Quadratic Functions Revision Notes for Grade 10 NSC Matric Mathematics

Quadratic Functions

Introduction to quadratic functions

A quadratic function is any function that can be written in the form $y = ax^2 + q$ , where $a$ and $q$ are constants. These functions create graphs called parabolas, which have a distinctive U-shaped or inverted U-shaped curve.

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The distinctive curved shape of parabolas makes quadratic functions easily recognisable on a graph. This U-shape appears in many real-world contexts, from the path of projectiles to the design of satellite dishes.

The simplest quadratic function is $y = x^2$ . This is our starting point for understanding how quadratic functions behave.

Functions of the form y = x²

When working with $f(x) = x^2$ , we can create a table of values to understand the function's behaviour:

From this table, we can see that the function produces the same output values for positive and negative inputs of the same magnitude. For example, $f(-2) = f(2) = 4$ .

When we plot these points and connect them with a smooth curve, we get our first parabola:

Key characteristics of y = x²

The basic quadratic function $y = x^2$ has several important properties that form the foundation for understanding all quadratic functions:

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Key Properties of y = x²:

Domain: All real numbers, written as $\{x : x \in \mathbb{R}\}$
Range: All non-negative real numbers, written as $\{y : y \geq 0, y \in \mathbb{R}\}$
Turning point: $(0, 0)$ - this is the minimum point
Axis of symmetry: The y-axis (line $x = 0$ )
Shape: Upward-opening parabola (forms a "smile")

Functions of the form y = ax² + q

The general form of a quadratic function is $y = ax^2 + q$ , where:

Parameter a affects the shape and direction of the parabola
Parameter q affects the vertical position of the parabola

The effect of parameter q

The value of $q$ creates a vertical shift of the basic parabola $y = x^2$ .

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Understanding Vertical Shifts:

When $q > 0$ : The graph shifts upward by $q$ units
When $q < 0$ : The graph shifts downward by $|q|$ units
When $q = 0$ : No vertical shift occurs

This means that all points on the parabola move the same distance in the same direction vertically.

The effect of parameter a

The value of $a$ determines both the direction and width of the parabola:

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Understanding Parameter a:

Direction:

When a > 0: The parabola opens upward (forms a "smile") and has a minimum turning point
When a < 0: The parabola opens downward (forms a "frown") and has a maximum turning point

Width: The size of $|a|$ affects the width:

Larger values of $|a|$ make the parabola narrower
Smaller values of $|a|$ make the parabola wider

Key characteristics of quadratic functions

Domain and range

Domain: For all quadratic functions of the form $y = ax^2 + q$ , the domain is $\{x : x \in \mathbb{R}\}$ because there are no restrictions on the x-values.

Range: The range depends on the values of $a$ and $q$ :

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Determining the Range:

If $a > 0$ : The range is $[q; \infty)$ (the parabola opens upward with minimum value $q$ )
If $a < 0$ : The range is $(-\infty; q]$ (the parabola opens downward with maximum value $q$ )

Intercepts

Understanding where a quadratic function intersects the axes is crucial for graphing and analysis.

Y-intercept: To find where the graph crosses the y-axis, substitute $x = 0$ into the equation: $y = a(0)^2 + q = q$

So the y-intercept is always at the point (0, q).

X-intercepts: To find where the graph crosses the x-axis, substitute $y = 0$ and solve for $x$ : $0 = ax^2 + q$ $ax^2 = -q$ $x^2 = -\frac{q}{a}$

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For real x-intercepts to exist, $-\frac{q}{a} \geq 0$ , which means $a$ and $q$ must have opposite signs. If $a$ and $q$ have the same sign, there are no real x-intercepts.

Turning points

The turning point (also called the vertex) of any quadratic function $y = ax^2 + q$ is at $(0, q)$ .

If $a > 0$ : This is a minimum point
If $a < 0$ : This is a maximum point

Axis of symmetry

For quadratic functions of the form $y = ax^2 + q$ , the axis of symmetry is always the y-axis, or the line $x = 0$ .

Sketching quadratic graphs

To sketch the graph of a quadratic function $y = ax^2 + q$ , follow these systematic steps:

Identify the sign of a (determines if parabola opens up or down)
Find the y-intercept (substitute $x = 0$ )
Find the x-intercepts (substitute $y = 0$ and solve)
Identify the turning point (always at $(0, q)$ )
Plot the points and sketch the curve

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Worked Example 1: Sketching y = 2x² - 4

Step 1: Examine the equation

$a = 2 > 0$ , so the parabola opens upward (minimum turning point)
$q = -4$

Step 2: Find the intercepts

Y-intercept: When $x = 0$ , $y = 2(0)^2 - 4 = -4$ . Point: $(0, -4)$
X-intercepts: When $y = 0$ , $0 = 2x^2 - 4$

$2x^2 = 4$

$x^2 = 2$

$x = \pm\sqrt{2}$

Points: $(-\sqrt{2}, 0)$ and $(\sqrt{2}, 0)$

Step 3: Identify the turning point From the standard form, the turning point is $(0, -4)$ .

Step 4: Sketch the graph

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Worked Example 2: Sketching g(x) = -½x² - 3

Step 1: Examine the equation

$a = -\frac{1}{2} < 0$ , so the parabola opens downward (maximum turning point)
$q = -3$

Step 2: Find the intercepts

Y-intercept: When $x = 0$ , $g(0) = -\frac{1}{2}(0)^2 - 3 = -3$ . Point: $(0, -3)$
X-intercepts: When $g(x) = 0$ , $0 = -\frac{1}{2}x^2 - 3$

$\frac{1}{2}x^2 = -3$

$x^2 = -6$

Since $x^2$ cannot be negative, there are no real x-intercepts.

Step 3: Identify the turning point The turning point is $(0, -3)$ .

Step 4: Sketch the graph

Exam tips

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Essential Exam Strategies:

Always check the sign of a first to determine if the parabola opens up or down
Remember that the turning point for $y = ax^2 + q$ is always at $(0, q)$
When finding x-intercepts, be careful with negative values under the square root - these indicate no real intercepts
The axis of symmetry for these functions is always $x = 0$
Use the symmetry property: if $(h, k)$ is on the graph, then $(-h, k)$ is also on the graph

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Key Points to Remember:

Quadratic functions have the form $y = ax^2 + q$ and create parabolic graphs
Parameter a determines the direction (positive = upward, negative = downward) and width of the parabola
Parameter q shifts the entire graph vertically by $q$ units
The turning point is always at $(0, q)$ and represents either a minimum (when $a > 0$ ) or maximum (when $a < 0$ )
All quadratic functions of this form have domain $\mathbb{R}$ and are symmetric about the y-axis

Quadratic Functions (Grade 10 NSC Matric Mathematics): Revision Notes