Maths › Algebra and functions › Graphs, proportion and transformations
Graphs, proportion and transformations
Curve sketching is reading, not drawing. Factors name the roots, squared factors touch instead of cross, reciprocal curves hug lines they never meet, and four standard transformations let one known graph stand in for a whole family of relatives.
Builds on Polynomials and the factor theorem.
IN THIS TOPIC
- Sketch cubics and quartics from factorised form, reading crossings and touches from root multiplicity.
- Sketch y = a/x and y = a/x2 with their asymptotes, and handle direct and inverse proportion.
- Apply the four standard transformations of y = f(x), singly and in combination.
WHAT YOU PROBABLY THINK
Replacing x by (x + a) moves the graph a units to the right.
Sketching from the factors
A polynomial in factorised form hands you its sketch. The leading term fixes the ends, a positive cubic runs from bottom-left to top-right, a positive quartic from top-left to top-right, and each factor plants a root. The power on the factor is the instruction. A simple factor crosses the axis; a squared factor touches it and turns back, as (x + 2)2 did in the last lesson's cubic.
WORKED EXAMPLE
A cubic read straight off its factors
Sketch y = (x + 1)(x − 2)2, marking every axis meeting.
Roots first. A crossing at x = −1 from the simple factor, a touch at x = 2 from the squared one.
The y-intercept comes from x = 0, namely 1 × 4 = 4.
A positive cubic with those features rises through −1, peaks, dips to kiss the axis at 2, then climbs away. No plotting table needed.
Intersections keep their meaning from the simultaneous equations lesson. Where two sketches cross, the pair of equations solves; counting crossings on a sketch counts solutions in advance.
Reciprocal curves and proportion
The graphs of y = a/x and y = a/x2 bring a new feature, the asymptote, a line the curve approaches without ever reaching. Both axes play that role here. Near x = 0 the curves blow up; for large x they flatten toward the axis. The difference between the two is sign. y = a/x takes the sign of x, one branch in each of two opposite quadrants, while y = a/x2 squares the bottom and keeps both branches on one side.
These are also the curves of proportion. Direct proportion, y = kx, is a straight line through the origin; inverse proportion, y = k/x, is the reciprocal curve; inverse square, y = k/x2, is its one-sided cousin. One data point fixes k, and the whole curve follows.
WORKED EXAMPLE
Inverse square proportion, pinned by one point
y is inversely proportional to x2, and y = 4 when x = 3. Find y when x = 2.
Write the relationship with a constant, y = k/x2, then substitute the known pair. k = 4 × 9 = 36.
So y = 36/x2, and at x = 2, y = 36/4 = 9.
Halving x did not double y; it did rather more, because the square in the denominator bites twice.
Transformations
Every change to a graph's equation moves its picture in a predictable way, and four moves cover the specification.
| Equation | Effect on the graph of y = f(x) |
|---|---|
| y = f(x) + a | translate a units up |
| y = f(x + a) | translate a units to the left |
| y = af(x) | stretch ×a vertically |
| y = f(ax) | stretch ×1/a horizontally |
The second row corrects the opening lie. Adding inside the bracket feeds the function an x from further right, so the picture slides left. Changes outside the bracket act on y and do what they say; changes inside act on x and do the opposite.
WORKED EXAMPLE
Tracking a single point
The curve y = f(x) passes through (2, 5). Find the corresponding point on y = f(x − 1) + 3.
Inside the bracket, x − 1 must equal 2, so x = 3. The picture has moved right by 1.
Outside, 3 is added to the output, lifting 5 to 8. The point is (3, 8).
Chasing one point through the brackets beats memorising rules, and it is how the mark scheme checks a sketch.
YOUR TURN
A transformed reciprocal
Sketch y = 1/(x − 2) + 1, stating the equations of its asymptotes and where it meets the axes, before opening the working.
Show the working
Start from y = 1/x and read the moves. Right 2, up 1, so the asymptotes travel too, to x = 2 and y = 1.
Axis meetings come from the equation. At x = 0, y = −1/2 + 1 = 1/2; setting y = 0 gives 1/(x − 2) = −1, so x = 1.
The curve meets the axes at (0, 1/2) and (1, 0), one branch below-left of the asymptote crossing, one above-right.
Transformed asymptotes are the skeleton of the sketch. Draw them dashed first and the branches hang off them naturally.
TRY IT UNSEEN
A combination, in the right order
The curve y = f(x) has a root at x = 0. Describe two correct sequences of transformations taking y = f(x) to y = f(2x + 6), and state where the root ends up.
Show the working
Read f(2x + 6) as f(2(x + 3)). One route translates 6 left, then stretches horizontally ×1/2. The other stretches ×1/2 first, then translates only 3 left.
Either way the root lands where 2x + 6 = 0, at x = −3.
Stretching first and then translating 6 left would park the root at −6, which is wrong, and checking it against 2x + 6 = 0 exposes it instantly.
Combinations inside the bracket do not commute with each other. Factorise the inside, or track a known point, and the order sorts itself out.
THE EXAM BIT
- Sketch from factors in three moves, ends from the leading term, roots from the brackets, y-intercept from x = 0. Label all three or lose the accuracy marks.
- A squared factor touches, a simple factor crosses. Examiners choose curves like x2(2x − 1)2 precisely to test that reading.
- Draw asymptotes dashed and state their equations in full, x = 2 and y = 1, never the bare numbers alone.
- Proportion questions want the constant found first. Write y = k/x2, fix k from the given pair, then answer everything else from the fitted equation.
- Inside the bracket means horizontal and opposite, outside means vertical and literal. When a combination looks ambiguous, factorise the inside and track one point.
CHECK YOURSELF
The curve y = 3/x is transformed to y = 3/x − 2. State the equations of the asymptotes of the new curve, and find where it crosses the x-axis.
Show a hint
The whole curve moves down 2, asymptotes included; then set y = 0.
Show the answer
Subtracting 2 outside translates the curve down 2, so the asymptotes are x = 0 and y = −2.
Setting y = 0 gives 3/x = 2, so x = 3/2, and the curve crosses at (3/2, 0).
The vertical asymptote never moved, because nothing happened inside the function, and that one-line reasoning is worth stating in the answer.
Factors are the sketch, simple roots cross and squared roots touch, and asymptotes are approached, never met.
Outside the bracket acts on y and does what it says; inside acts on x and does the opposite.
CHECK YOUR PROGRESS
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- Sketch cubics and quartics from factorised form, reading crossings and touches from root multiplicity.
- Sketch y = a/x and y = a/x2 with their asymptotes, and handle direct and inverse proportion.
- Apply the four standard transformations of y = f(x), singly and in combination.
No animated video for this topic yet; these notes stand alone.