Welcome to the World of Quadratics!

Hi there! Welcome to one of the most important chapters in Additional Mathematics. You’ve probably seen quadratic equations before in Elementary Math, but here in Add Math (4049), we go a bit deeper. We aren't just looking for "x"; we are looking at how these functions behave, how to find their highest or lowest points, and even how to predict if they will ever touch the ground!

Quadratic functions are used everywhere—from calculating the path of a basketball throw to designing the curves of a satellite dish. Don't worry if it seems a bit abstract at first; we will break it down step-by-step.

1. The Basics: What is a Quadratic Function?

A quadratic function is usually written in the General Form:
\( y = ax^2 + bx + c \)

The "shape" of the graph depends entirely on the value of \( a \):

  • If \( a > 0 \) (positive), the graph is a "Smiley Face" (U-shape). It has a minimum point.
  • If \( a < 0 \) (negative), the graph is a "Frown" (n-shape). It has a maximum point.

Quick Review: Think of positive as happy (smile) and negative as sad (frown). This simple trick helps you visualize the graph immediately!

2. Finding Max/Min Values: Completing the Square

In Add Math, we often need to find the exact coordinates of the "peak" or the "valley" of the curve. This point is called the Vertex. To find it, we use the method of Completing the Square to turn the General Form into the Vertex Form:

\( y = a(x - h)^2 + k \)

Where \( (h, k) \) is the maximum or minimum point.

Step-by-Step Recipe for Completing the Square:

Let's try an example: \( y = 2x^2 - 8x + 5 \)

  1. Factor out 'a' from the \( x^2 \) and \( x \) terms:
    \( y = 2(x^2 - 4x) + 5 \)
  2. Add and Subtract the square of half the coefficient of \( x \):
    Inside the bracket, take \(-4\), divide by 2 to get \(-2\), then square it to get \( (-2)^2 \).
    \( y = 2[x^2 - 4x + (-2)^2 - (-2)^2] + 5 \)
  3. Form the perfect square:
    \( y = 2[(x - 2)^2 - 4] + 5 \)
  4. Expand and Simplify:
    \( y = 2(x - 2)^2 - 8 + 5 \)
    \( y = 2(x - 2)^2 - 3 \)

Conclusion: The minimum value is \( -3 \) and it occurs when \( x = 2 \). The coordinates are \( (2, -3) \).

Common Mistake: Many students forget to multiply the "subtracted" number by the factor outside the bracket. In the example above, don't forget that the \(-4\) inside the bracket was multiplied by the \( 2 \) outside to become \(-8\)!

Key Takeaway:

The form \( y = a(x - h)^2 + k \) tells you everything: \( k \) is the max/min value, and \( h \) is the x-coordinate where it happens.

3. The Nature of Roots (The Discriminant)

Sometimes we need to know how many times a curve touches the x-axis (or another line) without actually drawing it. We use the Discriminant: \( D = b^2 - 4ac \).

  • If \( b^2 - 4ac > 0 \): The curve cuts the x-axis at two distinct real points.
  • If \( b^2 - 4ac = 0 \): The curve touches the x-axis at one point (it's a tangent). We call this "two equal real roots".
  • If \( b^2 - 4ac < 0 \): The curve never touches the x-axis. There are "no real roots".

Did you know? The word "Discriminant" comes from "discriminate," which means to tell the difference between things. It helps us tell the difference between these three types of graphs!

4. Conditions for "Always Positive" or "Always Negative"

This is a favorite exam topic! Sometimes a question asks for the conditions such that \( ax^2 + bx + c \) is always positive for all real values of \( x \).

Always Positive (The "Floating" Curve):

For a function to be always above the x-axis:

  1. It must be a Smiley Face: \( a > 0 \)
  2. It must never touch the x-axis: \( b^2 - 4ac < 0 \)

Always Negative (The "Drowning" Curve):

For a function to be always below the x-axis:

  1. It must be a Frown: \( a < 0 \)
  2. It must never touch the x-axis: \( b^2 - 4ac < 0 \)

Memory Aid: Notice that in both cases, the discriminant \( b^2 - 4ac \) is LESS than zero. Why? Because "always positive/negative" means the graph never crosses the x-axis!

5. Using Quadratics as Models

In the real world, many things follow a parabolic path. When solving modeling problems, follow these steps:

  1. Identify the variables: Usually, \( y \) is height or profit, and \( x \) is time or distance.
  2. Look for keywords: If the question says "maximum height," they want you to find the vertex (by completing the square).
  3. Interpret the intercepts: The y-intercept is usually the starting point (when \( x = 0 \)). The x-intercepts (roots) are usually when the object hits the ground.

Example: A ball is thrown and its height \( h \) is given by \( h = -5t^2 + 20t + 2 \). To find the maximum height, you would complete the square to find the vertex of this "frowning" curve.

Final Quick Review

  • General Form: \( y = ax^2 + bx + c \)
  • Vertex Form: \( y = a(x - h)^2 + k \). (Completing the square).
  • Max/Min: Look at the sign of \( a \).
  • Intersection/Roots: Use the discriminant \( b^2 - 4ac \).
  • Always Positive: \( a > 0 \) AND \( b^2 - 4ac < 0 \).
  • Always Negative: \( a < 0 \) AND \( b^2 - 4ac < 0 \).

Keep practicing! Quadratic functions are the foundation for much of the Calculus you will learn later. You've got this!