Graphs and Transformations Notes (JC1 H2 Math) — Free PDF

Q: How do you find the set of values y cannot take for a rational function?

Cross-multiply to get a quadratic in x, treating y as a constant. Then impose discriminant ≥ 0 and solve the resulting inequality in y. The interval where the discriminant is negative is the gap in the range, and its endpoints must equal the y-coordinates of the turning points shown by the GC.

Q: How should I describe transformations of graphs in H2 Math?

Use only the words translation, stretch and reflection, and state the direction, axis and scale factor in full. Examples are "translation of 3 units in the negative x-direction" and "stretch parallel to the x-axis, scale factor 1 ÷ 2". Words such as "shift", "move", "compress" and "shrink" are penalised.

Q: How do you sketch y = 1 ÷ f(x) from the graph of y = f(x)?

Zeros of f become vertical asymptotes, and vertical asymptotes of f become hole-approaches to the x-axis. A maximum (p, q) becomes a minimum (p, 1 ÷ q) and vice versa, and a horizontal asymptote y = k (k ≠ 0) becomes y = 1 ÷ k. The sign is preserved on every interval, and points where f(x) = ±1 are unchanged.

Q: How do you convert parametric equations to a cartesian equation?

For algebraic forms, make t the subject of the simpler equation and substitute it into the other. For trigonometric forms, isolate cos θ and sin θ, square both and add, using cos²θ + sin²θ = 1. Then carry any restriction on the parameter over to x or y; for example, 0 ≤ θ ≤ π gives y ≥ 0 for x = 3 cos θ, y = 2 sin θ.

Graphs and transformations covers the standard curves, rational functions, transformations of y = f(x), modulus and reciprocal graphs, and parametric equations in the 9758 H2 Mathematics syllabus. Curve sketching is examined directly in Paper 1. Later topics also draw on this small catalogue of curves, including rational functions, areas and volumes of revolution, and complex loci in the Argand diagram.

Every sketch is marked against a fixed checklist: shape and stationary points, intercepts, and asymptotes. Coordinates must be stated exactly rather than as GC decimals. These notes cover the algebra, the TI-84 procedures and the exact descriptive wording that mark schemes expect.

What these notes cover

•The SIA sketching checklist
•Parabolas y² = ax and x² = by
•The ellipse and the two hyperbolas
•Graphing implicit conics on the GC
•Linear over linear: the rectangular hyperbola
•Quadratic over linear: oblique asymptotes
•Range gaps by the discriminant method
•The four basic transformations and their order
•Modulus graphs y = |f(x)| and y = f(|x|)
•Reciprocal graphs y = 1 ÷ f(x)
•Parametric equations and cartesian conversion
•Common mistakes and exam technique

Key definitions

SIA checklist: The fixed list a marker checks on every sketch. S is shape and stationary points, with coordinates. I is intercepts, with coordinates. A is asymptotes, given as equations and drawn dashed and labelled. Both axes, the origin O and the equation of each curve must also be labelled.
Oblique asymptote: A slanted line y = Q(x) that the curve approaches as x → ±∞. Q(x) is the linear quotient obtained by long division of a quadratic-over-linear rational function.
Range gap (forbidden band): The interval of y-values the curve never takes. It is the interval where the discriminant of the quadratic in x, formed by cross-multiplying, is negative.
Parametric curve: A curve given by x = f(t), y = g(t) together with a stated range of the parameter t. The range is part of the definition, so the same formulae with a different range give a different curve.
Hole-approach: Where f has a vertical asymptote x = b, the graph of y = 1 ÷ f(x) approaches (b, 0) but excludes that point. This is shown with a small open circle.

Key formulas

Ellipse, centre the origin

$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

The intercepts are (±a, 0) and (0, ±b), and the curve is symmetric about both axes. a > b is not required. The case a = b = r gives the circle x² + y² = r².

Hyperbola asymptotes (both orientations)

$y = \pm \frac{b}{a} x$

Both hyperbolas share this asymptote pair. x² ÷ a² − y² ÷ b² = 1 opens east–west with vertices (±a, 0). y² ÷ b² − x² ÷ a² = 1 opens north–south with vertices (0, ±b). The positive term decides the opening direction.

Asymptotes of a linear-over-linear function

$y = \frac{ax + b}{cx + d}: \quad x = -\frac{d}{c}, \quad y = \frac{a}{c}$

These are the vertical and horizontal asymptotes, and they require c ≠ 0 and ad ≠ bc. The curve is a rectangular hyperbola with centre (−d ÷ c, a ÷ c). Its lines of symmetry have gradient ±1 through the centre.

Oblique asymptote via long division

$y = \frac{ax^2 + bx + c}{dx + e} = Q(x) + \frac{R}{dx + e}$

Q(x) is the linear quotient and R is the constant remainder. The oblique asymptote is y = Q(x) and the vertical asymptote is x = −e ÷ d. Show the division line by line.

Range gap via the discriminant

$Ax^2 + Bx + C = 0 \text{ has real roots} \iff B^2 - 4AC \ge 0$

After cross-multiplying, B and C contain y, so this becomes a quadratic inequality in y. The y-values where the discriminant is negative form the gap in the range.

Inner-argument rule

$f(bx + a) = f\left(b\left(x + \frac{a}{b}\right)\right)$

Factorise the inner argument before describing combined transformations. Either order is valid. If you translate first, translate by a units. If you stretch first (parallel to the x-axis, scale factor 1 ÷ b), then translate by a ÷ b units.

Worked example

The curve C has equation y = (x² − 3x + 7) ÷ (x − 2), x ≠ 2. Find the asymptotes, the axial intercepts and the turning points of C, and the set of values that y cannot take.

1Long division gives x² − 3x + 7 = (x − 1)(x − 2) + 5, so y = x − 1 + 5 ÷ (x − 2).
2The vertical asymptote is x = 2, where the denominator is zero, and the oblique asymptote is y = x − 1, since 5 ÷ (x − 2) → 0 as x → ±∞.
3When x = 0, y = 7 ÷ (−2) = −7 ÷ 2, so the y-intercept is (0, −7 ÷ 2).
4For x-intercepts, x² − 3x + 7 = 0 has discriminant (−3)² − 4(1)(7) = −19 < 0, so there are no x-intercepts.
5From the divided form, dy/dx = 1 − 5 ÷ (x − 2)², which is zero when (x − 2)² = 5, so x = 2 ± √5.
6At x = 2 + √5, y = 1 + 2√5 ≈ 5.47, which is a minimum point.
7At x = 2 − √5, y = 1 − 2√5 ≈ −3.47, which is a maximum point.
8For the range, cross-multiplying y(x − 2) = x² − 3x + 7 gives the quadratic in x: x² − (3 + y)x + (7 + 2y) = 0.
9For real x, (3 + y)² − 4(7 + 2y) ≥ 0, which simplifies to y² − 2y − 19 ≥ 0.
10The roots are y = (2 ± √(4 + 76)) ÷ 2 = 1 ± 2√5, and the quadratic opens upwards, so y ≤ 1 − 2√5 or y ≥ 1 + 2√5.
11Hence y cannot take values in 1 − 2√5 < y < 1 + 2√5 (approximately −3.47 < y < 5.47), which matches the turning-point y-values found earlier.

Watch out: The local maximum lies below the local minimum. This is normal for these curves, not an error. The gap endpoints are attained (discriminant = 0), and losing the equality case there is a common 1-mark leak.

Common mistakes

Direction of y = f(x + a)

Wrong: y = f(x + 3) has a +3 in it, so the graph moves 3 units in the positive x-direction.

Right: Replacing x by x + 3 is a translation of 3 units in the negative x-direction. If (5, 7) lies on y = f(x), the value 7 now occurs where x + 3 = 5, that is at x = 2.

Treating |f(x)| and f(|x|) as the same graph

Wrong: There is a modulus somewhere, so reflect something and hope for the best.

Right: |f(x)| acts on the output: keep everything on or above the x-axis, reflect the parts below upward, and draw sharp corners at crossing zeros. f(|x|) acts on the input: keep the right half, discard the left half entirely and mirror in the y-axis, so the result is an even graph.

Copying GC decimals onto the sketch

Wrong: CALC says X = 2.9999999, so the intercept is (2.9999999, 0), or (3.00, 0) to 3 s.f.

Right: Recover the exact value from the equation, such as x = 3 or y = 1 + 2√5, and label the sketch with that. A decimal where an exact value exists is penalised.

Questions students ask

How do you find the set of values y cannot take for a rational function?

Cross-multiply to get a quadratic in x, treating y as a constant. Then impose discriminant ≥ 0 and solve the resulting inequality in y. The interval where the discriminant is negative is the gap in the range, and its endpoints must equal the y-coordinates of the turning points shown by the GC.

How should I describe transformations of graphs in H2 Math?

Use only the words translation, stretch and reflection, and state the direction, axis and scale factor in full. Examples are "translation of 3 units in the negative x-direction" and "stretch parallel to the x-axis, scale factor 1 ÷ 2". Words such as "shift", "move", "compress" and "shrink" are penalised.

How do you sketch y = 1 ÷ f(x) from the graph of y = f(x)?

Zeros of f become vertical asymptotes, and vertical asymptotes of f become hole-approaches to the x-axis. A maximum (p, q) becomes a minimum (p, 1 ÷ q) and vice versa, and a horizontal asymptote y = k (k ≠ 0) becomes y = 1 ÷ k. The sign is preserved on every interval, and points where f(x) = ±1 are unchanged.

How do you convert parametric equations to a cartesian equation?

For algebraic forms, make t the subject of the simpler equation and substitute it into the other. For trigonometric forms, isolate cos θ and sin θ, square both and add, using cos²θ + sin²θ = 1. Then carry any restriction on the parameter over to x or y; for example, 0 ≤ θ ≤ π gives y ≥ 0 for x = 3 cos θ, y = 2 sin θ.

In the full PDF

•All 18 figures, including labelled sketches of the standard conics, rectangular hyperbolas and the |f(x)|, f(|x|) and 1 ÷ f(x) graphs
•Further worked examples: a downward parabola, two hyperbolas sharing one asymptote pair, an implicit conic on the GC, and a rectangular hyperbola with its lines of symmetry
•Worked examples on describing and reversing combined transformations, using the replacement chain and a sample-point check
•Step-by-step TI-84 procedures for implicit conics and parametric mode, including the silent truncation trap
•The complete reciprocal-graph feature table and the remaining common mistakes, such as crossing asymptotes and dropping the parameter range
•Section summaries and the exam habits checklist