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Grade 9 · lesson 4 of 4

Interpret Reflection, Stretch, and Shifts

The coefficient outside a squared expression controls output sign and vertical scale.

About 20 minutes, plus any extended practice

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Supplied materials

  • Paper or a browser scratch area; a calculator is optional unless rounding is requested.
  • Supplied teaching case: Compare y=−2(x−2)²+3 with y=x². Candidate answers: (A) Only shift left and down (B) Open upward and compress vertically by 2 (C) Reflect across x-axis, stretch vertically by 2, shift right and up (D) Reflect across y-axis, then shift left.
  • Independent check inputs: Compare y=−2(x−3)²+4 with y=x². Candidate answers: (A) Only shift left and down (B) Open upward and compress vertically by 2 (C) Reflect across x-axis, stretch vertically by 2, shift right and up (D) Reflect across y-axis, then shift left.

Learn the idea

Quadratic equations and graphs express the same underlying structure. A discriminant predicts real-root count, the quadratic formula calculates roots, and vertex form displays shifts, direction and stretching directly. The coefficient outside a squared expression controls output sign and vertical scale. A change inside the input shifts the graph horizontally, while an added outside constant shifts it vertically.

Here is the method on the supplied teaching case. The leading −2 reverses the output sign and doubles its magnitude. The input x−2 shifts right 2; +3 shifts up 3. The vertex is (2,3).

A negative leading coefficient makes the graph open downward. Read inside shifts in the opposite direction of their signs; describe the vertex as well as the scaling so the full transformation is clear.

  1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
  2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
  3. Solve the independent check with its changed inputs before revealing feedback.
  4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
Worked example

See the method in action

Teaching case: Compare y=−2(x−2)²+3 with y=x². Candidate answers: (A) Only shift left and down (B) Open upward and compress vertically by 2 (C) Reflect across x-axis, stretch vertically by 2, shift right and up (D) Reflect across y-axis, then shift left. Solution: The leading −2 reverses the output sign and doubles its magnitude. The input x−2 shifts right 2; +3 shifts up 3. The vertex is (2,3). The final answer is Reflect across x-axis, stretch vertically by 2, shift right and up. A negative leading coefficient makes the graph open downward. Read inside shifts in the opposite direction of their signs; describe the vertex as well as the scaling so the full transformation is clear.

Your turn

Check your reasoning

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Helpful tips
  • The coefficient outside a squared expression controls output sign and vertical scale.
  • A negative leading coefficient makes the graph open downward.
Compare y=−2(x−3)²+4 with y=x².

Feedback appears here after you check. This is practice, not a graded school assessment.

See the answer and explanation

Reflect across x-axis, stretch vertically by 2, shift right and up

The leading −2 reverses the output sign and doubles its magnitude. The input x−3 shifts right 3; +4 shifts up 4. The vertex is (3,4).

Take it further

Changed-input task: Compare y=−2(x−9)²+10 with y=x². Candidate answers: (A) Only shift left and down (B) Open upward and compress vertically by 2 (C) Reflect across x-axis, stretch vertically by 2, shift right and up (D) Reflect across y-axis, then shift left. Show the mathematical reason, not only the selected answer. Expected result: Reflect across x-axis, stretch vertically by 2, shift right and up. The leading −2 reverses the output sign and doubles its magnitude. The input x−9 shifts right 9; +10 shifts up 10. The vertex is (9,10).

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More practice: try new cases

Choose an answer for each problem. Explain your method on paper, then use the feedback to decide what to revisit.

More practice: try new cases
1. Compare y=−2(x−4)²+5 with y=x².
Need a hint?
  • The coefficient outside a squared expression controls output sign and vertical scale.
  • A negative leading coefficient makes the graph open downward.
2. Compare y=−2(x−5)²+6 with y=x².
Need a hint?
  • The coefficient outside a squared expression controls output sign and vertical scale.
  • A negative leading coefficient makes the graph open downward.
3. Compare y=−2(x−6)²+7 with y=x².
Need a hint?
  • The coefficient outside a squared expression controls output sign and vertical scale.
  • A negative leading coefficient makes the graph open downward.

Your choices are temporary. Feedback appears after you check all the answers.

Nothing here is saved to an account or sent to a teacher. Keep your own notes or download them before leaving. The score describes these questions; it does not certify subject mastery.

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