Drawing Graphs Revision Notes for Junior Cert Mathematics

Drawing Graphs

Graphing functions is a key skill in Junior Cycle Maths. It helps you visualise how functions behave by plotting their outputs on a coordinate plane. This topic is important because it connects algebra with geometry, allowing you to solve problems graphically.

Key Concepts:

Function: A function is a rule that assigns each input exactly one output. We often write it as $f(x)$ , where $x$ is the input.
Domain: The set of possible inputs (values of $x$ ).
Range: The set of possible outputs (values of $f(x$ ) ).
Graph of a Function: The visual representation of the function on a coordinate plane. The $x$ - $axis$ represents the inputs, and the $y$ - $axis$ represents the outputs.

Steps to Graph a Function:

Choose the Domain: Decide the range of $x$ -values you will use (e.g., $-3$ to $3$ ).
Create a Table of Values:

Substitute each $x$ value into the function to find the corresponding $y$ (or $f(x)$ ) value.
Write these pairs of $(x, y)$ values in a table.

Plot the Points: Mark each $(x, y)$ pair on the graph.
Draw the Graph: Connect the points with a smooth curve or straight line, depending on the function.

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Example 1: Graphing a Linear Function Let's graph the function $f(x) = 2x - 1$ .

Step 1: Choose the Domain

We'll use the values $x = -2, -1, 0, 1, 2$ .

Step 2: Create a Table of Values

$x$	$f(x) = 2x • 1$	$y$
$-2$	$2(-2) • 1 = -5$	$-5$
$-1$	$2(-1) • 1 = -3$	$-3$
$0$	$2(0) • 1 = -1$	$-1$
$1$	$2(1) • 1 = 1$	$1$
$2$	$2(2) • 1 = 3$	$3$

Step 3: Plot the Points

On graph paper, plot the points $(-2, -5) , (-1, -3), (0, -1) , (1, 1) ,$ and $(2, 3)$ .

Step 4: Draw the Graph

Draw a straight line through the points. This is the graph of the function $f(x) = 2x - 1$ . This is shown below:

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Example 2: Graphing a Quadratic Function Now, let's graph the function $f(x) = x^2 - 4$ .

Step 1: Choose the Domain

We'll use the values $x = -3, -2, -1, 0, 1, 2, 3$ .

Step 2: Create a Table of Values

$x$	$f(x) = x^2 • 4$	$y$
$-3$	$(-3)^2 • 4 = 5$	$5$
$-2$	$(-2)^2 • 4 = 0$	0
$-1$	$(-1)^2 • 4 = -3$	$-3$
$0$	$0^2 • 4 = -4$	$-4$
$1$	$1^2 • 4 = -3$	$-3$
$2$	$2^2 • 4 = 0$	$0$
$3$	$3^2 • 4 = 5$	$5$

Step 3: Plot the Points

Plot the points $(-3, 5), (-2, 0), (-1, -3), (0, -4), (1, -3), (2, 0)$ , and $(3, 5)$ .

Step 4: Draw the Graph

Draw a smooth curve through the points. This U-shaped curve is the graph of the quadratic function $f(x) = x^2 - 4$ .

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Exam Tips:

Use a ruler: Ensure your graphs are neat and accurate, especially for linear functions.
Check your work: Double-check your calculations for the table of values.
Understand the shape: Recognise that linear functions produce straight lines, quadratic functions produce U-shaped curves, and so on.
Label the Axes: Always label your $x$ - $axis$ and $y$ - $axis$ and provide a title if required.

Drawing Graphs (Junior Cert Mathematics): Revision Notes

Drawing Graphs

Key Concepts:

Steps to Graph a Function:

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