SECONDARY 3 MATHEMATICS CLASSROOM · CHAPTER 5 · GRAPHS OF FUNCTIONS · GRAPHICAL SOLUTION · POWER FUNCTIONS · EXPONENTIAL FUNCTIONS · G2/G3
Graphs of Functions: When Algebra Becomes Behaviour
A function rule does more than produce values. Its graph reveals shape, growth, symmetry, intersections and local steepness. A graph is therefore not a picture added after algebra; it is another mathematical representation of the same relationship.
Chapter 4 turned straight-line geometry into equations. Chapter 5 widens the field. Students now compare families of functions whose behaviour is no longer linear: powers, reciprocals, cubics and exponentials. They also use intersections to solve equations and tangents to estimate the gradient of a curve. The textbook sequence remains useful because it trains students to move repeatedly between rule, table, graph and solution.
Classroom rule: identify the function family → predict its broad shape → build or read a table of values → mark key points and restrictions → sketch smoothly → use intersections to solve → use a tangent for local gradient → verify the result against the algebra and graph.
Current syllabus boundary. The live G2/G3 syllabuses include power functions, exponential functions and estimating the gradient of a curve by drawing a tangent. G3 also expects stronger graphical-solution work. Schools may sequence individual graph families differently, so use the student’s current course as timing control.
Official reference: MOE G2 and G3 Mathematics Syllabuses.
Featured Answer: What Does a Graph Tell Us That an Equation Does Not Show Immediately?
A graph makes global behaviour visible at once. It can show where a function crosses an axis, where two functions intersect, whether values grow or decay, whether there is symmetry, whether a curve approaches but does not meet an axis, and how steeply the function changes at a particular point.
1. A Function Connects Input to Output
For y=f(x), each allowed x-value is an input and y is the corresponding output. A graph records all these input-output pairs as points (x,y).
2. Tables Are a Bridge Between Rule and Graph
For y=x², a table using x=−2,−1,0,1,2 gives y=4,1,0,1,4. The repeated outputs hint at symmetry before the graph is drawn.
3. Power Functions Form Families
The current syllabus includes functions of the form y=axⁿ for selected integer powers including n=−2,−1,0,1,2,3. Changing n changes the shape dramatically.
4. y=x² Is Even and Symmetric About the y-Axis
Because (−x)²=x², opposite x-values give the same output. The graph is a parabola with mirror symmetry about x=0.
5. y=x³ Is Odd and Changes Sign Through the Origin
Because (−x)³=−x³, the graph has rotational symmetry about the origin and passes through (0,0).
6. Reciprocal Functions Have Domain Restrictions
For y=1/x, x=0 is not allowed. The graph has separate branches and approaches the coordinate axes without meeting them.
7. y=1/x² Is Always Positive for x≠0
Squaring removes the sign before taking the reciprocal, so both branches lie above the x-axis.
8. Teacher Model 1: Compare Three Power Graphs
- y=x: straight line through origin;
- y=x²: upward parabola;
- y=x³: S-shaped curve through origin.
The equation exponent controls graph behaviour. The graph should be predicted before point plotting begins.
9. Exponential Functions Describe Multiplicative Change
For y=kaˣ with a>1, each increase of 1 in x multiplies the output by a. This is fundamentally different from a linear function, where equal changes in x produce equal additive changes in y.
10. Teacher Model 2: y=2ˣ
Using x=−2,−1,0,1,2 gives y=1/4,1/2,1,2,4. The graph remains positive, crosses the y-axis at 1 and grows increasingly rapidly to the right.
11. Exponential Graphs Approach the x-Axis
For y=aˣ with a>1, values become very small for large negative x but remain positive. The x-axis is approached as an asymptote rather than crossed.
12. Scaling by k Changes Vertical Size
For y=3·2ˣ, every y-value of 2ˣ is multiplied by 3. The y-intercept becomes 3 instead of 1.
13. Graphical Solution Means Solving by Intersections
To solve f(x)=g(x), draw or use the graphs y=f(x) and y=g(x). The x-coordinates of their intersections are the solutions.
14. Teacher Model 3: Solve x²=2x+3 Graphically
Graph y=x² and y=2x+3. Their intersections occur at x=−1 and x=3, matching the algebraic equation x²−2x−3=0.
15. One Graph Can Solve a Modified Equation
If the graph of y=f(x) is already given, solving f(x)=4 means finding where the curve meets the horizontal line y=4.
16. Horizontal and Sloping Comparison Lines Change the Question
f(x)=k uses a horizontal line y=k. f(x)=mx+c uses a straight line. The intersections represent simultaneous satisfaction of both relationships.
17. Graphical Answers Are Usually Approximate
Accuracy depends on scale, plotting and curve quality. A graphical root should therefore be read to an accuracy justified by the graph.
18. Gradient of a Curve Changes From Point to Point
A straight line has one constant gradient. A curve can be steep in one region and shallow in another.
19. A Tangent Estimates Local Gradient
Draw a tangent that touches the curve at the required point and follows the curve’s local direction. Then choose two well-separated points on the tangent and calculate rise/run.
20. Teacher Model 4: Tangent Gradient
If two convenient points on a tangent are (1,3) and (5,11), estimated gradient=(11−3)/(5−1)=2.
21. Use Points on the Tangent, Not Necessarily on the Curve
The tangent line is the object whose gradient is being estimated. Choosing distant points on that line improves numerical stability.
22. Scale Choice Affects Graph Quality
A cramped scale can hide curvature and make intersections difficult to read. A suitable scale uses the graphing area efficiently and keeps intervals simple.
23. Smooth Curves Are Not Dot-to-Dot Polygons
For a continuous function, plotted points guide a smooth curve. Joining every pair by straight segments can misrepresent behaviour.
24. Domain Restrictions Must Appear in the Graph
For reciprocal functions, x=0 is excluded. A table or graph that quietly inserts a value there is mathematically invalid.
25. Graph Shape Can Check Algebra
If y=x³ is drawn as a parabola or y=2ˣ crosses the x-axis, the sketch contradicts the function family and should be corrected before any graphical solution is trusted.
26. Misconception Clinic: Every Curve Is a Parabola
Repair: identify the function family before sketching. Power, reciprocal and exponential graphs have distinct structures.
27. Misconception Clinic: 1/x Is Defined at x=0
Repair: division by zero is undefined, so the graph must exclude x=0.
28. Misconception Clinic: Intersections Give y-Values as the Equation Solutions
Repair: when solving f(x)=g(x), the required solutions are normally the x-coordinates of intersections unless the question asks otherwise.
29. Misconception Clinic: Use Two Curve Points to Find Tangent Gradient
Repair: the estimated local gradient comes from the tangent line at the specified point.
30. Guided Practice A: Function Families
- State the broad shape of y=x².
- State the symmetry of y=x³.
- State the excluded x-value for y=1/x.
- State whether y=2ˣ can be zero.
Answers
Upward parabola. Rotational symmetry about origin. x=0. No; it remains positive.
31. Guided Practice B: Tables and Values
- For y=x³, find y at x=−2,−1,0,1,2.
- For y=3·2ˣ, find y at x=−1,0,1,2.
Answers
−8,−1,0,1,8. 1.5,3,6,12.
32. Guided Practice C: Graphical Solutions
- If y=f(x) intersects y=5 at x≈−1.2 and x≈3.4, solve f(x)=5.
- If y=x³ and y=4x intersect at x=−2,0,2, solve x³=4x.
Answers
x≈−1.2 or 3.4. x=−2,0,2.
33. Guided Practice D: Tangent Gradient
A tangent passes through convenient points (−1,2) and (3,10). Estimate its gradient.
Solution
(10−2)/(3−(−1))=8/4=2.
34. Challenge Practice: One Graph, Several Questions
Suppose a graph of y=f(x) is supplied. A strong reader should be able to answer different questions from the same curve: f(2), solve f(x)=0, solve f(x)=4, estimate the gradient at x=1 and compare where f(x) is increasing or decreasing.
35. Assessment Method: Predict Before Plotting
Write the expected family behaviour first: intercepts, sign, symmetry, restrictions and end behaviour where appropriate. Plotting then becomes verification rather than blind construction.
36. Assessment Method: Read Intersections in Context
If a graph models time, length or another physical quantity, reject intersection values outside the stated domain.
37. Assessment Method: Make Tangent Triangles Large
Using widely separated tangent points reduces the effect of reading errors from the graph scale.
38. Oral Classroom Check
- How is a function graph built from input-output pairs?
- What is distinctive about y=x²?
- What is distinctive about y=x³?
- Why is x=0 excluded from y=1/x?
- How does exponential growth differ from linear growth?
- What do intersections represent when solving f(x)=g(x)?
- Why are graphical solutions approximate?
- How do you estimate the gradient of a curve?
- Why should tangent points be far apart?
- How can graph shape expose an algebraic or plotting error?
39. Exit Ticket
- State the symmetry of y=x².
- State the symmetry of y=x³.
- State the excluded x-value for y=1/x².
- Find y when x=3 for y=2ˣ.
- Explain how to solve f(x)=6 from a supplied graph.
- Explain how to solve f(x)=2x+1 graphically.
- State what the x-coordinates of intersections represent.
- A tangent contains (2,5) and (8,17). Find its gradient.
- Explain why y=2ˣ does not cross the x-axis.
- Name one reason a graphical solution may differ slightly from an algebraic value.
Exit-ticket solutions
y-axis symmetry. Rotational symmetry about origin. x=0. 8. Find where y=f(x) meets y=6. Draw/use y=f(x) and y=2x+1 and read intersection x-values. They are the solutions. Gradient=(17−5)/(8−2)=2. 2ˣ is positive for every real x. Graph scale, plotting or reading accuracy.
40. The Seven-Day Return Cycle
- Day 0: identify function families and build tables.
- Day 1: reciprocal restrictions and exponential behaviour.
- Day 3: graphical solution from intersections.
- Day 7: mixed graph-reading question including one tangent-gradient estimate.
41. The Full Graph Routine
identify family → predict behaviour → calculate key values → choose scale → plot → sketch smoothly → read intersections → draw tangent if needed → verify with algebra and domain.
42. Connect Back to Chapter 4
Return to Secondary 3 Chapter 4: Coordinate Geometry when plotting, gradient or straight-line equations are unstable. Graphical solution depends on reading coordinates and comparison lines accurately.
43. Specialist Companions
- Secondary 3 Mathematics Learning Guide | Quadratic, Power and Exponential Graphs
- Secondary 3 Mathematics Learning Guide | Graphical Solutions, Intersections and Tangent Gradients
- Secondary 3 Mathematics Learning Guide | Functions, Graphs and Coordinate Relationships
44. Why This Chapter Matters for Chapter 6
Graphs train the learner to interpret a relationship rather than merely manipulate symbols. Chapter 6 returns to trigonometry and extends right-triangle ratios into obtuse angles and non-right triangles. The same representational discipline remains: identify what the quantities mean before selecting the formula.
45. Ready for Chapter 6?
- distinguish key power-function families;
- recognise reciprocal restrictions;
- describe exponential growth;
- construct graphs from tables accurately;
- solve equations from graph intersections;
- use a supplied graph for multiple equation forms;
- estimate gradient using a tangent;
- check graph scale, shape and domain before trusting an answer.
If one item is weak, return to the smallest section that owns it and solve a changed example. When the route is stable, continue to Chapter 6: Further Trigonometry.
