Functions (Grade 11 NSC Matric Mathematics): Revision Notes

Quadratic Functions

A quadratic function is a type of polynomial function that creates a U-shaped curve called a parabola. These functions are essential in many areas of mathematics and have distinctive characteristics that make them easy to identify and work with.

Basic form: y = ax² + q

The simplest form of a quadratic function is $y = ax^2 + q$, where $a$ and $q$ are constants.

Effects of parameter a

The value of a determines the shape and direction of the parabola:

• When a > 0: The parabola opens upward (smiles) and has a minimum turning point

  • As $|a|$ increases, the parabola becomes narrower
  • As $|a|$ approaches 0, the parabola becomes wider

• When a < 0: The parabola opens downward (frowns) and has a maximum turning point

  • As $|a|$ increases, the parabola becomes narrower
  • As $|a|$ approaches 0, the parabola becomes wider

Effects of parameter q

The value of q causes a vertical shift of the parabola:

• When q > 0: The parabola shifts upward by q units
• When q < 0: The parabola shifts downward by |q| units
• q is also the y-intercept of the parabola

General form: y = a(x + p)² + q

The complete form of a quadratic function is $y = a(x + p)^2 + q$, where $a$, $p$, and $q$ are constants.

Effects of parameter p

The value of p causes a horizontal shift of the parabola:

• When p > 0: The parabola shifts to the left by p units
• When p < 0: The parabola shifts to the right by |p| units

Combined effects

The parameters work together to transform the basic parabola $y = x^2$:

• a: Controls width and direction (up or down opening)
• p: Controls horizontal position (left or right shift)
• q: Controls vertical position (up or down shift)

Key characteristics

Understanding the key characteristics of quadratic functions allows you to analyze and sketch them effectively.

Domain and range

For any quadratic function $f(x) = a(x + p)^2 + q$:

• Domain: $\{x : x \in \mathbb{R}\}$ (all real numbers)

  • There is no value of x for which the function is undefined

• Range: Depends on the value of a

  • If $a > 0$: Range is $\{y : y \geq q, y \in \mathbb{R}\}$ or $[q; \infty)$
  • If $a < 0$: Range is $\{y : y \leq q, y \in \mathbb{R}\}$ or $(-\infty; q]$

Intercepts

Y-intercept

To find the y-intercept, substitute $x = 0$ into the equation.

For example, for $g(x) = (x - 1)^2 + 5$:

• $g(0) = (0 - 1)^2 + 5 = 1 + 5 = 6$
• The y-intercept is $(0; 6)$.

X-intercepts

To find the x-intercepts, substitute $y = 0$ and solve for x.

For example, for $g(x) = (x - 1)^2 + 5$:

• $0 = (x - 1)^2 + 5$
• $(x - 1)^2 = -5$

Since this equation has no real solutions, the graph lies above the x-axis and has no x-intercepts.

Turning point

The turning point (or vertex) of $f(x) = a(x + p)^2 + q$ is at $(-p; q)$.

• If $a > 0$: The turning point is a minimum with range $[q; \infty)$
• If $a < 0$: The turning point is a maximum with range $(-\infty; q]$

Converting to vertex form

To find the turning point when the function is in standard form $y = ax^2 + bx + c$, use completing the square:

Axis of symmetry

The axis of symmetry for $f(x) = a(x + p)^2 + q$ is the vertical line $x = -p$.

Sketching quadratic functions

To sketch graphs of the form $f(x) = a(x + p)^2 + q$, determine these five characteristics:

1. Sign of a (determines if parabola opens up or down)
2. Turning point $(-p; q)$
3. Y-intercept (substitute $x = 0$)
4. X-intercepts (substitute $y = 0$ and solve)
5. Domain and range

Transformations and shifts

Understanding how to apply transformations to quadratic functions is essential for working with more complex problems.

Horizontal shifts

• Right shift by m units: Replace $x$ with $(x - m)$
• Left shift by m units: Replace $x$ with $(x + m)$

Vertical shifts

• Upward shift by n units: Replace $y$ with $(y - n)$
• Downward shift by n units: Replace $y$ with $(y + n)$

Finding equations from graphs

When you need to find the equation of a quadratic function from its graph, you can use different approaches depending on what information is given.

When x-intercepts are given

Use $y = a(x - x_1)(x - x_2)$ where $x_1$ and $x_2$ are the x-intercepts.

When turning point is given

Use $y = a(x + p)^2 + q$ where $(-p; q)$ is the turning point.