Parent Functions and Transformations

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

Lesson 2 of 43 parts Practice Flashcards

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Parent Functions

Objective: Explain parent functions, apply the rule, and justify relevant restrictions.

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. Identify the stated domain before using the rule, preserve restrictions during algebraic simplification, and interpret the result in the units or graph features requested. Use the worked example to connect the symbolic rule with a concrete calculation.

Formulas

$g(x)=f(x−h)+k$

Worked examples

Try itFor g(x)=(x−7)²+5, what is the vertex x-coordinate?Show answer

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

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Parent Functions

Objective: Explain parent functions, apply the rule, and justify relevant restrictions.

Formulas

$g(x)=f(x−h)+k$

Worked examples

Try itFor g(x)=(x−7)²+5, what is the vertex x-coordinate?Show answer

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

Horizontal and Vertical Shifts

Objective: Explain horizontal and vertical shifts, apply the rule, and justify relevant restrictions.

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. Identify the stated domain before using the rule, preserve restrictions during algebraic simplification, and interpret the result in the units or graph features requested. Use the worked example to connect the symbolic rule with a concrete calculation.

Formulas

$g(x)=f(x−h)+k$

Worked examples

Try itFor g(x)=(x−7)²+5, what is the vertex x-coordinate?Show answer

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

Reflections and Stretches

Objective: Explain reflections and stretches, apply the rule, and justify relevant restrictions.

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. Identify the stated domain before using the rule, preserve restrictions during algebraic simplification, and interpret the result in the units or graph features requested. Use the worked example to connect the symbolic rule with a concrete calculation.

Formulas

$g(x)=−f(x−h)+k$

Worked examples

Try itFor g(x)=−(x−7)²+5, find its maximum value.Show answer

Step 1: −(x−7)² is at most zero. Step 2: It equals zero at x=7. Step 3: The maximum is 5.

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