Hint
Inside changes affect x first; outside changes affect the output. Keep the final upward translation after the stretch and reflection.
Worked solution
(a)
First replace x by x−3, moving the graph right 3. Multiply outputs by −2: a vertical stretch of factor 2 and reflection in the x-axis, in either order. Finally add 5 to move it up 5.
Answer: Translate right 3; stretch vertically by factor 2 and reflect in the x-axis; then translate up 5.
(b)
\(h(x)=-2(x-3)^2+5\), so its maximum is 5 at \(\displaystyle x=3\). The downward branch has no lower bound.
Answer: Vertex \((3,5)\); range \(y\le5\).
(c)
\(\displaystyle -2(x-3)^{2}+5\ge -3\) gives \(\displaystyle (x-3)^{2}\le 4\). Hence \(\displaystyle -2\le x-3\le 2\), so \(\displaystyle 1\le x\le 5\).
Answer: \(1\le x\le5\).
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official examination-board mark scheme. Award stated criteria once each, up to the part maximum. Accept valid equivalent methods. Follow-through applies only where explicitly stated; do not award an accuracy criterion for an incorrect final result.
(a) · 3 marks
- 1 mark: Translate right 3.
- 1 mark: Apply a vertical stretch of factor 2 and an x-axis reflection.
- 1 mark: Translate up 5 after the output multiplication; the full sequence must produce the stated function.
Alternative methods: Any other explicitly ordered sequence producing exactly −2f(x−3)+5 is accepted, as is the coordinate rule (u,v)→(u+3,−2v+5) with a correct description.
Common error: Reflecting in the y-axis or stretching an already-translated vertical output without adjusting the translation.
(b) · 2 marks
- 1 mark: Give vertex (3,5).
- 1 mark: Give range \(\displaystyle y\le 5\).
Alternative methods: Transforming the vertex (0,0) and the original range is accepted.
Common error: Giving a minimum or the range \(\displaystyle y\ge 5\).
(c) · 3 marks
- 1 mark: Rearrange to \(\displaystyle (x-3)^{2}\le 4\) with the correct inequality sign.
- 1 mark: Find boundary values 1 and 5.
- 1 mark: Give the inclusive interval between them.
Alternative methods: A graph/level intersection or expanded quadratic inequality is accepted.
Common error: Failing to reverse the inequality on division by −2.
Total: 8 marks.