towards the y-axis while maintaining the M point. When a is negative, the paranoma curves downwards (vice versa). When c increases, the y-intercept increases without curve change. When c is negative, only the y-intercept changes. When b increases, the M point moves further away from the x-axis while maintaining a y-intercept of c. When b is negative, the M point lies to the right of the y-axis (vice versa).

17. Finding Equation using x-axis points (u) Identify minimum point below x axis Find x with coordinates on x axis (y = 0) A (-8, 0) and B (2, 0) Find the factorisation of the equation (y +ve) y = (x + 8)(x – 2) Expand the equation y = x2 + 6x – 16 A (-8, 0) B (2, 0)

18. Finding Equation using x-axis points (n) Identify maximum point above x axis Find x with coordinates on x axis (y = 0) A (-8, 0) and B (2, 0) Find the factorisation of the equation (y –ve) -y = (x + 8)(x – 2) Expand the equation -y = x2 + 6x – 16 y = -x2 - 6x + 16 A (-8, 0) B (2, 0)

19. Finding Equation using minimum point (u) Identify minimum point above x axis Find minimum point at y and find x y = 2 x = - 4 Find equation on x axis (y +ve) y = (x + 4) 2 y = x2 + 8x + 16 Derive actual equation Add the minimum point y = x2 + 8x + 16 + 2 y = x2 + 8x + 18 -4, 2

20. Finding Equation using maximum point (u) Identify maximum point below x axis Find maximum point at y and find x y = - 2 x = - 4 Find equation on x axis (y –ve) -y = (x + 4) 2 y = -x2 – 8x – 16 Derive actual equation Add the maximum point y = -x2 – 8x – 16 – 2 y = -x2 – 8x – 18 -4, -2

21. Finding a min/max point using x-axis points Find the x value of the midpoint of A and B x = = -1 Substitute x = -1 into the equation y = (x + 3)(x – 1) y = (2)(-2) y = -4 Derive the minimum point M = (-1, -4) -3 + 1 2 A (-3, 0) B (1, 0) M

22. Finding a min/max point using the equation Example Equation : y = x2+ 2x - 3 Find the eqn. of line of symmetry : x= - x = - = -1 Find y by substitution : y = (-1) 2 + 2(-1) - 3 = -4 Minimum point = (-1, -4) b 2a 2 2 (1) A (-3, 0) B (1, 0) M

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