Section B — Chapters 23–39

Functions II: Linear and Quadratic Functions

Junior Cycle — 1st Year

✓By the end of this lesson students will be able to define a function and identify its input and output.
✓By the end of this lesson students will be able to distinguish between linear and quadratic functions.
✓By the end of this lesson students will be able to construct a table of values and plot the graph of a given linear or quadratic function.
✓By the end of this lesson students will be able to interpret information from the graph of a function, including finding values and identifying intercepts.
✓By the end of this lesson students will be able to use the graph of a function to find the values of x for which f(x) = 0.

Key concepts

Function

A function is a rule that assigns exactly one output value to each input value. We often write a function as f(x), where 'x' is the input and 'f(x)' is the output. For example, if f(x) = x + 3, when the input is 2, the output is f(2) = 2 + 3 = 5.

Linear Function

A linear function is a function whose graph is a straight line. The highest power of x in a linear function is 1. The general form of a linear function is y = mx + c, where m and c are constants. 'm' is the slope (gradient) of the line, and 'c' is the y-intercept (where the line crosses the y-axis).

f(x) = mx + c

Quadratic Function

A quadratic function is a function whose graph is a U-shaped or n-shaped curve called a parabola. The highest power of x in a quadratic function is 2. The general form of a quadratic function is y = ax^2 + bx + c, where a, b, and c are constants, and a is not equal to 0.

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

Graphing a Function

To graph a function, we choose a range of input values (x-values), calculate the corresponding output values (f(x) or y-values) using the function's rule, and then plot these (x, y) coordinate pairs on a coordinate plane. Finally, we connect the points to form the graph.

Interpreting Graphs

Interpreting a graph means understanding the information it presents. You can find the output (y-value) for a given input (x-value) by locating x on the x-axis, moving vertically to the graph, and then horizontally to the y-axis. Similarly, you can find the input(s) for a given output.

Using Graphs to Solve f(x) = 0

When we solve f(x) = 0, we are looking for the x-values where the output of the function is zero. On a graph, this corresponds to the points where the graph crosses or touches the x-axis. These points are called the x-intercepts or roots of the function.

Key facts to remember

1A function assigns exactly one output for each input.
2The graph of a linear function (e.g., f(x) = mx + c) is a straight line.
3The graph of a quadratic function (e.g., f(x) = ax^2 + bx + c) is a parabola (a U-shaped or n-shaped curve).
4To find f(x) = 0 graphically, look for the points where the graph crosses or touches the x-axis. These are the x-intercepts.
5The y-intercept is the point where the graph crosses the y-axis (where x = 0).
6A table of values is essential for plotting the graph of a function accurately.

Worked examples

Example 1

Draw the graph of the function f(x) = 2x - 1 for x values from -2 to 3. Use your graph to find the value of x for which f(x) = 0.

ICreate a table of values for x from -2 to 3:

IIx | f(x) = 2x - 1 | (x, f(x))

III-2 | 2(-2) - 1 = -5 | (-2, -5)

IV-1 | 2(-1) - 1 = -3 | (-1, -3)

V0 | 2(0) - 1 = -1 | (0, -1)

VI1 | 2(1) - 1 = 1 | (1, 1)

VII2 | 2(2) - 1 = 3 | (2, 3)

VIII3 | 2(3) - 1 = 5 | (3, 5)

9Plot the points (-2, -5), (-1, -3), (0, -1), (1, 1), (2, 3), (3, 5) on a coordinate plane.

10Draw a straight line through the plotted points using a ruler.

11To find the value of x for which f(x) = 0, locate where the graph crosses the x-axis (where y = 0). From the graph, the line crosses the x-axis at x = 0.5.

Answer

x = 0.5

Ensure your axes are clearly labelled and scaled appropriately.

Example 2

Draw the graph of the function f(x) = x^2 - 4x + 3 for x values from 0 to 4. Use your graph to find the values of x for which f(x) = 0.

ICreate a table of values for x from 0 to 4:

IIx | f(x) = x^2 - 4x + 3 | (x, f(x))

III0 | (0)^2 - 4(0) + 3 = 3 | (0, 3)

IV1 | (1)^2 - 4(1) + 3 = 1 - 4 + 3 = 0 | (1, 0)

V2 | (2)^2 - 4(2) + 3 = 4 - 8 + 3 = -1 | (2, -1)

VI3 | (3)^2 - 4(3) + 3 = 9 - 12 + 3 = 0 | (3, 0)

VII4 | (4)^2 - 4(4) + 3 = 16 - 16 + 3 = 3 | (4, 3)

VIIIPlot the points (0, 3), (1, 0), (2, -1), (3, 0), (4, 3) on a coordinate plane.

9Draw a smooth U-shaped curve (parabola) through the plotted points.

10To find the values of x for which f(x) = 0, locate where the graph crosses the x-axis (where y = 0). From the graph, the curve crosses the x-axis at x = 1 and x = 3.

Answer

x = 1, x = 3

Remember to draw a smooth curve for quadratic functions, not straight line segments.

Example 3

The graph of a function f(x) is shown below (imagine a simple parabola opening upwards, with vertex at (2, -1) and x-intercepts at (1,0) and (3,0), y-intercept at (0,3)). Use the graph to answer the following:\na) Find f(0).\nb) Find the value(s) of x for which f(x) = 0.\nc) Find the minimum value of f(x).

Ia) To find f(0), locate x = 0 on the x-axis. Move vertically to the graph, then horizontally to the y-axis. The graph passes through (0, 3), so f(0) = 3.

IIb) To find the value(s) of x for which f(x) = 0, locate where the graph crosses the x-axis. The graph crosses the x-axis at x = 1 and x = 3.

IIIc) The minimum value of f(x) is the lowest point on the graph. From the graph, the lowest point is (2, -1). So the minimum value of f(x) is -1.

Answer

a) f(0) = 3, b) x = 1, x = 3, c) Minimum value of f(x) = -1

Always read values carefully from the graph, using a ruler if necessary for accuracy.

Common mistakes

✗Mixing up the x and y coordinates when plotting points.
✗Incorrectly calculating f(x) values, especially with negative numbers or squaring.
✗Drawing a quadratic graph with straight line segments instead of a smooth curve.
✗Not extending the graph sufficiently to see all x-intercepts or the turning point.
✗Misinterpreting f(x) = 0 as finding the y-intercept instead of the x-intercepts.

Exam tips

★Always use a ruler for drawing linear graphs and for reading values from any graph.
★Label your axes clearly (x-axis, y-axis) and indicate the scale used on each axis.
★When plotting points, use a sharp pencil to make small, clear crosses (x) or dots.
★Double-check your calculations in the table of values before plotting to avoid errors.

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