Kenneth ChengMathematical modeling
← Senior Mathematics Studio

LEARN · EXPLAIN · REVISE

Translate this graph 3 units to the left and 4 units down. Which equation describes the new graph?

Read the idea, work independently, then explain what changed.

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TOPIC 01

Quadratic equations and functions

A left shift replaces x by x + 3 in the original expression. It does not replace x by x − 3.

PREDICT → MOVE → EXPLAIN

What does each coefficient change?

Before moving a slider, predict the opening, vertex and intercepts. Then compare the graph with your prediction.

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Try it. Leave your reasoning visible.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Transfer#Your turn

Translate this graph 3 units to the left and 4 units down. Which equation describes the new graph?

y=(x−1)2+2y=(x-1)^2+2
  1. First equationy=(x+2)2−2y=(x+2)^2-2
  2. Second equationy=(x−4)2−2y=(x-4)^2-2
  3. Third equationy=(x+2)2+6y=(x+2)^2+6
  4. Fourth equationy=(x−2)2−2y=(x-2)^2-2

Working and explanation

BUILD THE REASONING

Hint 1
Track the vertex (1, 2) before writing the new equation.
Hint 2
Subtract 3 from the horizontal coordinate and 4 from the vertical coordinate.
Worked solution

  1. Move the vertex with the graph.

    (1,2)⟼(1−3,2−4)=(−2,−2)(1,2)\longmapsto(1-3,2-4)=(-2,-2)

  2. Keep the opening and scale; use the new vertex.

    y=(x−(−2))2−2=(x+2)2−2y=(x-(-2))^2-2=(x+2)^2-2

A: y = (x + 2)² − 2.

Checks and common pitfalls: A left shift replaces x by x + 3 in the original expression. It does not replace x by x − 3.

Think first. Reveal a hint when the class is ready.

Focus on one question

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Curriculum and source notes ↗