Quadratic Functions Guide

A quadratic function is a polynomial function of degree 2, commonly expressed as $f(x) = ax^2 + bx + c$. Quadratic functions are extensively utilised in various disciplines for modelling real-world phenomena, such as projectile motion in physics or optimisation in economics and engineering.

Definition of Quadratic Functions

• Quadratic Function: A polynomial function of degree 2.
• General form:
  • $f(x) = ax^2 + bx + c$, where $a \neq 0$.
  • Components:
    • $a$: Coefficient influencing the direction and width of the parabola.
    • $b$: Coefficient impacting the vertex's horizontal position.
    • $c$: Constant term indicating the y-intercept.

Key Features of Quadratic Functions

• Parabolas:

  • Quadratic functions' graphs form symmetrical U-shapes.

• Vertex:

  • The parabola's turning point.
  • Formula: $\left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)$

• Axis of Symmetry:

  • The vertical line through the vertex at $x = -\frac{b}{2a}$.

• Roots/Intercepts:

  • Roots: Solutions to $f(x) = 0$.
  • Y-intercept: The point where the function intersects the y-axis when $x = 0$.

Standard Form

• Definition: The standard form is $y = ax^2 + bx + c$. Understanding these coefficients is essential for solving quadratic equations and analysing their graphs.

• Coefficient Analysis:

  • $a$: Determines the parabola's direction and width:
    • Opens upwards if $a > 0$.
    • Opens downwards if $a < 0$.
  • $b$: Alters the horizontal position of the vertex, affecting symmetry.
  • $c$: Denotes where the graph intercepts the y-axis.

Vertex Form

• Definition: Vertex form is $y = a(x-h)^2 + k$, facilitating easy vertex identification at $(h, k)$.

• Conversion Process:

  1. Start with $y = ax^2 + bx + c$.
  2. Factor $a$ from $x^2$ and $x$ terms.
  3. Complete the square within these terms.
  4. Rearrange to fit the vertex form.

• Example:

  • Convert $y = 2x^2 + 8x + 6$:
    1. Factor out $2$: $y = 2(x^2 + 4x) + 6$.
    2. Complete the square: $y = 2(x^2 + 4x + 4 - 4) + 6$ = $y = 2((x+2)^2 - 4) + 6$.
    3. Simplify: $y = 2(x+2)^2 - 8 + 6$ = $y = 2(x+2)^2 - 2$.
    4. Therefore, the vertex form is $y = 2(x+2)^2 - 2$ with vertex at $(-2, -2)$.

Factored Form

• Definition: The factored form $y = a(x-p)(x-q)$ clearly indicates the roots or x-intercepts.
• Applications: Useful for swiftly locating x-intercepts.

Graphical Characteristics

• Orientation and Width:

  • Coefficient $a$ influences orientation:
    • Opens upwards: if $a > 0$.
    • Opens downwards: if $a < 0$.
  • $a$ also impacts width:
    • Larger $|a|$: The parabola is narrower.
    • Smaller $|a|$: The parabola is wider.

• Importance of $a$:

  • It defines both the slope and the direction.

Plotting Parabolas

Step-by-Step Instruction

• Identify Components:

  • Vertex: The point of symmetry on the parabola.
  • Axis of Symmetry: A vertical line dividing the parabola into two equal halves.
  • Y-intercept: Point of intersection with the y-axis, $x=0$.

• Axis of Symmetry Calculation:

  • Determine the line of symmetry using $x = -\frac{b}{2a}$.

• Drawing the Parabola:

  • Compute points near the axis of symmetry to draw the graph.

Example Exercise

• Worked Example: Graph $y = 2x^2 - 4x + 1$:
  • Find the axis of symmetry: $x = -\frac{b}{2a} = -\frac{-4}{2(2)} = \frac{4}{4} = 1$
  • Calculate the vertex: At $x = 1$, $y = 2(1)^2 - 4(1) + 1 = 2 - 4 + 1 = -1$
  • So vertex is at $(1, -1)$
  • Y-intercept: When $x = 0$, $y = 2(0)^2 - 4(0) + 1 = 0 + 0 + 1 = 1$
  • Plot these key points and additional points on either side of the vertex to sketch the parabola.

Solving Quadratic Equations

Factorisation Method

Factorisation: Decompose into simpler binomials.

• Example: Solve $x^2 - 5x + 6 = 0$.
  • Find factors of 6 that sum to -5: -2 and -3
  • Factored form: $(x-2)(x-3) = 0$.
  • By the zero product property: $x-2 = 0$ or $x-3 = 0$
  • Solutions: $x=2$, $x=3$.

Quadratic Formula

Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

• Applicable to all quadratic equations.
• Example: Solve $2x^2 + 4x + 1 = 0$.
  • Identify $a=2$, $b=4$, $c=1$
  • Substitute into formula: $x = \frac{-4 \pm \sqrt{16 - 4(2)(1)}}{2(2)} = \frac{-4 \pm \sqrt{16 - 8}}{4} = \frac{-4 \pm \sqrt{8}}{4}$
  • Simplify: $x = \frac{-4 \pm 2\sqrt{2}}{4} = -1 \pm \frac{\sqrt{2}}{2}$
  • Solutions: $x \approx -0.29$ and $x \approx -1.71$

Completing the Square

Reframes equations to facilitate graph comprehension.

• Example: Solve $x^2 + 6x + 8 = 0$.
  • Rearrange: $x^2 + 6x = -8$
  • Half the coefficient of $x$: $\frac{6}{2} = 3$
  • Square this value: $3^2 = 9$
  • Add and subtract this value: $x^2 + 6x + 9 - 9 = -8$
  • Rewrite as perfect square: $(x+3)^2 - 9 = -8$
  • Solve: $(x+3)^2 = 1$
  • Therefore: $x+3 = \pm 1$
  • Solutions: $x = -3+1 = -2$ or $x = -3-1 = -4$

X-Intercepts from Graphs

Estimate solutions by locating x-intercepts graphically.

• Visualise: The graph of $x^2 - 4 = 0$ crosses at $x = 2$ and $x = -2$.

Discriminant in Quadratic Equations

• Discriminant: Assesses the nature of roots, $\Delta = b^2 - 4ac$.

Scenarios Based on $\Delta$

• $\Delta > 0$: Two distinct real roots.
• $\Delta = 0$: One repeated root; graph touches the x-axis.
• $\Delta < 0$: No real roots, resulting in complex numbers.

Worked Example

• Example 1 ($\Delta > 0$): $2x^2 - 4x + 1 = 0$
  • $\Delta = (-4)^2 - 4(2)(1) = 16 - 8 = 8$
  • Since $\Delta > 0$, there are two distinct real roots.
• Example 2 ($\Delta = 0$): $x^2 - 4x + 4 = 0$
  • $\Delta = (-4)^2 - 4(1)(4) = 16 - 16 = 0$
  • Since $\Delta = 0$, there is one repeated root.
  • The equation can be factored: $(x-2)^2 = 0$
  • Solution: $x = 2$
• Example 3 ($\Delta < 0$): $x^2 + x + 1 = 0$
  • $\Delta = 1^2 - 4(1)(1) = 1 - 4 = -3$
  • Since $\Delta < 0$, there are no real roots, only complex roots.

Graphical Demonstrations

Illustrations depict different root scenarios:

Use of Technology in Quadratic Functions

Graphing Software

• Facilitate real-time graph manipulation.
• Provide insights into parabola characteristics like intercepts and vertices.

Interactive Experimentation

• Modification & Observation: Alter coefficients to observe graph dynamics.

Coding & Spreadsheets

• Enable visualisation of functions and their transformations.

Skills and Benefits

• Encourages the development of analytical skills and showcases real-world applications of quadratic functions.