Quadratic Graphs (Edexcel A-Level Mathematics): Revision Notes

Quadratic graphs are the graphical representation of quadratic functions, which are polynomial functions of degree $2$. The general form of a quadratic function is: $\\ y = ax^2 + bx + c$

where $\ a\ , \ b\ , and \ c$ are constants, and $\ x$ is the independent variable. The graph of a quadratic function is a curve called a parabola.

To sketch the graph of a quadratic function, follow these steps:

1. Determine the Shape: Identify whether the parabola opens upwards or downwards based on the sign of $\ a.$
2. Find the Vertex: Calculate the $\ x$-coordinate using $x = -\frac{b}{2a},$ then substitute this back into the equation to find the $\ y$-coordinate.
3. Identify the Axis of Symmetry: Draw the axis of symmetry through the vertex.
4. Find the $Y$-Intercept: Set $\ x$= 0 and calculate the$\ y-intercept \ (0, c)$.
5. Find the $X$-Intercepts (if any): Solve $\ ax^2 + bx + c = 0$ using the quadratic formula to find the roots.
6. Plot Additional Points: If needed, choose additional values of $x$ and calculate the corresponding $\ y$ values to help in sketching the graph.
7. Draw the Parabola: Sketch the curve, ensuring it passes through the vertex, intercepts, and follows the direction determined by the sign of$\ a$.

📑 Example Sketch the graph of the quadratic function: $\\ y = 2x^2 - 4x + 1$

Step-by-Step Solution:

8. Shape: Since $\ a = 2 > 0,$ the parabola opens upwards.

9. Vertex:

• Calculate the $\ x$-coordinate:

• Calculate the $y$-coordinate by substituting $\ x$ = 1 into the equation:

• The vertex is $\ (1, -1).$

10. Axis of Symmetry: The axis of symmetry is $\ x = 1$.

11. $Y$-Intercept:

• Set $\ x = 0$:

• The $\ y$-intercept is $\ (0, 1).$

12. $X$-Intercepts:

• Solve $\ 2x^2 - 4x + 1 = 0$ using the quadratic formula:

• The $\ x-$intercepts are at $x = \frac{2 + \sqrt{2}}{2}\ and \ x = \frac{2 - \sqrt{2}}{2}.$

13. Plot Additional Points: For example, if $\ x = 2$:

• $\ y = 2(2)^2 - 4(2) + 1 = 8 - 8 + 1 = 1$
• The point $(2, 1)$ can be plotted.

14. Draw the Parabola: Connect the points smoothly, ensuring the curve passes through the vertex $\ (1, -1),$ the $\ y-intercept \ (0, 1)$, and the $\ x-intercepts \ \left(\frac{2 + \sqrt{2}}{2}, 0\right)$ and $\ \left(\frac{2 - \sqrt{2}}{2}, 0\right).$

• Quadratic graphs are parabolas with important features like the vertex, axis of symmetry, and intercepts.
• The shape of the parabola is determined by the coefficient $\ a,$ and the position of the vertex and intercepts provides a clear understanding of the graph's behaviour.
• To sketch a quadratic graph, calculate key points like the vertex,$\ x$-intercepts, and $\ y-$intercepts, and use these to draw the curve.