A graph is not a picture added after the algebra. It is another representation of the same relationship. A root becomes an x-intercept. A repeated root becomes a touching point. A turning point becomes a minimum or maximum. A changing gradient becomes visible as the curve steepens or flattens.
This Secondary 3 Mathematics Learning Guide develops quadratic, power and exponential graphs as families rather than isolated sketches. It explains key features, transformations, intersections and tangent gradients, and it shows how graphical and algebraic methods can check each other.
The official 2027 SEC G3 Mathematics syllabus includes sketching quadratic graphs in useful forms, power functions, exponential functions and estimation of the gradient of a curve by drawing a tangent. This guide is G3-oriented and uses original teaching examples.
This article builds on the earlier Functions, Graphs and Coordinate Relationships guide inside the Secondary Mathematics Hub. Use diagnostic · quadratics · power graphs · exponentials · tangent gradients · practice · answers.
Function, Equation and Graph
The function y = x² − 5x + 6 gives an output y for each allowed input x. The equation x² − 5x + 6 = 0 asks which inputs make the output zero. The graph shows all input-output pairs at once.
Therefore the roots x = 2 and x = 3 correspond to the x-intercepts (2, 0) and (3, 0). The same algebraic relationship is being viewed through different mathematical objects.
This distinction prevents a common error: a graph question may ask for coordinates while an equation question asks only for x-values. Read the requested form of the answer.
A Six-Question Diagnostic
For y = (x − 2)² − 3, state the turning point. For y = (x − 1)(x + 4), state the x-intercepts. What is the y-intercept of y = x² + 2x − 8? Is y = x³ an increasing or decreasing function overall? Does y = 2ˣ ever equal zero? Finally, what geometric object is used to estimate the gradient of a curve at one point?
The answers are (2, −3); x = 1 and −4; −8; increasing; no; and a tangent. A wrong turning point usually points to completed-square interpretation. A zero claimed for 2ˣ suggests confusion between approaching the x-axis and crossing it.
Quadratic Graphs Are Parabolas
A quadratic function has the general form y = ax² + bx + c with a ≠ 0. Its graph is a parabola. When a > 0, the parabola opens upward and has a minimum. When a < 0, it opens downward and has a maximum.
The graph is symmetric about a vertical line through its turning point. This symmetry provides a strong check when plotting or interpreting coordinates.
Completed-Square Form Reveals the Turning Point
In y = (x − p)² + q, the square term is never negative. Its smallest value is zero, occurring when x = p. Therefore the minimum point is (p, q).
Similarly, y = −(x − p)² + q has a maximum at (p, q) because the negative square is never positive.
For y = (x − 3)² − 5, the turning point is (3, −5) and the axis of symmetry is x = 3. The graph is the basic y = x² shape shifted three units right and five units down.
Factorised Form Reveals the Roots
In y = (x − a)(x − b), the output is zero when x = a or x = b. Therefore the x-intercepts are (a, 0) and (b, 0).
For y = (x − 2)(x + 6), the roots are 2 and −6. The axis of symmetry lies midway between them at x = (2 + (−6))/2 = −2.
Substituting x = −2 gives y = (−4)(4) = −16, so the turning point is (−2, −16). Three representations—factorisation, midpoint symmetry and substitution—work together.
Worked Example 1: Sketch From Factorised Form
Sketch y = −(x − 1)(x − 5). The roots are x = 1 and x = 5. Their midpoint is x = 3. Because the leading coefficient is negative, the parabola opens downward.
At x = 3, y = −(2)(−2) = 4, so the maximum point is (3, 4). At x = 0, y = −(−1)(−5) = −5, so the y-intercept is (0, −5).
A useful sketch therefore shows x-intercepts (1, 0) and (5, 0), maximum (3, 4), y-intercept (0, −5), and symmetry about x = 3. Exact artistic scale is less important than correct structure when only a sketch is requested.
Worked Example 2: Convert to Completed-Square Form
Find the turning point of y = x² + 6x + 2. Completing the square gives x² + 6x + 2 = (x + 3)² − 9 + 2 = (x + 3)² − 7.
Therefore the minimum point is (−3, −7). The axis of symmetry is x = −3.
The y-intercept is 2. The x-intercepts solve (x + 3)² = 7, so x = −3 ± √7. These roots lie equal distances from the symmetry line, as expected.
Repeated Roots and Tangency to the x-Axis
The function y = (x − 4)² has only one distinct root, x = 4. Its graph touches the x-axis at (4, 0) and turns back rather than crossing.
This is the graphical meaning of a repeated root. Algebraically the factor x − 4 occurs twice; geometrically the x-axis is tangent to the parabola at the turning point.
Intersections Solve Two Relationships Simultaneously
If y = f(x) and y = g(x), their intersection points satisfy f(x) = g(x). Graphically, the curves share the same x and y values there. Algebraically, solving f(x) = g(x) finds the same x-coordinates.
This makes graphs useful for estimating solutions to equations that may be difficult to solve exactly. It also means algebra can verify a graph estimate when exact solving is possible.
Worked Example 3: Line Meets a Quadratic
Find the intersection points of y = x² − 2x − 3 and y = x + 1. Set the expressions equal: x² − 2x − 3 = x + 1.
Rearrange to x² − 3x − 4 = 0, giving (x − 4)(x + 1) = 0. Therefore x = 4 or x = −1.
Using y = x + 1 gives y = 5 or y = 0. The intersection points are (4, 5) and (−1, 0).
This example links the graph work to Quadratic Equations and Word Problems.
Power Functions Form Several Distinct Families
A power function can be written y = axⁿ for a fixed exponent n. Different exponents create very different shapes. The family includes familiar lines, parabolas, cubic curves and reciprocal-type graphs.
| Function | Key behaviour |
|---|---|
| y = x | straight line through the origin |
| y = x² | upward parabola, symmetric about y-axis |
| y = x³ | increasing S-shaped curve through origin |
| y = 1/x | two branches; undefined at x=0 |
| y = 1/x² | positive two-branch curve; undefined at x=0 |
| y = 1 | horizontal line |
The exponent controls symmetry and long-run behaviour. Even positive powers such as x² give matching outputs for x and −x. Odd positive powers such as x³ reverse sign with x.
Reciprocal Graphs Have Asymptotic Behaviour
For y = 1/x, the function is undefined at x = 0. As x approaches zero from the positive side, y becomes very large and positive; from the negative side, y becomes very large in magnitude and negative.
As |x| becomes large, 1/x approaches zero without becoming zero. The axes x = 0 and y = 0 act as asymptotes in this graph.
For y = 1/x², both branches are positive because x² is positive for nonzero real x. The graph is symmetric about the y-axis.
Worked Example 4: Compare x² and x³
At x = −2, x² = 4 while x³ = −8. At x = 2, x² = 4 while x³ = 8. The square loses the sign because two negative factors produce a positive product; the cube retains it because three negative factors produce a negative result.
Thus y = x² is symmetric about the y-axis, while y = x³ has rotational symmetry about the origin. The algebraic exponent explains the graphical symmetry.
Exponential Functions Put the Variable in the Exponent
In y = 2ˣ, the base 2 is fixed and x changes. This is fundamentally different from y = x², where x is the base and the exponent is fixed.
For y = 2ˣ, when x increases by 1, the output doubles. Values include 2⁻² = 1/4, 2⁻¹ = 1/2, 2⁰ = 1, 2¹ = 2 and 2² = 4.
The graph is always positive, crosses the y-axis at (0, 1), increases rapidly for positive x, and approaches the x-axis as x becomes very negative without reaching it.
Worked Example 5: A Simple Exponential Model
A simplified model is P = 150(2ᵗ), where t is measured in time units. Find P at t = 0, 1, 2 and 3.
The values are 150, 300, 600 and 1,200. The output doubles whenever t increases by one.
This is a mathematical growth model, not a claim that a real population or investment must double forever. Real applications require assumptions and often additional constraints.
Quadratic Growth and Exponential Growth Are Different
The function x² grows by squaring the input. The function 2ˣ grows by repeatedly multiplying the output by a fixed factor as the input increases by equal steps.
At small positive x, either function may be larger. At x = 2 they are equal at 4. At x = 4, x² = 16 and 2ˣ = 16. At x = 10, x² = 100 while 2ˣ = 1,024. Exponential growth eventually outpaces the quadratic in this comparison.
Graphs make that change in relative growth visible. Tables provide exact sampled values. Equations describe the rule compactly. No single representation has to do every job.
Transformations: Changing the Rule Changes the Graph Predictably
Starting from y = x², replacing x by x − 3 shifts the graph three units right: y = (x − 3)². Adding 2 shifts it two units up: y = x² + 2. Multiplying the entire function by −1 reflects it in the x-axis.
Similar ideas apply to other graph families. For example, y = 2ˣ + 3 is the graph of y = 2ˣ shifted three units upward. Its horizontal asymptotic level becomes y = 3 rather than y = 0.
The exact transformation vocabulary used by schools may vary in emphasis. The important idea is that changing a formula in a controlled way produces a predictable change in its graph.
Gradient of a Curve at a Point
A straight line has one constant gradient. A curve changes gradient from point to point. To estimate the gradient at one point, draw a tangent—a straight line that locally follows the direction of the curve at that point.
Then choose two well-separated points on the tangent line, not necessarily points on the original curve, and calculate rise/run. Using widely separated tangent points reduces the relative effect of reading error.
The answer is an estimate because the tangent is drawn and coordinates are read from a graph. State suitable precision rather than reporting more decimal places than the diagram supports.
Worked Example 6: Estimate a Tangent Gradient
Suppose a tangent drawn to a curve at a point passes approximately through (1, 2.5) and (5, 10.1). Estimate the gradient.
Gradient ≈ (10.1 − 2.5)/(5 − 1) = 7.6/4 = 1.9.
The two chosen coordinates lie on the tangent. They do not have to be original data points on the curve. The quality of the estimate depends on how accurately the tangent represents the local direction.
Positive, Zero and Negative Gradient on a Curve
Where a curve rises from left to right, the tangent gradient is positive. Where it falls, the gradient is negative. At a smooth turning point of a parabola, the tangent is horizontal and its gradient is zero.
This gives another check on the turning point. For y = (x − 3)² − 5, the graph decreases before x = 3 and increases after x = 3. At x = 3, the tangent is horizontal.
Worked Example 7: Read Change From a Contextual Graph
A simplified distance-time curve becomes steeper as time increases. What does that suggest about speed? The gradient of a distance-time graph represents speed under the usual interpretation. A steeper positive gradient therefore suggests a larger speed.
If the curve becomes flatter, speed is decreasing. A horizontal section has zero gradient and suggests no change in distance during that interval.
This interpretation depends on the axes and units. A steep graph does not mean “fast” unless the vertical and horizontal quantities make gradient represent speed.
Graph Scale Can Change Visual Impression
Changing axis scales can make the same data look steeper or flatter. Therefore do not judge gradient from appearance alone. Read the numerical scale and calculate the ratio of changes.
This is one reason graph interpretation is mathematical judgement rather than picture reading. Labels, units, scales and domain all constrain what the graph means.
Domain Matters
The function y = x² is defined for all real x, but a contextual model may use only x ≥ 0 because x represents time after a starting point. The mathematical graph and the modelled domain are not automatically identical.
Similarly, a reciprocal function excludes x = 0. An exponential function such as 2ˣ is defined for every real x and always produces positive output. These domain and range features are part of graph understanding.
Four Common Errors
Reading the sign inside a completed square incorrectly: y = (x + 4)² has turning point x = −4, not 4. Repair by solving x + 4 = 0.
Treating roots as coordinate pairs or vice versa: x = 2 is a root; (2, 0) is the corresponding intercept. Repair by reading what the question requests.
Assuming 2ˣ crosses the x-axis because it gets close: positive exponential values approach zero but do not become zero. Repair by evaluating the function and understanding its range.
Using two points on the curve instead of on the tangent: that finds an average gradient between curve points, not the tangent gradient at one point. Repair by drawing and using the tangent line itself.
A Graph-Sketch Checklist
- Identify the graph family.
- Find exact intercepts when they are available.
- Find or infer the turning point for a quadratic.
- State the axis of symmetry when relevant.
- Check opening direction or long-run behaviour.
- Respect domain restrictions and asymptotes.
- Label important coordinates.
This checklist focuses on structure rather than plotting many unnecessary points. A table is useful when needed, but the equation often reveals major features directly.
Independent Practice
1. State the turning point and axis of symmetry of y = (x − 4)² + 2.
2. State the turning point of y = −(x + 3)² + 5.
3. For y = (x − 2)(x + 6), state the roots and axis of symmetry.
4. For y = −(x − 1)(x − 7), find the maximum point.
5. Write x² + 8x + 7 in completed-square form and state the minimum point.
6. Find the intersection points of y = x² − 1 and y = x + 1.
7. State the symmetry of y = x² and y = x³.
8. State the domain restriction of y = 1/x.
9. Compare the signs of y = 1/x and y = 1/x² when x is negative.
10. Evaluate 3ˣ at x = −1, 0, 1 and 2.
11. Explain why y = 3ˣ has no x-intercept.
12. A tangent passes through approximately (2, 5.2) and (8, 17.8). Estimate its gradient.
13. A curve has a horizontal tangent at x = 4. What is the tangent gradient there?
14. Explain why changing graph scale can alter visual steepness without changing the mathematical gradient.
15. A model y = 100(1.5)ᵗ is used for t ≥ 0. Find y at t = 0, 1 and 2 and state the model’s domain restriction.
Explained Answers
1. Turning point (4, 2); axis x = 4.
2. x + 3 = 0 gives x = −3. Maximum point is (−3, 5).
3. Roots are 2 and −6. Their midpoint is −2, so the axis is x = −2.
4. Roots are 1 and 7, so the axis is x = 4. Substituting x = 4 gives −(3)(−3) = 9. Maximum point: (4, 9).
5. x² + 8x + 7 = (x + 4)² − 9. Minimum point: (−4, −9).
6. Set x² − 1 = x + 1. Then x² − x − 2 = 0, so x = 2 or −1. Corresponding y-values are 3 and 0. Intersections: (2, 3) and (−1, 0).
7. y = x² is symmetric about the y-axis. y = x³ has rotational symmetry about the origin.
8. x ≠ 0 because division by zero is undefined.
9. When x is negative, 1/x is negative while 1/x² is positive.
10. Values are 1/3, 1, 3 and 9.
11. 3ˣ is positive for every real x, so it never equals zero.
12. Gradient ≈ (17.8 − 5.2)/(8 − 2) = 12.6/6 = 2.1.
13. A horizontal tangent has gradient 0.
14. Visual angle depends on axis scaling. Mathematical gradient uses actual coordinate changes, so the numerical ratio can stay unchanged even when the drawn line looks steeper or flatter.
15. y-values are 100, 150 and 225. The stated model uses t ≥ 0, even though the exponential expression itself is defined for all real t.
A Reliable Graph-Checking Routine
Check important coordinates by substitution. Check roots by setting y = 0. Check the turning point against symmetry. Check the y-intercept by setting x = 0. For a reciprocal or exponential graph, check domain and sign before trusting the shape.
For tangent gradients, inspect whether the chosen points really lie on the tangent and whether the axis scales have been read correctly. A calculator can divide the changes, but it cannot decide whether the tangent was drawn sensibly.
Teacher and Parent Prompts
Ask “What feature can you read directly from this form?” before requesting a full sketch. For a factorised quadratic, that may be the roots. For completed-square form, it may be the turning point. For a reciprocal graph, it may be the excluded input.
For extension, give the graph features first and ask the learner to construct a possible equation. Roots 2 and 5 with an upward-opening parabola suggest y = a(x − 2)(x − 5) for positive a. This reverses the usual direction of reasoning.
Questions Students Often Ask
Do I need many plotted points for a sketch? Not always. Use exact structural features first, then enough additional points to confirm the shape if needed.
Why is the sign inside (x − p)² reversed when reading the turning point? Because the square is zero when x − p = 0, so x = p. For (x + 4)², that means x = −4.
Can an exponential graph cross the x-axis? A basic positive-base exponential such as 2ˣ remains positive and does not cross y = 0. Vertical shifts can change where the graph lies relative to the axis.
Is tangent gradient exact? When found from a drawn tangent on a graph, it is an estimate. Exact gradient methods belong to later calculus work.
Continue the Secondary 3 Learning Route
Continue with Linear Inequalities and Number-Line Reasoning for solution regions, Set Language, Venn Diagrams and Counting for classification and overlap, and Matrices, Operations and Information Representation for structured numerical representation.
Graph competence means being able to move between rule, table, curve, coordinate and interpretation without losing the relationship. Return to the Secondary Mathematics Hub for the complete learning route.
