A quadratic graph is a relationship made visible. Its equation tells us how y depends on x; its factorised form can reveal roots; its expanded form shows coefficients; its turning point marks a maximum or minimum; and its axis of symmetry explains why pairs of x-values can produce the same y-value.
This thirty-seventh Secondary 4 Mathematics Learning Guide develops quadratic functions as a graph-and-algebra system. It belongs to the Secondary Mathematics Hub and S1–S4 Capability Map.
It deepens the earlier Quadratic Equations and Algebraic Fractions, Algebra, Functions and Graphs Under Mixed-Topic Conditions and Power, Reciprocal and Exponential Graphs guides by focusing specifically on the geometry of a quadratic graph.
The basic quadratic form
A quadratic function can be written as:
y=ax²+bx+c, where a≠0.
If a>0, the parabola opens upward and the turning point is a minimum. If a<0, the parabola opens downward and the turning point is a maximum.
The coefficient a also affects width: larger |a| makes the parabola narrower; smaller non-zero |a| makes it wider.
Worked Example 1 | Read the y-intercept
For y=2x²−5x−3, find the y-intercept.
At the y-axis, x=0:
y=2(0)²−5(0)−3=−3.
The y-intercept is therefore (0,−3).
Roots are x-intercepts
A root of a quadratic is an x-value for which y=0. On the graph, roots are where the parabola crosses or touches the x-axis.
If a quadratic factorises as y=a(x−p)(x−q), then x=p and x=q are roots.
Worked Example 2 | Roots from factorised form
Find the roots of y=(x−4)(x+2).
Set y=0:
(x−4)(x+2)=0.
Therefore x=4 or x=−2. The x-intercepts are (4,0) and (−2,0).
The axis of symmetry lies midway between two roots
For a parabola with two roots p and q, the vertical axis of symmetry is halfway between them:
x=(p+q)/2.
This is a geometric consequence of the left-right symmetry of the parabola.
Worked Example 3 | Axis of symmetry and turning point
For y=(x−4)(x+2), find the axis of symmetry and turning point.
Roots are 4 and −2, so:
x=(4+(−2))/2=1.
Substitute x=1:
y=(1−4)(1+2)=−3×3=−9.
Turning point=(1,−9).
Because the coefficient of x² is positive, this is a minimum.
Expanded and factorised forms reveal different information
| Form | What it reveals quickly |
|---|---|
| ax²+bx+c | Coefficient signs, y-intercept c |
| a(x−p)(x−q) | Roots p and q |
| graph | Turning point, symmetry, range, intersections |
Strong algebra means switching form for a purpose, not expanding everything automatically.
Worked Example 4 | Move between forms
Expand y=(x−3)(x+5).
y=x²+2x−15.
The factorised form reveals roots 3 and −5. The expanded form reveals y-intercept −15. Both describe the same graph.
One repeated root means the graph touches the x-axis
If y=(x−3)², there is one repeated root x=3. The graph touches the x-axis at (3,0) and turns there instead of crossing.
Worked Example 5 | Repeated root
For y=x²−6x+9:
x²−6x+9=(x−3)².
The graph has a repeated root and turning point at (3,0).
No real roots means no x-axis intersection
A quadratic can remain entirely above or below the x-axis. For example, y=x²+4 has minimum value 4, so it never reaches y=0 in the real plane.
The graph tells us immediately that the equation x²+4=0 has no real solution.
Worked Example 6 | Minimum from graph structure
For y=(x−2)²+5, what is the minimum value of y?
Because a square is never negative, (x−2)²≥0.
Minimum y=5, occurring at x=2.
The turning point is (2,5).
Symmetric x-values produce equal y-values
If the axis of symmetry is x=3, then x=1 and x=5 are equally far from the axis and therefore have the same y-value.
This can be used to complete tables quickly or check plotted points.
Worked Example 7 | Use symmetry in a table
A quadratic has axis of symmetry x=4. If y=11 at x=2, what is y at x=6?
Both x-values are 2 units from x=4.
Therefore y=11 at x=6.
Intersections solve equations graphically
Where y=x²−4 and y=2x meet, the same x-value satisfies both equations. Therefore graphical intersections solve:
x²−4=2x.
Rearranging gives x²−2x−4=0. The graph and algebra represent the same simultaneous condition.
Worked Example 8 | Quadratic meets a horizontal line
Find the x-values where y=x²−5x+6 meets y=2.
Set x²−5x+6=2:
x²−5x+4=0.
(x−1)(x−4)=0, so x=1 or x=4.
The horizontal line y=2 crosses the parabola at two points whose x-coordinates are 1 and 4.
Quadratics can model area
Suppose a rectangle has sides x and 12−x. Its area is:
A=x(12−x)=−x²+12x.
The downward-opening parabola tells us the area has a maximum.
Worked Example 9 | Maximum rectangular area
For A=−x²+12x, roots are x=0 and x=12. The axis of symmetry is halfway:
x=(0+12)/2=6.
A(6)=−36+72=36.
The maximum area is 36 square units, attained when both sides are 6.
Quadratic inequalities can be read from the graph
For y=(x−2)(x−5), the graph crosses the x-axis at x=2 and x=5 and opens upward.
Therefore y<0 between the roots and y>0 outside them.
Worked Example 10 | Read a sign interval
Solve (x−2)(x−5)<0.
2<x<5.
The graph is below the x-axis only between the two roots.
Common failure modes
| Error | Cause | Repair |
|---|---|---|
| Calls y-intercept a root | Axes confused | Root means y=0; y-intercept means x=0 |
| Finds axis of symmetry by averaging y-values | Horizontal and vertical information mixed | Average root x-values |
| Assumes every quadratic has two real roots | Graph-position possibilities ignored | Allow two, one repeated, or no real roots |
| Uses only expanded form | Representation switching weak | Factor when roots are useful |
| Misses maximum/minimum meaning | Turning point treated as decoration | Relate opening direction to extremum |
| Plots asymmetric points around the vertex | Parabolic symmetry ignored | Use the axis as a check |
Independent practice
- Find the roots of y=(x−6)(x+1).
- Find the axis of symmetry of the graph in Question 1.
- Find the turning point of y=(x−6)(x+1).
- State the minimum value of y=(x+3)²−4.
- Solve x²−7x+10=0.
- Solve (x−1)(x−4)<0 using graph structure.
Explained answers
1. x=6 or −1.
2. x=(6−1)/2=2.5.
3. At x=2.5, y=(−3.5)(3.5)=−12.25. Turning point=(2.5,−12.25).
4. Minimum y=−4, at x=−3.
5. (x−5)(x−2)=0, so x=2 or 5.
6. Upward-opening parabola is below the x-axis between roots: 1<x<4.
Final thought
A quadratic function is easier when algebra and graph are treated as one object. Roots tell where the graph meets the x-axis, symmetry locates the centre of the curve, and the turning point reveals the extreme value.
Use the form that exposes the feature you need: roots, intercept, symmetry, turning point or model behaviour.
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