DPST1012 Chap.3 Transformations and Combinations of Functions
Transformations and Combinations of Functions
Moving a graph without redrawing it
The second half of the functions topic shows how new functions are built from old ones. A vertical translation adds a constant to the output, y = f(x) + c, and moves the graph up or down. A horizontal translation changes the input, y = f(x − h), and moves the graph sideways, right when h is positive.
The worksheets give a drone flight as the context: adding 10 metres to the altitude raises the path, while replacing t by t − 5 flies the same path five seconds later.
Sums, scalar multiples, compositions, reciprocals
Adding two functions adds their outputs at each input, and the worksheet models a 'feels like' temperature as a maximum temperature plus a daily adjustment.
Multiplying by a constant scales every output, which is how a price in US dollars becomes a price in Australian dollars at a fixed exchange rate. Composition chains two rules, as when a day maps to a UV index and the UV index maps to a sunscreen decision.
The reciprocal 1/f(x) turns each output into its reciprocal, the idea behind a fair payout multiplier in games of chance.
Working from tables as well as formulas
Many questions give a function only as a short table. Vertical changes edit the output row and horizontal changes edit the input row, and confusing the two is the most common error.
Compositions are evaluated from the inside out, writing the middle value down. The order of composition matters: f(g(x)) and g(f(x)) are usually different functions, and a composition is undefined wherever the inner output is not in the outer function's domain.
What this chapter covers
- 01
Vertical translations y = f(x) + c
- 02
Horizontal translations y = f(x − h)
- 03
Combining a shift and a stretch
- 04
Sum of functions and scalar multiples
- 05
Composition f ∘ g and g ∘ f
- 06
Reciprocal of a function
- 07
Transforming a function given as a table
Worked example · free
Describe a transformed graph and compose two rules
- 1g replaces x by x − 1, a shift 1 unit right, then adds 3, a shift 3 units up.
- 1The vertex of f moves from (0, 0) to (1, 3).
- 2(f ∘ h)(x) = f(3x + 2) = (3x + 2)² = 9x² + 12x + 4.
- 2(h ∘ f)(−2) = h(f(−2)) = h(4) = 3(4) + 2 = 14.
Key terms
- Vertical Translation
- A shift of a whole graph up or down, produced by adding a constant to every output.
- Horizontal Translation
- A shift of a whole graph left or right, produced by replacing x with x − h in the rule.
- Scalar Multiple
- A new function whose outputs are a fixed constant times the outputs of the original function.
- Composite Function
- A function formed by applying one rule to the output of another, written f ∘ g for f(g(x)).
- Reciprocal Function
- The function 1/f(x), which replaces each output of f by 1 divided by that output, wherever f is not zero.
Transformations and Combinations of Functions FAQ
Why does f(x − 3) move a graph to the right and not the left?
The new graph has to reach each old value three units later along the x-axis, because x − 3 only equals the old input when x is 3 larger. Arriving later means sitting further right on the axis.
Does the order of a composition matter?
Almost always. f(g(x)) applies g first and g(f(x)) applies f first, and the two results are usually different. Test them at one value, such as x = 1, to see the difference immediately.
How is a sum of functions different from a composition?
A sum adds two outputs computed from the same input. A composition feeds the output of one rule into the other as its input. Writing the middle value down keeps the two ideas apart.
What happens to points on the x-axis when a graph is stretched vertically?
They stay where they are. A vertical stretch multiplies every height by the same constant, and zero times any constant is still zero, so the x-intercepts are unchanged.
How do I describe a combination of shifts in words?
Name each move in the order it is applied, with its size and direction: for (x + 2)² − 4, a shift of 2 units left followed by 4 units down. Then confirm the description by moving one known point, such as the vertex.
Exam move
Draw y = x² once, then sketch five transformed versions on the same axes from their rules alone, checking each vertex by substitution; repeat with a function given only as a table. Say each move aloud as you sketch it, so the order of the moves becomes automatic.
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