Strand 5 — Functions

Interpreting Graphs

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

✓By the end of this lesson students will be able to identify the roots (x-intercepts) of a function from its graph.
✓By the end of this lesson students will be able to determine the maximum or minimum point of a function from its graph.
✓By the end of this lesson students will be able to interpret real-life graphs, including distance-time and speed-time graphs.
✓By the end of this lesson students will be able to relate features of a graph (e.g., slope, intercepts) to the context of a real-life situation.

Key concepts

Roots of a Function

The roots of a function are the x-values where the graph of the function crosses or touches the x-axis. At these points, the y-value of the function is zero. They are also known as x-intercepts.

Maximum Point

The maximum point of a graph is the highest point on the graph within a given interval or for the entire function. At this point, the y-value is at its greatest.

Minimum Point

The minimum point of a graph is the lowest point on the graph within a given interval or for the entire function. At this point, the y-value is at its least.

Distance-Time Graph

A graph that shows how the distance travelled by an object changes over time. The slope (gradient) of a distance-time graph represents the speed of the object. A horizontal line means the object is stationary. A steeper slope means a faster speed.

Speed = Change in Distance / Change in Time

Speed-Time Graph

A graph that shows how the speed of an object changes over time. The slope (gradient) of a speed-time graph represents the acceleration of the object. A horizontal line means constant speed. The area under a speed-time graph represents the total distance travelled.

Acceleration = Change in Speed / Change in Time

Key facts to remember

1Roots are the x-values where the graph crosses or touches the x-axis (y=0).
2The maximum point is the highest point on the graph.
3The minimum point is the lowest point on the graph.
4On a distance-time graph, the slope (gradient) represents speed.
5On a distance-time graph, a horizontal line segment indicates that the object is stationary.
6On a speed-time graph, the slope (gradient) represents acceleration.
7On a speed-time graph, a horizontal line segment indicates that the object is moving at a constant speed.
8The area under a speed-time graph represents the total distance travelled.

Worked examples

Example 1

The graph of a function y = f(x) is shown below. The graph is a parabola opening upwards, crossing the x-axis at x = -1 and x = 3, and having its lowest point at (1, -4). Use the graph to answer the following questions: (a) What are the roots of the function? (b) What is the minimum point of the function?

I(a) To find the roots, identify the points where the graph crosses the x-axis (where y = 0). From the description, these are x = -1 and x = 3.

II(b) To find the minimum point, identify the lowest point on the graph. From the description, this is (1, -4).

Answer

(a) x = -1, x = 3 (b) (1, -4)

Example 2

A distance-time graph shows a person walking. The graph starts at (0,0), goes to (2 hours, 100 km), then to (4 hours, 100 km), and finally to (6 hours, 200 km). (a) What distance did the person travel in the first 2 hours? (b) What was the person's speed during the first 2 hours? (c) How long did the person stop for? (d) What was the person's speed during the last 2 hours?

I(a) Read the distance (y-axis) at time = 2 hours. The graph shows 100 km.

II(b) Speed is the slope of the distance-time graph. For the first 2 hours: Speed = (Change in Distance) / (Change in Time) = (100 - 0) km / (2 - 0) hours = 100 km / 2 hours = 50 km/h.

III(c) The person stopped when the distance did not change, meaning the graph is horizontal. This occurs between t = 2 hours and t = 4 hours. Duration = 4 - 2 = 2 hours.

IV(d) For the last 2 hours (from t = 4 to t = 6): Speed = (Change in Distance) / (Change in Time) = (200 - 100) km / (6 - 4) hours = 100 km / 2 hours = 50 km/h.

Answer

(a) 100 km (b) 50 km/h (c) 2 hours (from t=2 to t=4) (d) 50 km/h

Remember that the slope of a distance-time graph represents speed.

Example 3

A speed-time graph shows a car accelerating from rest. The graph starts at (0 seconds, 0 m/s), goes to (5 seconds, 20 m/s), then to (10 seconds, 20 m/s). (a) What was the acceleration of the car during the first 5 seconds? (b) What was the speed of the car after 7 seconds? (c) What was the total distance travelled by the car in the first 10 seconds?

I(a) Acceleration is the slope of the speed-time graph. For the first 5 seconds: Acceleration = (Change in Speed) / (Change in Time) = (20 - 0) m/s / (5 - 0) s = 20 m/s / 5 s = 4 m/s².

II(b) After 7 seconds, the graph shows the speed is constant at 20 m/s (between t=5 and t=10). So, the speed is 20 m/s.

III(c) The total distance travelled is the area under the speed-time graph. This area can be split into a triangle (0-5s) and a rectangle (5-10s). Area of triangle = (1/2) × base × height = (1/2) × 5 s × 20 m/s = 50 m. Area of rectangle = length × width = (10 - 5) s × 20 m/s = 5 s × 20 m/s = 100 m. Total distance = 50 m + 100 m = 150 m.

Answer

(a) 4 m/s² (b) 20 m/s (c) 150 m

The area under a speed-time graph represents the distance travelled.

Common mistakes

✗Confusing x-intercepts (roots) with y-intercepts.
✗Mixing up maximum and minimum points, especially when a graph has both.
✗Interpreting the slope of a distance-time graph as acceleration instead of speed.
✗Interpreting the slope of a speed-time graph as speed instead of acceleration.
✗Forgetting that the area under a speed-time graph gives the distance travelled.

Exam tips

★Always label axes carefully when drawing graphs and read them accurately when interpreting.
★Pay close attention to the units used on the axes (e.g., km/h, m/s) as they are crucial for correct interpretation.
★For real-life graphs, consider what each section of the graph means in the context of the problem (e.g., increasing slope, horizontal line).
★Use a ruler to help read values accurately from graphs during exams to avoid estimation errors.

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