Characteristics of Quadratics (HSC SSCE Mathematics Advanced): Revision Notes

Characteristics of Quadratics

What is a quadratic function?

A quadratic function is a function that can be written in the form:

$f(x) = ax^2 + bx + c$

where $a$, $b$, and $c$ are constants, and $a \neq 0$.

The graph of a quadratic function is always a smooth U-shaped curve called a parabola. The simplest quadratic function is $y = x^2$.

Key features of parabolas

Every parabola has several important features that help us understand and sketch the function.

The parabola shape

The graph of $y = x^2$ demonstrates the basic parabola shape. Here are some values:

$x$	$-4$	$-3$	$-2$	$-1$	$0$	$1$	$2$	$3$	$4$
$y$	$16$	$9$	$4$	$1$	$0$	$1$	$4$	$9$	$16$

Notice how the $y$-values are symmetrical around $x = 0$. This creates the characteristic U-shape.

Vertex (turning point)

The vertex is the turning point of the parabola. It represents either the minimum value (when the parabola opens upward) or the maximum value (when the parabola opens downward).

For the basic parabola $y = x^2$, the vertex is at the point $(0, 0)$.

Concavity

The concavity of a parabola describes the direction it opens:

• Concave up: The parabola opens upward like a smile. This happens when $a > 0$ (the coefficient of $x^2$ is positive). The vertex is a minimum point.
• Concave down: The parabola opens downward like a frown. This happens when $a < 0$ (the coefficient of $x^2$ is negative). The vertex is a maximum point.

The sign of $a$ in the quadratic function $f(x) = ax^2 + bx + c$ determines the concavity.

Intercepts

Intercepts are points where the parabola crosses the axes.

y-intercept: The point where the curve crosses the $y$-axis. This occurs when $x = 0$. To find it, substitute $x = 0$ into the equation. The $y$-intercept is always at $(0, c)$ where $c$ is the constant term.

x-intercepts: The points where the curve crosses the $x$-axis. These occur when $y = 0$. A parabola can have:

• Two x-intercepts
• One x-intercept (when the vertex touches the x-axis)
• No x-intercepts (when the parabola doesn't reach the x-axis)

Axis of symmetry

The axis of symmetry is a vertical line that divides the parabola into two mirror images. It always passes through the vertex.

For a quadratic function $y = ax^2 + bx + c$, the axis of symmetry has the equation:

$x = \frac{-b}{2a}$

where:

• $a$ is the coefficient of $x^2$
• $b$ is the coefficient of $x$

Finding key features: worked example

Let's identify the key features of the parabola $y = x^2 - 4x + 3$.

Factored form and x-intercepts

When a quadratic function is written in factored form, finding the x-intercepts becomes straightforward.

The factored form of a quadratic function is:

$y = a(x - r)(x - s)$

where:

• $r$ and $s$ are the x-intercepts
• $a$ is the stretch factor (determines how wide or narrow the parabola is and its orientation)

Finding x-intercepts from factored form

To find the x-intercepts, set $y = 0$ and solve:

$$y = a(x - r)(x - s) \\ 0 = a(x - r)(x - s) \\ 0 = (x - r)(x - s)$$

Now set each factor equal to zero:

First factor:

$$x - r = 0 \\ x = r$$

Second factor:

$$x - s = 0 \\ x = s$$

Therefore, the x-intercepts are at $x = r$ and $x = s$.

Worked example: finding x-intercepts

The discriminant and number of roots

The discriminant is a mathematical tool that tells us how many x-intercepts a parabola has without actually solving the equation.

For a quadratic function $f(x) = ax^2 + bx + c$, the discriminant is:

$\Delta = b^2 - 4ac$

where:

• $\Delta$ (Greek letter delta) represents the discriminant
• $a$ is the coefficient of $x^2$
• $b$ is the coefficient of $x$
• $c$ is the constant term

Interpreting the discriminant

The value of the discriminant tells us about the x-intercepts (roots):

The quadratic formula

The discriminant appears in the quadratic formula, which gives the roots of any quadratic equation $ax^2 + bx + c = 0$ where $a \neq 0$:

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

The term under the square root ($b^2 - 4ac$) is the discriminant.

Worked example: using the discriminant

Another worked example

Domain and range

Domain

The domain of a function is the set of all possible input values (x-values).

For any quadratic function, the domain is all real numbers:

$\text{Domain} = \mathbb{R}$

This means you can substitute any real number for $x$ in a quadratic function.

Range

The range of a function is the set of all possible output values (y-values).

For a quadratic function $f(x) = ax^2 + bx + c$:

If $a > 0$ (concave up): The parabola has a minimum value at the vertex. The range is:

$\text{Range} = [k, \infty)$

where $k$ is the y-coordinate of the vertex.

If $a < 0$ (concave down): The parabola has a maximum value at the vertex. The range is:

$\text{Range} = (-\infty, k]$

where $k$ is the y-coordinate of the vertex.

Worked example: domain and range

Graphing quadratics

To sketch the graph of a parabola showing its key features, follow these steps:

1. Calculate $\Delta$ to determine the number of x-intercepts (if $\Delta \geq 0$), or determine them by factorising
2. Find the y-intercept: Set $x = 0$, giving $f(0) = c$
3. Find the vertex: Calculate $x = \frac{-b}{2a}$, then find the corresponding y-value
4. Determine concavity: Check whether $a > 0$ (concave up) or $a < 0$ (concave down)
5. Plot the key points and sketch the parabola

Complete worked example

Exam tips

Remember!