Using Graphs | Mathematics for GCSE/IGCSE - Year 11 PDF Download

Solving Equations Using Graphs

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) intersects the x-axis.
• If asked to use the graph of y = f(x) to solve a different equation, the question will indicate to do so by drawing a suitable straight line.
• Rearrange the equation to be solved into the form f(x) = mx + c and draw the line y = mx + c.
• Solutions are the x-coordinates where the line y = mx + c intersects the curve y = f(x).
• For example, given the curve for y = x³ + 2x² + 1 and asked to solve x³ + 2x² − x − 1 = 0;
  • Rearrange x³ + 2x² − x − 1 = 0 to x³ + 2x² + 1 = x + 2.
  • Draw the line y = x + 2 on the graph of y = x³ + 2x² + 1.
  • Read the x-values where the line and the curve intersect. In this case, there are three 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 the straight-line forms y = mx + c.
• Find where the lines intersect (cross).
• The solutions to the simultaneous equations are the x and y coordinates of the intersection point.
• For example, to solve 2x − y = 3 and 3x + y = 4 simultaneously:
  • First, plot both equations on the graph.
  • Identify the point of intersection, which in this case is (2, 1).
  • The solution is x = 2 and y = 1.

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

• For example, to solve y = x² + 4x − 12 and y = 1 simultaneously:
  • First, plot both equations on the same graph.
• Identify the two points of intersection by reading off the scale. In this case, they are approximately (−6.1,1) and (−2.1,1)(−2.1,1) to one decimal place.
• The solutions from the graph are approximately x = −6.1 and y = 1, and x = −2.1 and y = 1.
• Note that there are two pairs of x, y solutions.
• To find exact solutions, use algebra.

Finding Gradients of Tangents

• The gradient of a graph at any point is equal to the gradient of the tangent to the curve at that point.
• A tangent is a line that just touches a curve without crossing it.

How do I estimate the gradient under a graph?

• To estimate the gradient:
  • Draw a tangent to the curve.
  • Calculate the gradient of the tangent using the formula: Gradient = Rise ÷ Run.

• In the example above, the gradient at x = 4 would be
• It's important to note that when we visually estimate the gradient by drawing a tangent, it is not exact. 
  • To find the precise gradient, we would need to use differentiation.

What does the gradient represent?

• In a y-x graph, the gradient signifies the rate of change of y concerning x. 
• This concept finds applications in various real-world scenarios.
  • For instance, in a distance-time graph, the gradient represents the rate of change of distance with respect to time, which equates to speed.
  • Similarly, in a speed-time graph, the gradient corresponds to the rate of change of speed with respect to time, which denotes acceleration.