EDEXCEL GCSE FOUNDATION MATHS - What are Quadratic Graphs

Edexcel GCSE Foundation Maths: What are Quadratic Graphs?

Introduction

Quadratic graphs are the visual representations of quadratic equations. A quadratic equation is an equation with a highest power of 2, like x² + 2x - 3 = 0. These graphs have a distinctive U-shape or upside-down U-shape, known as a parabola.

Key Features of Quadratic Graphs

1. Shape: They always have a parabolic shape, either opening upwards or downwards.

2. Turning Point: This is the highest or lowest point on the graph, also known as the vertex. The turning point determines whether the graph opens upwards or downwards.

3. Axis of Symmetry: A vertical line that divides the graph into two symmetrical halves. It passes through the turning point.

4. Roots: These are the points where the graph intersects the x-axis. They are also called solutions or x-intercepts. A quadratic equation can have two, one, or no roots.

Understanding the Equation

The standard form of a quadratic equation is y = ax² + bx + c.

• 'a' determines the shape:
  • If 'a' is positive, the parabola opens upwards.
  • If 'a' is negative, the parabola opens downwards.
  • The larger the absolute value of 'a', the narrower the parabola.
• 'b' affects the position of the turning point:
  • A larger 'b' value shifts the turning point to the left or right.
• 'c' is the y-intercept:
  • This is the point where the graph crosses the y-axis.

Drawing Quadratic Graphs

1. Find the y-intercept: This is simply the value of 'c'. Plot this point on the y-axis.

2. Find the turning point: The x-coordinate of the turning point is given by x = -b/2a. Substitute this value into the equation to find the y-coordinate.

3. Find the roots (if any): You can use the quadratic formula or factorization to find the roots.

4. Plot the points: Plot the y-intercept, turning point, and any roots you found.

5. Draw the curve: Sketch a smooth, symmetrical parabola that passes through all the points you plotted.

Examples

Example 1: y = x² + 2x - 3

• y-intercept: c = -3
• Turning point: x = -2/2(1) = -1. Substitute x = -1 into the equation to get y = -4.
• Roots: Using factorization, we find (x+3)(x-1) = 0, so x = -3 and x = 1.

Example 2: y = -x² + 4

• y-intercept: c = 4
• Turning point: x = 0/2(-1) = 0. Substitute x = 0 into the equation to get y = 4.
• Roots: There are no real roots in this case because the parabola opens downwards and its vertex is above the x-axis.

Applications

Quadratic graphs have various applications in real-world scenarios, including:

• Physics: Modeling projectile motion.
• Engineering: Designing bridges and buildings.
• Economics: Predicting market trends.

Conclusion

Understanding quadratic graphs is crucial for success in Edexcel GCSE Foundation Maths. By grasping the key features, the standard equation, and methods of drawing them, you'll be able to analyze and interpret these graphs effectively. Remember to practice drawing various examples to solidify your understanding.