Strand 5 — Functions

Graphing Functions

1st Year · 2nd Year · 3rd Year (Junior Cert)

✓By the end of this lesson students will be able to understand the concept of a function and its various representations.
✓By the end of this lesson students will be able to construct a table of values for a given function.
✓By the end of this lesson students will be able to plot points accurately on a coordinate plane.
✓By the end of this lesson students will be able to draw and interpret graphs of linear functions.
✓By the end of this lesson students will be able to draw and interpret graphs of quadratic functions, identifying key features like roots and turning points.

Key concepts

Function

A function is a rule that assigns exactly one output value (usually denoted by y or f(x)) to each input value (usually denoted by x). Functions can be represented using words, tables, diagrams, graphs, and formulae.

Coordinate Plane

A two-dimensional plane formed by the intersection of a horizontal number line (the x-axis) and a vertical number line (the y-axis) at a point called the origin (0, 0). Points are located on this plane using ordered pairs (x, y), where x is the horizontal position and y is the vertical position.

Table of Values

A table used to list input (x) values and their corresponding output (y) values for a given function. Each row in the table forms an ordered pair (x, y) that can be plotted as a point on a graph.

Linear Function

A function whose graph is a straight line. It can be written in the general form y = mx + c, where 'm' represents the slope (or gradient) of the line and 'c' represents the y-intercept (the point where the line crosses the y-axis).

y = mx + c

Quadratic Function

A function whose graph is a U-shaped or inverted U-shaped curve called a parabola. It can be written in the general form y = ax^2 + bx + c, where 'a', 'b', and 'c' are constants and 'a' cannot be zero. The parabola has a turning point, which is either its minimum or maximum point.

y = ax^2 + bx + c

Key facts to remember

1A function assigns exactly one output for each input.
2The coordinate plane uses an x-axis (horizontal) and a y-axis (vertical) to locate points.
3Linear functions always produce straight-line graphs.
4Quadratic functions always produce U-shaped (or inverted U-shaped) graphs called parabolas.
5The roots of a function are the x-values where its graph crosses the x-axis (where y = 0).
6The y-intercept is the point where the graph crosses the y-axis (where x = 0).
7The turning point of a parabola is its minimum or maximum point.

Worked examples

Example 1

Draw the graph of the function f(x) = 2x + 1 for x values from -3 to 3. From your graph, identify the y-intercept.

ICreate a table of values by substituting each x-value into the function f(x) = 2x + 1:

IIx | f(x) = 2x + 1 | y

III--|---------------|---

IV-3| 2(-3) + 1 | -5

V-2| 2(-2) + 1 | -3

VI-1| 2(-1) + 1 | -1

VII0 | 2(0) + 1 | 1

VIII1 | 2(1) + 1 | 3

92 | 2(2) + 1 | 5

103 | 2(3) + 1 | 7

11Plot the points (-3, -5), (-2, -3), (-1, -1), (0, 1), (1, 3), (2, 5), (3, 7) on a coordinate plane.

12Join the plotted points with a straight line using a ruler.

13Identify the y-intercept: This is the point where the graph crosses the y-axis. From the table and graph, when x = 0, y = 1. So, the y-intercept is (0, 1).

Answer

A straight line passing through the plotted points. The y-intercept is (0, 1).

Always use a ruler to draw straight lines for linear functions. Ensure your axes are labelled and a suitable scale is chosen.

Example 2

Draw the graph of the function y = x^2 - 4x + 3 for x values from -1 to 5. From your graph, identify the roots and the turning point.

ICreate a table of values for x from -1 to 5:

IIx | x^2 | -4x | +3 | y = x^2 - 4x + 3

III--|-----|-----|----|------------------

IV-1| 1 | 4 | 3 | 8

V0 | 0 | 0 | 3 | 3

VI1 | 1 | -4 | 3 | 0

VII2 | 4 | -8 | 3 | -1

VIII3 | 9 | -12 | 3 | 0

94 | 16 | -16 | 3 | 3

105 | 25 | -20 | 3 | 8

11Plot the points (-1, 8), (0, 3), (1, 0), (2, -1), (3, 0), (4, 3), (5, 8) on a coordinate plane.

12Join the plotted points with a smooth curve to form a parabola.

13Identify the roots: These are the x-values where the graph crosses the x-axis (where y = 0). From the graph, the roots are x = 1 and x = 3.

14Identify the turning point: This is the lowest point on this parabola. From the graph, the turning point is (2, -1).

Answer

A parabola with roots at x = 1 and x = 3, and a turning point at (2, -1).

For quadratic graphs, always draw a smooth curve, not straight line segments between points. The curve should pass through all plotted points.

Common mistakes

✗Incorrectly calculating the y-values in the table of values, leading to wrong points.
✗Plotting points incorrectly on the coordinate plane (e.g., mixing up x and y coordinates).
✗Using a ruler to connect points for a quadratic graph, resulting in a jagged line instead of a smooth curve.
✗Not labelling axes or including a clear scale on the graph.
✗Failing to extend the graph to the full range of x-values specified in the problem.

Exam tips

★Always use a sharp pencil and a ruler for drawing graphs, especially for linear functions.
★Label your x-axis and y-axis clearly, and indicate the scale used on both axes.
★Double-check your calculations for the table of values before plotting points.
★For quadratic graphs, ensure your curve is smooth and passes through all the points you have plotted.
★Read the question carefully to ensure you identify all required features from the graph, such as roots, turning points, or intercepts.

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