A graph can solve an equation because an equation is a condition on coordinates. Roots are points where a graph meets the x-axis. Simultaneous solutions are intersections of two graphs. A tangent estimates how quickly a curved relationship is changing at one point.
This Secondary 3 Mathematics Learning Guide develops graphical solutions, intersections, roots, simultaneous equations, tangent gradients and rate-of-change interpretation. It deepens the earlier Quadratic, Power and Exponential Graphs guide by focusing on what graphs can solve and measure.
The current 2027 SEC G3 Mathematics syllabus listed by SEAB includes graphical representation and solving, alongside estimation of the gradient of a curve using a tangent. This guide keeps algebraic, numerical and graphical interpretations connected.
A Root Is an x-Coordinate Where y=0
For y=f(x), a root of f(x)=0 occurs where the graph crosses or touches the x-axis. The corresponding coordinate is (root,0).
This means solving an equation and reading an intercept are two representations of the same condition.
Worked Example 1: Roots of a Quadratic
For y=x²−5x+6, factorisation gives (x−2)(x−3). Therefore the roots are x=2 and x=3.
Graphically, the parabola meets the x-axis at (2,0) and (3,0). If a plotted graph gave estimates near 2.0 and 3.0, the two methods would agree.
Touching the Axis Represents a Repeated Root
For y=(x−4)², the graph touches the x-axis at x=4 and turns back. Algebraically, x=4 is a repeated root.
A graph can therefore reveal not only where y=0 but also whether the curve crosses or merely touches the axis.
Intersections Solve Two Equations Simultaneously
If y=f(x) and y=g(x), an intersection point satisfies both equations at once. Its x-coordinate solves f(x)=g(x), and its y-coordinate is the common output.
This is the graphical meaning of simultaneous equations.
Worked Example 2: Line Meets Line
Find the intersection of y=3x−2 and y=−x+10.
Set the equations equal: 3x−2=−x+10. Then 4x=12, so x=3 and y=7.
The intersection is (3,7). On a graph, that is the unique point lying on both lines.
Worked Example 3: Line Meets Quadratic
Find the intersection points of y=x²−4 and y=x+2.
Set x²−4=x+2, giving x²−x−6=0. Factorise: (x−3)(x+2)=0.
Thus x=3 or x=−2. Corresponding y-values from y=x+2 are 5 and 0. The intersections are (3,5) and (−2,0).
Graphical Solutions Are Often Estimates
When a solution is read from a printed or hand-drawn graph, the result depends on plotting accuracy, scale and line thickness. It should usually be reported to a precision supported by the diagram.
An algebraic solution may be exact while a graphical solution is approximate. Neither representation is automatically superior; each has a different job.
Worked Example 4: Approximate Root From a Sign Change
Suppose a graph of y=f(x) has f(1.7)=−0.4 and f(1.8)=0.2, with the curve continuous between these points. A root lies between 1.7 and 1.8.
A graph might estimate the crossing around x≈1.77, depending on the curve and scale. The sign change brackets the root but does not by itself prove the exact value.
Graph Scale Controls Reading Precision
If one small square represents 1 unit, reading 2.37 from the graph may be unjustified. If one small square represents 0.05 units, greater precision may be reasonable.
The diagram determines the defensible precision. Extra decimal places from visual interpolation do not create extra evidence.
Gradient of a Straight Line Is Constant
For a straight line, gradient = vertical change / horizontal change. Any two distinct points on the line give the same ratio.
For y=2.5x−4, gradient is 2.5 everywhere. The line rises 2.5 units in y for every 1 unit increase in x.
A Curve Does Not Have One Gradient
A curve changes direction. The gradient near one point may differ from the gradient near another.
To estimate the gradient at a particular point, draw a tangent line that matches the local direction of the curve there, then calculate the gradient of that tangent.
Worked Example 5: Tangent Gradient
A tangent drawn to a curve passes approximately through (2,3.4) and (8,15.1).
Gradient ≈ (15.1−3.4)/(8−2)=11.7/6=1.95.
The chosen points should lie on the tangent, not merely on the original curve. Using points far apart on the tangent reduces the relative effect of reading error.
Positive, Zero and Negative Tangent Gradient
A positive tangent gradient means the curve is increasing at that point. A negative gradient means it is decreasing. A horizontal tangent has gradient zero.
At the smooth turning point of an upward-opening parabola, the tangent is horizontal. The curve changes from decreasing to increasing and the tangent gradient is zero.
Worked Example 6: Turning Point Interpretation
For y=(x−5)²−7, the turning point is (5,−7). The graph decreases for x values before 5 and increases after 5.
At x=5, the tangent is horizontal, so its gradient is 0.
Gradient Can Represent a Rate
The meaning of gradient depends on the axes. On a distance-time graph, distance/time has units of speed. On a cost-quantity graph, gradient may represent cost per item. On a temperature-time graph, gradient may represent degrees per minute.
Always interpret gradient with units, not as a bare number.
Worked Example 7: Distance-Time Graph
A straight segment of a distance-time graph passes through (2 h,90 km) and (5 h,270 km).
Gradient=(270−90)/(5−2)=180/3=60 km/h.
The units confirm the interpretation: kilometres divided by hours.
Average Gradient and Instantaneous Gradient Are Different Ideas
The gradient of a chord joining two points on a curve gives an average rate of change over an interval. The gradient of a tangent estimates the rate of change at one point.
Do not use two arbitrary curve points when the question asks for tangent gradient. That computes a different quantity.
Worked Example 8: Average Versus Tangent
Suppose a quantity rises from 20 to 50 while time goes from 2 to 8. The average rate over that interval is (50−20)/(8−2)=5 units per time unit.
The tangent gradient at time 5 could be larger or smaller than 5 depending on the curvature. The average rate does not determine the instantaneous rate.
Intersections Can Encode Thresholds
If one graph represents cost and another revenue, their intersection can represent a break-even point under the model. If one graph represents two travel plans, an intersection may represent equal distance or equal time depending on the axes.
Interpret the coordinate in the context rather than reporting x and y without meaning.
Worked Example 9: Equal-Cost Model
Plan A costs C=12+3n. Plan B costs C=24+2n, where n is number of sessions.
At intersection, 12+3n=24+2n, so n=12. Cost is 48. Therefore the plans cost the same at (12,48) in the (sessions,cost) coordinate model.
Calculator Tables Can Support Graph Reasoning
When plotting a function, a table of values can identify sign changes, approximate intersections and suitable graph windows. The calculator supports representation; it does not replace interpretation.
A poor graph window can hide roots or intersections. Before concluding that no solution exists, check whether the relevant region is visible.
Common Errors
Giving an intercept coordinate when only the root is requested: distinguish x-value from coordinate pair.
Using curve points instead of tangent points: this gives average gradient rather than tangent gradient.
Reporting too many decimal places from a graph: match precision to the scale.
Forgetting units on a contextual gradient: gradient has vertical-axis units divided by horizontal-axis units.
Independent Practice
1. Find the roots of y=x²−7x+12.
2. Find the intersection of y=2x+3 and y=−x+12.
3. Find the intersections of y=x² and y=2x+3.
4. Explain why a graph touching the x-axis can represent a repeated root.
5. A tangent passes through (1,2.6) and (7,14.0). Estimate its gradient.
6. A curve has a horizontal tangent at x=3. What is the tangent gradient?
7. A distance-time line passes through (1 h,40 km) and (4 h,220 km). Find the speed represented.
8. A cost graph has vertical axis dollars and horizontal axis items. What are the units of gradient?
9. Explain the difference between the gradient of a chord and the gradient of a tangent.
10. Plan A is C=10+4n and Plan B is C=25+n. Find the break-even session count and cost.
Explained Answers
1. (x−3)(x−4)=0, so roots 3 and 4.
2. 2x+3=−x+12 gives x=3, y=9; intersection (3,9).
3. x²=2x+3 gives x²−2x−3=0=(x−3)(x+1), so intersections (3,9) and (−1,1).
4. The function reaches y=0 but does not change sign across the touching point, matching a repeated factor.
5. (14.0−2.6)/(7−1)=11.4/6=1.9.
6. 0.
7. (220−40)/(4−1)=60 km/h.
8. dollars per item.
9. Chord gradient is average change between two curve points; tangent gradient estimates change at one point.
10. 10+4n=25+n gives n=5; cost=30.
Continue the Secondary 3 Learning Route
Continue with Angles, Parallel Lines, Polygons and Symmetry, Histograms, Cumulative Frequency, Box Plots and Standard Deviation, and Mathematical Reasoning, Proof and Communication.
