Function Transformations and Equivalent Representations

7.6: Function Transformations and Equivalent Representations. Original editorial SAT preparation with three worked examples, ten flashcards, and thirty practice questions. Educator-review draft.

Lesson 6 of 83 parts Practice Flashcards

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Vertical transformations

Objective: Evaluate a vertically stretched and shifted function.

Prerequisite skills: function evaluation and graph interpretation. In g(x)=af(x)+k, multiply each original output by a, then add k. A negative a reflects the outputs across the x-axis before scaling. The input value does not change. Common error: changing the input instead of the output in a vertical transformation. Restrictions and interpretation: Horizontal changes act inside the input; vertical changes act on the output.

Formulas

$g(x)=af(x)+k$

Worked examples

Try itIf f(2) = 24 and g(x) = 3f(x) − 5, what is g(2)?Show answer

Use the same input 2: g(2)=3f(2)−5=3(24)−5=67. Therefore the requested answer is 67.

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Vertical transformations

Objective: Evaluate a vertically stretched and shifted function.

Formulas

$g(x)=af(x)+k$

Worked examples

Try itIf f(2) = 24 and g(x) = 3f(x) − 5, what is g(2)?Show answer

Use the same input 2: g(2)=3f(2)−5=3(24)−5=67. Therefore the requested answer is 67.

Horizontal shifts and scaling

Objective: Track a known point through an input shift or scale.

Replacing x by x−h shifts a graph right by h because the new x must be h larger to reproduce the original argument. Replacing x by x+h shifts left. In f(cx), a point at original input u moves to x=u/c for c≠0. Thus |c|>1 compresses horizontally, and c<0 also reflects across the y-axis. These rules change input positions, unlike a vertical multiplier outside f. Common error: moving right for an inside plus or multiplying a horizontal coordinate by c in f(cx). Restrictions and interpretation: Horizontal changes act inside the input; vertical changes act on the output.

Formulas

$If (u,v) is on f, then (u+h,v) is on f(x−h) and (u/c,v) is on f(cx), c≠0$

Worked examples

Try itThe point (23, 9) on y=f(x) is mapped by the horizontal transformation y=f(2x). What is its transformed x-coordinate?Show answer

The original argument is 23. Set 2x=23, giving x=11.5. This compresses horizontal coordinates by 1/2; the output remains 9. Therefore the requested answer is 11.5.

Equivalent nonlinear representations

Objective: Compare expanded and transformed expressions.

An equivalent representation agrees at every allowed input. Use algebraic expansion to prove equality; a few matching graph points are helpful checks but do not establish an identity on their own. Completing a square can reveal a transformation. Common error: matching only the leading coefficient when comparing expressions. Restrictions and interpretation: Horizontal changes act inside the input; vertical changes act on the output. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.

Formulas

$a(x−h)²+k = ax²−2ahx+ah²+k$

Worked examples

Try itIf 2(x − 25)² + 28 = 2x² + bx + c, what is c?Show answer

Expand the square: x²−50x+625. Multiply by 2 and add 28; the constant is c=2(625)+28=1278. Therefore the requested answer is 1278.

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