Graphs Of Equations

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Graphs Of Equations

  1. 1. Graphs of Equations<br />Mathematics Chapter 8 and 9<br />
  2. 2. Linear Equations<br />Graphs of Equations<br />y<br />x<br />
  3. 3. Axes of x and y<br />The x-axis is the horizontal line of a graph.<br />Positive values of x lie after the y-axis while negative values of x lie before the y-axis.<br />The y-axis is the vertical line of a graph.<br />Positive values of y lie after the x-axis while negative values of y lie before the x-axis.<br />
  4. 4. Graphs of y = c<br />The lines of y are horizontal and x = 0.<br /> y<br />y = 7<br />y = 4<br />x<br />
  5. 5. Graphs of x = a<br />The lines of x are vertical and y = 0.<br /> y<br />x = 3<br />x = 5<br />x<br />
  6. 6. Graphs of y = mx<br />Example : y = 3x<br /> y<br />y = 6<br />y = 3<br />x<br />x = 2<br />x = 1<br />
  7. 7. Graphs of y = mx + c<br />Example : y = 2x + 3<br /> y<br />y = 9<br />y = 7<br />y = 5<br />y = 0<br />x<br />x = 1 / 2 / 3<br />x = -1.5<br />
  8. 8. Using graphs in Simultaneous Linear Equations<br />Example : y = 2x + 3, y = -x + 4<br /> y<br />5<br />3<br />x<br />0<br />1<br />1/3 , 32/3<br />Answer : x = 1/3and y = 3 2/3 <br />
  9. 9. Simultaneous Linear Equations with no solution<br />Lines are parallel<br />2x - 2y = 4<br /> y<br />x - y = 1<br />x<br />
  10. 10. Simultaneous Linear Equations with infinite solutions<br />Lines are identical<br /> y<br />x – y = 2<br />2x – 2y = 4<br />x<br />
  11. 11. Length of Line Segments<br /> x2 y2– x1 y1 = (x2 – x1) 2 + (y2 – y1) 2 <br /> y<br />(By Pythagoras Theorem)<br />y 2<br />y 2 - y 1<br />y1<br />x 2 - x 1<br />x<br />x2<br />x1<br />
  12. 12. Gradient of Straight line<br /> rise y2 – y1<br /> run x2 – x1<br /><ul><li>Gradient = = </li></ul> y<br />y 2<br />y 2 - y 1<br />y1<br />x 2 - x 1<br />x<br />x2<br />x1<br />
  13. 13. Equations by intercept of y = mx<br />Gradient = m<br />y – c<br />x – 0<br /> y<br />= m<br />B (x, y)<br />y – c = mx<br />y = mx + c<br />A (0, c)<br />x<br />
  14. 14. Equations by intercept of y = mx<br />Find the gradient of A (4, 5) and B (5, 8)<br />Complete the equation with the gradient<br />y = 3x + c<br />Fill in x and y of one of the two points<br />5 = 3(4) + c<br />Work out the value of c<br />c = -7<br />Complete the equation with c<br />y = 3x – 7 <br />8 – 5<br />5 – 4 <br />= 3<br />
  15. 15. Quadratic Equations<br />Graphs of Equations<br />y<br />x<br />
  16. 16. Basic Concepts (y = ax2+ bx – c)<br />When a increases, the line bends nearer towards the y-axis while maintaining the M point.<br />When a is negative, the paranoma curves downwards (vice versa).<br />When c increases, the y-intercept increases without curve change.<br />When c is negative, only the y-intercept changes.<br />When b increases, the M point moves further away from the x-axis while maintaining a y-intercept of c.<br />When b is negative, the M point lies to the right of the y-axis (vice versa).<br />
  17. 17. Finding Equation using x-axis points (u)<br />Identify minimum point below x axis<br />Find x with coordinates on x axis (y = 0)<br />A (-8, 0) and B (2, 0)<br />Find the factorisation of the equation (y +ve)<br />y = (x + 8)(x – 2)<br />Expand the equation<br />y = x2 + 6x – 16<br />A (-8, 0)<br />B (2, 0)<br />
  18. 18. Finding Equation using x-axis points (n) <br />Identify maximum point above x axis<br />Find x with coordinates on x axis (y = 0)<br />A (-8, 0) and B (2, 0)<br />Find the factorisation of the equation (y –ve)<br />-y = (x + 8)(x – 2)<br />Expand the equation<br />-y = x2 + 6x – 16<br />y = -x2 - 6x + 16<br />A (-8, 0)<br />B (2, 0)<br />
  19. 19. Finding Equation using minimum point (u)<br />Identify minimum point above x axis<br />Find minimum point at y and find x<br />y = 2<br />x = - 4<br />Find equation on x axis (y +ve)<br />y = (x + 4) 2<br />y = x2 + 8x + 16<br />Derive actual equation<br />Add the minimum point<br />y = x2 + 8x + 16 + 2<br />y = x2 + 8x + 18<br />-4, 2<br />
  20. 20. Finding Equation using maximum point (u)<br />Identify maximum point below x axis<br />Find maximum point at y and find x<br />y = - 2<br />x = - 4<br />Find equation on x axis (y –ve)<br />-y = (x + 4) 2<br />y = -x2 – 8x – 16<br />Derive actual equation<br />Add the maximum point<br />y = -x2 – 8x – 16 – 2<br />y = -x2 – 8x – 18<br /> -4, -2<br />
  21. 21. Finding a min/max point using x-axis points <br />Find the x value of the midpoint of A and B<br />x = = -1<br />Substitute x = -1 into the equation<br />y = (x + 3)(x – 1)<br />y = (2)(-2)<br />y = -4<br />Derive the minimum point<br />M = (-1, -4)<br /> -3 + 1<br /> 2<br />A (-3, 0)<br />B (1, 0)<br />M<br />
  22. 22. Finding a min/max point using the equation <br />Example Equation : y = x2+ 2x - 3<br />Find the eqn. of line of symmetry : x= -<br />x = - = -1<br />Find y by substitution : y = (-1) 2 + 2(-1) - 3 = -4<br />Minimum point = (-1, -4)<br />b<br /> 2a<br />2<br />2 (1)<br />A (-3, 0)<br />B (1, 0)<br />M<br />
  23. 23. END<br />

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