Parent Functions and Transformations

Explore parent functions, horizontal and vertical shifts, reflections and stretches.

Lesson 3 of 43 parts Practice Flashcards

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Parent Functions

Objective: Explain parent functions and use it to solve or interpret a problem.

Linear, absolute-value, quadratic, and square-root parent functions provide baseline shapes; transformations alter position or scale without changing the underlying input-output rule type. It makes shifts, reflections, and stretches easier to identify.

Worked examples

Try itParent Functions and Transformations — practice 1.1: For g(x)=(x−5)²+7, what is the vertex x-coordinate?Show answer

Step 1: The squared part is zero at x=5. Step 2: The vertex is (5,7). Step 3: The requested x-coordinate is 5.

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Parent Functions

Objective: Explain parent functions and use it to solve or interpret a problem.

Worked examples

Try itParent Functions and Transformations — practice 1.1: For g(x)=(x−5)²+7, what is the vertex x-coordinate?Show answer

Step 1: The squared part is zero at x=5. Step 2: The vertex is (5,7). Step 3: The requested x-coordinate is 5.

Horizontal and Vertical Shifts

Objective: Explain horizontal and vertical shifts and use it to solve or interpret a problem.

In f(x−h)+k, h moves the graph right for positive h and k moves it up. The apparent minus sign inside produces a right shift. The old reference input zero occurs at x=h.

Worked examples

Try itParent Functions and Transformations — practice 2.1: For g(x)=(x−8)²+3, what is the vertex x-coordinate?Show answer

Step 1: The squared part is zero at x=8. Step 2: The vertex is (8,3). Step 3: The requested x-coordinate is 8.

Reflections and Stretches

Objective: Explain reflections and stretches and use it to solve or interpret a problem.

Multiplying an output by a negative reflects over the x-axis; a factor with magnitude greater than one stretches vertically. Distinguish outside from inside transformations. No, it reflects the output direction.

Worked examples

Try itParent Functions and Transformations — practice 3.1: For g(x)=−(x−11)²+4, find its maximum value.Show answer

Step 1: −(x−11)² is at most zero. Step 2: It equals zero at x=11. Step 3: The maximum is 4.

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