Using Graphs (Cambridge O Level Maths)

How do we use graphs to solve equations?

• Solutions are always read off the x-axis
• Solutions of f(x) = 0 are where the graph of y = f(x) crosses the x-axis
• If asked to use the graph of y = f(x) to solve a different equation (the question will say something like “by drawing a suitable straight line”) then:
  • Rearrange the equation to be solved into f(x) = mx + c and draw the line y = mx + c
  • Solutions are the x-coordinates of where the line (y = mx + c) crosses the curve (y = f(x))

  • E.g. if given the curve for y = x³ + 2x² + 1 and asked to solve x³ + 2x²  − x − 1 = 0, then;

    1. rearrange x³ + 2x²  − x − 1 = 0 to x³ + 2x² + 1 = x + 2
    2. draw the line y = x + 2 on the curve y = x³ + 2x² + 1
    3. read the x-values of where the line and the curve cross (in this case there would be 3 solutions, approximately x = -2.2, x = -0.6 and x = 0.8);

• Note that solutions may also be called roots

How do we use graphs to solve linear simultaneous equations?

• Plot both equations on the same set of axes using straight line graphs y = mx + c
• Find where the lines intersect (cross)
  • The x and y solutions to the simultaneous equations are the x and y coordinates of the point of intersection
• e.g. to solve 2x - y = 3 and 3x + y = 4 simultaneously, first plot them both (see graph)
  
  • find the point of intersection, (2, 1)
  • the solution is x = 2 and y = 1

How do we use graphs to solve simultaneous equations where one is quadratic?

• e.g. to solve y = x² + 4x − 12 and y = 1 simultaneously, first plot them both (see graph)
  
  • find the two points of intersection (by reading off your scale), (-6.1 , 1) and (-2.1, 1) to 1 decimal place
  • the solutions from the graph are approximately x = -6.1 and y = 1 and x = 2.1 and y = 1
    • note their are two pairs of x, y solutions 
    • to find exact solutions, use algebra

What is the gradient of a graph?

• The gradient of a graph at any point is equal to the gradient of the tangent to the curve at that point
• Remember that a tangent is a line that just touches a curve (and doesn’t cross it)

How do I estimate the gradient under a graph?

• To find an estimate for the gradient:
  • Draw a tangent to the curve
  • Find the gradient of the tangent using Gradient = RISE ÷ RUN

  

  • • In the example above, the gradient at x = 4 would be 
• It is an estimate because the tangent has been drawn by eye and is not exact
  • (To find the exact gradient we would need to use differentiation)

What does the gradient represent?

• In a y-x graph, the gradient represents the rate of change of y against x
• This has many practical applications, for example;
  • in a distance-time graph, the gradient (rate of change of distance against time) is the speed
  • in a speed-time graph, the gradient (rate of change of speed against time) is the acceleration