Stage 9 - Q1 - 2 marks
For input \(x=-4\), find the output and write the function equation.
Hint
The order of the machine boxes controls the algebra.
Stage 9 - Q2 - 2 marks
For input \(x=-3\), find the output and write the function equation.
Hint
The order of the machine boxes controls the algebra.
Stage 9 - Q3 - 2 marks
For input \(x=-\frac{3}{2}\), find the output and write the function equation.
Hint
The order of the machine boxes controls the algebra.
Stage 9 - Q4 - 2 marks
For input \(x=5\), find the output and write the function equation.
Hint
The order of the machine boxes controls the algebra.
Stage 9 - Q5 - 4 marks
Complete the table and draw a mapping for \(y=x^2+4\). Explain why two inputs may share an output.
Hint
A function gives one output per input; different inputs may still share an output.
Stage 9 - Q6 - 4 marks
Write equations for “square, then multiply by 4” and “multiply by 4, then square”. Are they equivalent?
Hint
Expand the square of the product.
Stage 9 - Q7 - 3 marks
A pupil says \(y=5x^2\) is always positive. Is the statement correct for every real input?
Hint
Test zero as well as negative inputs.
Stage 9 - Q8 - 3 marks
Prove that \(y=16x^2\) and \(y=(4x)^2\) give the same output for every real \(x\).
Hint
Use the index law for a product.
Stage 9 - Q9 - 4 marks
The rule is \(y=kx^2\). Find \(k\) from the table and verify all entries.
Hint
Use a nonzero input to avoid division by zero.
Stage 9 - Q10 - 5 marks
Challenge: The table is produced by “square, then multiply by \(k\)”. Find \(k\), then design two different two-step machines of “multiply, then square” that give the same outputs for every real input.
Hint
The multiplier itself may be negative even though its square is positive.