Quadratic and Other Graphs

Beyond straight lines, GCSE Mathematics requires you to recognise, plot, and interpret quadratic graphs, cubic graphs, reciprocal graphs, and exponential graphs. This lesson covers all the graph types in the Edexcel GCSE (1MA1) specification.

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Key Vocabulary

Term	Meaning
Parabola	The U-shaped (or ∩-shaped) curve of a quadratic graph
Turning point (vertex)	The highest or lowest point on a parabola
Line of symmetry	The vertical line through the turning point
Root	Where the graph crosses the x-axis (y = 0)
Asymptote	A line that a graph approaches but never reaches

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Quadratic Graphs: $y = ax^{2} + bx + c$y=ax2+bx+c

Shape

• If a > 0: the parabola opens upward (U-shape) — it has a minimum turning point.
• If a < 0: the parabola opens downward (∩-shape) — it has a maximum turning point.

Key Features

• Roots: the x-values where y = 0 (where the graph crosses the x-axis). Found by solving $ax^{2} + bx + c = 0$ax2+bx+c=0.
• y-intercept: the value of c (set x = 0).
• Line of symmetry: x = −b/(2a).
• Turning point: substitute x = −b/(2a) into the equation to find y. [Or use completing the square: $y = a(x + p)^{2} + q$y=a(x+p)2+q gives turning point (−p, q).]

Worked Example 1

Sketch $y = x^{2} - 4x + 3$y=x2−4x+3.

Roots: $x^{2} - 4x + 3 = 0 \to (x - 1)(x - 3) = 0 \to x = 1$x2−4x+3=0→(x−1)(x−3)=0→x=1 and x = 3

y-intercept: When x = 0: y = 3. So the graph passes through (0, 3).

Line of symmetry: $x = -(-4)/(2 \times 1) = 2$x=−(−4)/(2×1)=2

Turning point: $y = (2)^{2} - 4(2) + 3 = 4 - 8 + 3 = -1$y=(2)2−4(2)+3=4−8+3=−1. Turning point = (2, −1) — a minimum.

Worked Example 2

Sketch $y = -x^{2} + 2x + 8$y=−x2+2x+8.

Roots: $-x^{2} + 2x + 8 = 0 \to x^{2} - 2x - 8 = 0 \to (x - 4)(x + 2) = 0 \to x = 4$−x2+2x+8=0→x2−2x−8=0→(x−4)(x+2)=0→x=4 and x = −2

y-intercept: (0, 8)

Turning point: $x = -2/(2 \times -1) = 1$x=−2/(2×−1)=1. y = −1 + 2 + 8 = 9. Turning point = (1, 9) — a maximum (since a = −1 < 0).

Worked Example 3 — Using Completing the Square [H]

Find the turning point of $y = x^{2} + 6x + 5$y=x2+6x+5 by completing the square.

$y = (x + 3)^{2} - 9 + 5 = (x + 3)^{2} - 4$y=(x+3)2−9+5=(x+3)2−4

Turning point = (−3, −4) — a minimum.

Worked Example 4 — Plotting from a Table

Plot $y = x^{2} - 2x - 3$y=x2−2x−3 for $-2 \leq x \leq 4$−2≤x≤4.

x	−2	−1	0	1	2	3	4
$x^{2}$x2	4	1	0	1	4	9	16
−2x	4	2	0	−2	−4	−6	−8
−3	−3	−3	−3	−3	−3	−3	−3
y	5	0	−3	−4	−3	0	5

Plot the points and join with a smooth curve (NOT straight line segments).

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Cubic Graphs: $y = ax^{3} + bx^{2} + cx + d$y=ax3+bx2+cx+d

Shape

• If a > 0: the graph goes from bottom-left to top-right (like a stretched S).
• If a < 0: the graph goes from top-left to bottom-right (reflected S).

Key Features

• May have 1 or 3 real roots (x-intercepts).
• The y-intercept is d (set x = 0).
• May have two turning points (a local maximum and a local minimum) or none.

Worked Example 5

Sketch $y = x^{3} - 3x$y=x3−3x.

Roots: $x^{3} - 3x = 0 \to x(x^{2} - 3) = 0 \to x = 0$x3−3x=0→x(x2−3)=0→x=0, $x = \sqrt{3} \approx 1.73$x=3​≈1.73, $x = -\sqrt{3} \approx -1.73$x=−3​≈−1.73

y-intercept: (0, 0)

The graph passes through the origin and crosses the x-axis at $\pm\sqrt{3}$±3​. It has a local maximum at approximately (−1, 2) and a local minimum at approximately (1, −2).

Worked Example 6

Sketch y = (x − 1)(x + 2)(x − 3).

Roots: x = 1, x = −2, x = 3

y-intercept: (0 − 1)(0 + 2)(0 − 3) = (−1)(2)(−3) = 6. So (0, 6).

The coefficient of $x^{3}$x3 is positive (expanding gives $x^{3} +$x3+ …), so the curve goes from bottom-left to top-right.

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Reciprocal Graphs: y = a/x

Shape

• Two separate curves (hyperbolas), one in the first quadrant and one in the third quadrant (if a > 0).
• If a < 0, the curves are in the second and fourth quadrants.
• Asymptotes: the x-axis (y = 0) and the y-axis (x = 0). The graph never touches or crosses these lines.

Worked Example 7

Sketch y = 3/x.

x	−3	−1	−0.5	0.5	1	3
y	−1	−3	−6	6	3	1

The graph has two branches: one in Q1 (where both x and y are positive) and one in Q3 (where both are negative). It never crosses either axis.

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Exponential Graphs: $y = a^{x} [H]$y=ax[H]

Shape

• If a > 1: the graph rises steeply to the right and approaches 0 as $x \to -\infty$x→−∞.
• The graph always passes through (0, 1) since $a^{0} = 1$a0=1.
• The x-axis is an asymptote (the graph never reaches y = 0).

Worked Example 8

Sketch $y = 2^{x}$y=2x for $-3 \leq x \leq 4$−3≤x≤4.

x	−3	−2	−1	0	1	2	3	4
y	0.125	0.25	0.5	1	2	4	8	16

The curve passes through (0, 1), rises steeply for positive x, and approaches 0 for negative x.

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Recognising Graph Shapes

Equation type	Shape	Key feature
y = mx + c	Straight line	Constant gradient
$y = ax^{2} + bx + c$y=ax2+bx+c	Parabola (U or ∩)	One turning point
$y = ax^{3} +$y=ax3+ …	S-shaped curve	Up to two turning points
y = a/x	Hyperbola (two branches)	Two asymptotes
$y = a^{x}$y=ax	Exponential curve	Passes through (0, 1); asymptote y = 0

Graph Transformation Map

flowchart TD
    BASE[Base graph y = f of x] --> T1{Transformation type?}
    T1 -->|y = f of x + a| VS[Vertical shift up by a]
    T1 -->|y = f of x + a inside| HS[Horizontal shift LEFT by a]
    T1 -->|y = a times f of x| VST[Vertical stretch factor a]
    T1 -->|y = f of ax| HST[Horizontal stretch factor 1 over a]
    T1 -->|y = minus f of x| RX[Reflection in x-axis]
    T1 -->|y = f of minus x| RY[Reflection in y-axis]
    VS --> APPLY[Apply to all points]
    HS --> APPLY
    VST --> APPLY
    HST --> APPLY
    RX --> APPLY
    RY --> APPLY

This map summarises the transformations Higher candidates must recognise. Note the counter-intuitive rule: changes inside the bracket (e.g. f(x + a)) shift the graph in the opposite direction to what the sign suggests.

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Using Graphs to Solve Equations