This lesson covers two distinct but important types of graphs that frequently appear in exams: solving simultaneous equations and analyzing quadratic functions.

1. Solving Simultaneous Equations Graphically

This involves drawing two straight lines on a graph and finding where they cross.

The Core Idea: The solution to a pair of simultaneous equations is the (x, y) coordinate of the point where their graphs intersect.

How to Solve Graphically

1. Rearrange the Equations: Make 'y' the subject in both equations (i.e., put them in the form y = mx + c).

2. Create Tables of Values: For each equation, choose three simple x-values (like 0, 1, 2) and calculate the corresponding y-values. Three points are enough to draw a straight line.

3. Draw the Axes and Plot Points: Choose a suitable scale for your x and y axes. Carefully plot the points from your tables for both equations.

4. Draw the Lines: Use a ruler to draw a straight line through the three points for each equation. Extend the lines until they intersect.

5. Find the Solution: Write down the coordinates (x, y) of the intersection point. This is your answer.

Exam Tip: Always verify your answer by substituting the x and y values back into the original equations to see if they hold true. This can save you from losing marks due to a small plotting error.

2. Graphs of Quadratic Functions (Parabolas)

This is a major part of the exam paper. You will be given a function like y = ax² + bx + c, asked to complete a table of values, draw the graph, and answer several questions about its properties.

The Shape: The graph is a smooth curve called a parabola.

• If a is positive (e.g., y = x²...), the parabola opens upwards (like a 'U') and has a minimum point.

• If a is negative (e.g., y = -x²...), the parabola opens downwards (like an 'n') and has a maximum point.

Key Features to Find from the Graph

• Maximum/Minimum Value: The y-coordinate of the turning point.

• Coordinates of the Turning Point: The coordinates (x, y) of the highest or lowest point on the curve.

• Equation of the Axis of Symmetry: The vertical line that divides the parabola into two mirror images. The equation is always x = (the x-coordinate of the turning point).

• Roots of the Equation ax² + bx + c = 0: These are the x-coordinates of the points where the graph intersects the x-axis.

• Interval of values for x where the function is positive (y > 0): The range of x-values where the curve is above the x-axis.

• Interval of values for x where the function is negative (y < 0): The range of x-values where the curve is below the x-axis.

• Interval of values for x where the function is increasing/decreasing:

  • Increasing: As you move from left to right along the x-axis, the part of the curve that is going upwards.

  • Decreasing: As you move from left to right, the part of the curve that is going downwards.

3. Special Forms (Finding Properties Without Drawing)

Sometimes you need to find properties just from the equation.

1. If the function is in the form y = ±(x + b)² + c:

   • The coordinates of the turning point are (-b, c). (Note the sign change for b!).

   • The axis of symmetry is x = -b.

   • The maximum/minimum value is c.

2. If the function is in the form y = ±(x + a)(x + b):

   • The roots are where y=0, so the roots are x = -a and x = -b.

   • The axis of symmetry is exactly halfway between the roots: x = (-a - b) / 2.

Exam Tips & Common Mistakes

• Mistake 1: Joining points with a ruler! A parabola is a smooth curve. Never use a ruler to connect the plotted points. Draw a freehand curve through them.

• Mistake 2: Not reading the scale. Always check the scale on the x-axis and y-axis before reading coordinates.

• Mistake 3: Giving the wrong coordinate. When asked for the "minimum value", give the y-coordinate. When asked for "roots", give the x-coordinates.

• Exam Tip: For the best curve, plot all the points given in the table. Use a sharp pencil. If your points don't form a smooth U-shape, double-check your calculations in the table – you've likely made an error there.