Quadratics and Graphing Study Notes

Quadratic functions and their graphs are a fundamental topic in GCSE Maths. They describe curves known as parabolas, which have many applications in real-world scenarios, such as the trajectory of a thrown ball or the design of satellite dishes.

What is a Quadratic Expression/Equation? 🧐

A quadratic expression is a mathematical expression where the highest power of the variable (usually \( x \)) is 2. A quadratic equation sets a quadratic expression equal to a value, often zero.

• The general form of a quadratic expression is \( ax^2 + bx + c \), where \( a, b, \) and \( c \) are constants, and \( a \neq 0 \).
• The general form of a quadratic equation is \( ax^2 + bx + c = 0 \).

Example Quadratic Expressions:

• \( x^2 + 5x + 6 \) (Here, \( a=1, b=5, c=6 \))
• \( 2x^2 - 3x + 1 \) (Here, \( a=2, b=-3, c=1 \))
• \( 4x^2 - 9 \) (Here, \( a=4, b=0, c=-9 \))

Graphing Quadratic Functions (Parabolas) 📈

When you plot a quadratic function on a coordinate plane, the graph always forms a U-shaped or n-shaped curve called a parabola. The function is typically written as \( y = ax^2 + bx + c \).

Creating a Table of Values 📊

To graph a quadratic function, you first need to create a table of values:

1. Choose a range of \( x \)-values (e.g., from -3 to 3).
2. Substitute each \( x \)-value into the quadratic equation to find the corresponding \( y \)-value.
3. Record the \( (x, y) \) coordinate pairs.

Example: For \( y = x^2 - 2x - 3 \)

Let's calculate some \( y \) values:

• If \( x = -2 \): \( y = (-2)^2 - 2(-2) - 3 = 4 + 4 - 3 = 5 \)
• If \( x = 0 \): \( y = (0)^2 - 2(0) - 3 = 0 - 0 - 3 = -3 \)
• If \( x = 4 \): \( y = (4)^2 - 2(4) - 3 = 16 - 8 - 3 = 5 \)

A full table might look like this:

\( x \)	\( x^2 \)	\( -2x \)	\( -3 \)	\( y \)
-2	4	4	-3	5
-1	1	2	-3	0
0	0	0	-3	-3
1	1	-2	-3	-4
2	4	-4	-3	-3
3	9	-6	-3	0
4	16	-8	-3	5

Plotting Points and Drawing the Curve ✏️

1. Plot all the \( (x, y) \) coordinate pairs from your table on a graph.
2. Carefully draw a smooth curve connecting the points. Do NOT use a ruler; parabolas are curved.
3. Extend the curve slightly beyond your plotted points with arrows to show it continues.

Key Features of a Parabola 📌

Every parabola has distinct features that are important to identify:

• Turning Point (Vertex): This is the lowest point (minimum) of a U-shaped parabola or the highest point (maximum) of an n-shaped parabola. It's where the curve changes direction.
• Line of Symmetry: A vertical line that passes through the turning point, dividing the parabola into two mirror-image halves. Its equation is always \( x = \text{a constant} \).
• y-intercept: The point where the parabola crosses the y-axis. This occurs when \( x=0 \). For \( y = ax^2 + bx + c \), the y-intercept is always \( (0, c) \).
• Roots (x-intercepts): The points where the parabola crosses the x-axis. These are the \( x \)-values where \( y=0 \). The roots are the solutions to the quadratic equation \( ax^2 + bx + c = 0 \). A parabola can have two, one, or no real roots.

Solving Quadratic Equations by Graphing 🔍

One way to find the solutions (roots) of a quadratic equation is by looking at its graph.

1. To solve \( ax^2 + bx + c = 0 \): Find the \( x \)-coordinates where the parabola \( y = ax^2 + bx + c \) crosses the x-axis (where \( y=0 \)). These are your solutions.
2. To solve \( ax^2 + bx + c = k \) (where \( k \) is a constant):
   • Draw the graph of \( y = ax^2 + bx + c \).
   • Draw the horizontal line \( y = k \).
   • The \( x \)-coordinates of the points where the parabola and the line intersect are the solutions to the equation.

Solutions found by graphing are often approximate, especially if the roots are not integers.

Solving Quadratic Equations by Factorising (Simple Cases) ✅

For some quadratic equations, particularly those where \( a=1 \), you can find the solutions by factorising the expression.

To solve \( x^2 + bx + c = 0 \):

1. Find two numbers that multiply to give \( c \) and add to give \( b \). Let these numbers be \( p \) and \( q \).
2. Rewrite the quadratic equation in factored form: \( (x+p)(x+q) = 0 \).
3. For the product of two factors to be zero, at least one of the factors must be zero. So, set each factor equal to zero and solve for \( x \):
   • \( x+p = 0 \implies x = -p \)
   • \( x+q = 0 \implies x = -q \)

Example: Solve \( x^2 + 5x + 6 = 0 \)

We need two numbers that multiply to 6 and add to 5. These numbers are 2 and 3.

So, the equation factorises to \( (x+2)(x+3) = 0 \).

Setting each factor to zero:

• \( x+2 = 0 \implies x = -2 \)
• \( x+3 = 0 \implies x = -3 \)

The solutions are \( x = -2 \) and \( x = -3 \).

Identifying Quadratic Graphs from Equations 💡

You can tell some things about a parabola just by looking at its equation \( y = ax^2 + bx + c \):

• Shape:
  • If \( a > 0 \) (positive), the parabola is U-shaped (opens upwards). It has a minimum turning point.
  • If \( a < 0 \) (negative), the parabola is n-shaped (opens downwards). It has a maximum turning point.
• y-intercept: The value of \( c \) tells you where the graph crosses the y-axis. The y-intercept is always \( (0, c) \).