Q: What does IP EMaths Notes (Upper Sec, Year 3-4): 08) Functions and Transformations cover? A: Interpret function notation, compose mappings, and relate algebraic changes to graph translations or stretches.
Function language appears in calculator prompts, inverse-relationship questions, and graph sketching. Track how algebraic edits move or stretch the curve.
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Key ideas
f(x) gives the output when input x is substituted.
Composite functions: (f∘g)(x)=f(g(x))
.
Common transformations:
y=f(x)+k: shift up by k.
y=f(x−a): shift right by a.
y=af(x): vertical stretch by factor a.
y=f(ax): horizontal compression by factor a.
Worked example - Transformation tracker
The graph of y=x2 is transformed to y=3(x−2)2+5. Describe the sequence of transformations.
Start with y=x2.
x↦x−2 translates the graph 2 units to the right.
Multiplying by 3 stretches it vertically by factor 3 (parabola becomes narrower).
Adding 5 shifts the graph up by 5 units.
So the sequence is: translate right by 2, stretch vertically by 3, then shift up by 5.
Worked example - Compose functions carefully
Let f(x)=2x−3 and g(x)=x2+1. Find (f∘g)(x) and evaluate (g∘f)(2).
Compute (f∘g)(x)=f(g(x)):
First find g(x): x2+1.
Substitute into f: f(g(x))=2(x2+1)−3=2x2+2−3=2x2−1
Compute (g∘f)(2)=g(f(2)):
f(2)=2(2)−3=1
Thus (f∘g)(x)=2x2−1 and (g∘f)(2)=2.
Worked example - Find an inverse function
Given h(x)=23x−5, determine h−1(x) and state the transformation that links h and h−1.
Let y=23x−5.
Swap x and y: x=23y−5. This reflects that an inverse reverses the original mapping-the output y from h becomes the input for h−1, while the original input x becomes the output we now solve for.
Solve for y:
Multiply by 2: 2x=3y−5.
Add 5: 3y=2x+5.
Divide by 3: y=32x+5
Therefore h−1(x)=32x+5.
The original function multiplies by 3, subtracts 5, then divides by 2. The inverse reverses these steps: multiply by 2, add 5, then divide by 3. Graphically, h and h−1 are reflections of each other across the line y=x.
Practice Quiz
Consolidate function evaluation, compositions, and transformation language with auto-marked prompts.
Try this
Given f(x)=2x−1 and g(x)=x2+3, find (g∘f)(x) and state the value when x=−2.
IP EMaths Notes (Upper Sec, Year 3-4): 08) Functions and Transformations
