2.2.1 Quadratic Graphs

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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.

1. Key Features of Quadratic Graphs

a) Shape of the Parabola

Upward Opening: If $\ a > 0$ , the parabola opens upwards, resembling a " $U$ " shape.
Downward Opening: If $\ a < 0$ , the parabola opens downwards, resembling an upside-down " $U$ ."

b) Vertex

The vertex of the parabola is the point where the graph changes direction. It represents either the minimum point (if $\ a > 0$ ) or the maximum point (if $\ a < 0)$ of the function.

The $\ x$ -coordinate of the vertex can be found using the formula: $\\ x = -\frac{b}{2a}$
The $\ y$ -coordinate of the vertex is found by substituting the $\ x$ -coordinate into the quadratic equation: $\\ y = a\left(-\frac{b}{2a}\right)^2 + b\left(-\frac{b}{2a}\right) + c$

Simplifying this gives the vertex $\ \left(-\frac{b}{2a}, \text{value of } y\right).$

c) Axis of Symmetry

The parabola is symmetric about a vertical line called the axis of symmetry, which passes through the vertex. The equation of the axis of symmetry is: $\\ x = -\frac{b}{2a}$

d) Y-Intercept

The $\ y$ -intercept is the point where the graph crosses the $\ y$ -axis. This occurs when $\ x$ = 0: $\\ y = c$

So, the $\ y$ -intercept is $\ (0, c)$ .

e) X-Intercepts (Roots)

The $x$ -intercepts (also called the roots or zeros of the function) are the points where the graph crosses the $\ x$ -axis. These occur when $\ y$ = 0. To find the $\ x$ -intercepts, solve the quadratic equation: $\\ ax^2 + bx + c = 0$

The solutions can be found using the quadratic formula: $\\ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

If $\ b^2 - 4ac > 0$ , there are two distinct real roots, and the parabola crosses the $\ x$ -axis at two points.
If $\ b^2 - 4ac = 0$ , there is one real root (a repeated root), and the parabola touches the $x$ -axis at one point (the vertex).
If $\ b^2 - 4ac < 0$ , there are no real roots, and the parabola does not cross the $x-axis.$

2. Sketching a Quadratic Graph

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

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

3. Example

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

Step-by-Step Solution:

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

Vertex:

Calculate the $\ x$ -coordinate:

\\ x = -\frac{-4}{2(2)} = \frac{4}{4} = 1

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

\\ y = 2(1)^2 - 4(1) + 1 = 2 - 4 + 1 = -1

The vertex is $\ (1, -1).$

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

$Y$ -Intercept:

Set $\ x = 0$ :

\\ y = 2(0)^2 - 4(0) + 1 = 1

The $\ y$ -intercept is $\ (0, 1).$

$X$ -Intercepts:

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

\\ x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(2)(1)}}{2(2)} = \frac{4 \pm \sqrt{16 - 8}}{4} = \frac{4 \pm \sqrt{8}}{4} = \frac{4 \pm 2\sqrt{2}}{4} = \frac{2 \pm \sqrt{2}}{2}

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

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.

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 $\ \left(\frac{2 - \sqrt{2}}{2}, 0\right).$

4. Summary

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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.

Quadratic Graphs Simplified Revision Notes for A-Level Edexcel Maths Pure

2.2.1 Quadratic Graphs

1. Key Features of Quadratic Graphs

a) Shape of the Parabola

b) Vertex

c) Axis of Symmetry

d) Y-Intercept

e) X-Intercepts (Roots)

2. Sketching a Quadratic Graph

3. Example

4. Summary

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