Porisma

Graph transformations

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By the end, you can

  • Track translations, stretches and reflections
  • Keep horizontal and vertical effects distinct
  • Relate absolute-value and reciprocal graphs to an original curve

about 45 minutes for the lesson · practise at your own pace.

The central idea

For y=Af(B(x−h))+ky=A f(B(x-h))+k, a point (u,v)(u,v) on the original graph becomes (h+u/B,k+Av)(h+u/B,k+Av) when B≠0B\ne0. This point mapping gives the transformation order without guessing from the formula. Absolute values reflect selected portions; reciprocals turn zeros into potential singularities.

Horizontal translation and point mapping

For a>0a>0:

EquationTranslationPoint mapping
y=f(x−a)y=f(x-a)a units in the positive x-direction(p,q)↦(p+a,q)(p,q)\mapsto(p+a,q)
y=f(x+a)y=f(x+a)a units in the negative x-direction(p,q)↦(p−a,q)(p,q)\mapsto(p-a,q)

The introductory examples change x2x^2 into (x+1)2(x+1)^2 by moving left one unit and into (x−2)2(x-2)^2 by moving right two. Solve for a known point’s new input to verify direction.

Explore

Use the worked example below, then change one input or assumption. Predict which part of the answer must change before calculating.

Related interactive example

Explore this related example, then return to the prediction task above.

Scale, then translate

Step through the two transformations. The faint curve is where the graph was before the step.

Teal: y = f(x). No comparison overlay.

Start: y = f(x)

The printed graph, with A(4,3)A(4,3), B(1,−1)B(1,-1), C(3,0)C(3,0) and asymptotes x=2x=2, y=1y=1.

  • Graph after the step you have selected
  • Images of the key points A, B and C
  • Vertical and horizontal asymptotes of the current graph
Ideas to try
  1. Look at point C. It lies on the xx-axis, so doubling does nothing to it. Then +1 lifts it to y=1y=1.
  2. Which coordinates change at each step?
Open the full interactive problem →

Worked example

From y=x2y=x^2 to y=2(x−3)2+1y=2(x-3)^2+1, translate right 3, stretch vertically by 2, then translate up 1. The vertex moves from (0,0)(0,0) to (3,1)(3,1); (1,1)(1,1) moves to (4,3)(4,3).

Replacing x by 2x halves horizontal distances; it does not double them.

Check your understanding

Where does (4,5)(4,5) move under y=f(2x)+3y=f(2x)+3?

Practice questions

Practice section 1 · Questions 1–6

Move the parabola horizontally · Move a reciprocal asymptote to the right · +4 more

Practice section 2 · Questions 7–12

Halve horizontal distances · Scale output distances about the x-axis · +4 more

Practice section 3 · Questions 13–18

Reflect input coordinates in the y-axis · Reflect a rational graph in the y-axis · +4 more

Practice section 4 · Questions 19–24

Complete the square before choosing transformations · Transform a cubic and find its exact intercepts · +4 more

Practice section 5 · Questions 25–30

Fold a rational graph above the x-axis · Sketch the modulus of a shifted straight line · +4 more

Practice section 6 · Questions 31–36

Compare curves with their reciprocals · Reciprocate three supplied sketches · +4 more

Practice section 7 · Questions 37–42

Start from the actual given graph f(x)+3 · Transform symbolic coordinates without choosing k · +4 more

Practice section 8 · Questions 43–48

Recover f from a transformed input graph before reciprocating · Transform a piecewise curve and crop to the requested interval · +4 more

Practice section 9 · Questions 49–53

Compare modulus, negative scaling and reciprocal graphs · Undo a three-step transformation of an implicit curve · +3 more

More practice

Ready to try this on your own?

For “Graph transformations”, try a question without the working and explain your method. This records your confidence, saved in this browser only.

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