Quadratic: y=ax^2 +bx + c (Junior Cert Mathematics): Revision Notes

Quadratic Functions

What is a Quadratic Function?

A quadratic function is a function that includes an $x^2$ ($x$ squared). This is different from a straight line because the $x^2$ makes the graph curve into a U-shape. This U-shaped curve is called a parabola.

The simplest quadratic function is:

$f(x) = x^2$

This means that for any value of $x$, you square it to get the value of $f(x)$. For example, if $x = 2$, then $f(x) = 2^2 = 4$.

Why Do Quadratic Functions Create Parabolas?

When you plot the points for a quadratic function like $f(x) = x^2$ on a graph, you'll notice that the graph doesn't form a straight line like with linear functions. Instead, it forms a curve called a parabola. This happens because the $x^2$ term causes the values of $f(x)$ to increase more rapidly as $x$ gets larger.

A parabola is a symmetrical curve, which means it looks the same on both sides of a vertical line called the axis of symmetry. The lowest point on a parabola that opens upwards is called the vertex.

Discovering the Parabola: Plotting $f(x) = x^2$

Let's explore how this works by looking at the function $f(x) = x^2$. We'll start by calculating the outputs f(x)$$-values for some inputs (x$$-values):

$\begin{array}{|c|c|} \hline x & f(x) = x^2 \\ \hline -3 & 9 \\ -2 & 4 \\ -1 & 1 \\ 0 & 0 \\ 1 & 1 \\ 2 & 4 \\ 3 & 9 \\ \hline \end{array}$

Now, let's plot these points on a graph:

1. Plot the Points: Each pair of $x$ and $f(x)$ values gives you a point on the graph. For example, when $x = -3$, $f(x) = 9$, so you plot the point $(-3, 9)$.
2. Draw the Curve: Once you've plotted all the points, connect them with a smooth curve. You should see that the curve is $U$-shaped and symmetrical. This is your parabola.
3. Notice the Symmetry: The graph is symmetrical around the y$$-axis, which is the axis of symmetry in this case.

Graphing Functions When the Coefficient of $x^2$Is Negative

If the coefficient of $x^2$ is negative in a quadratic function, e.g., $f(x) = -3x^2 + 4$, then the graph of the function will take the shape of an upside-down parabola as shown below:

Using the Table Function on a Calculator

• Find the table button – it might say TABLE or be written above another key (you may need to press SHIFT or 2nd first).
• Enter your equation – type it in exactly as you see it, e.g. 2x + 3.
• Set the starting value – choose the first x value the table should show (e.g. start at 0).
• Set the step value – choose how much x changes each time (e.g. step of 1 means x goes 0, 1, 2, 3…).
• View the table – the calculator will list x values in one column and the matching y (answer) values in another.
• Scroll up or down – use arrow keys to see more values.
• Change settings if you need different starting points or steps.
• Use it to spot patterns – the table can help you see how y changes as x changes.

Summary

• A quadratic function is a function that includes $x^2$.
• The graph of a quadratic function forms a U-shaped curve called a parabola.
• If the $x^2$ term is negative, then the parabola will be an upside down U-shape.
• The simplest quadratic function is $f(x) = x^2$.
• You can discover the shape of the parabola by plotting points on a graph.