Transformations and Modeling with Quadratics

Develop quadratic shift, feasible vertex, units in a model.

Lesson 4 of 43 parts Practice Flashcards

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Quadratic Shift

Objective: Explain quadratic shift, apply the rule, and justify relevant restrictions.

A rule a(x−h)²+k translates the parent curve and changes its vertical scale. Read h from the inside and k from the outside. The vertex output or vertical shift. 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

$y=A(x−h)²+k; vertex=(h,k)$

Worked examples

Try itFor y=3(x−7)²+5, find the vertex y-coordinate.Show answer

Step 1: The squared term is zero at x=7. Step 2: Then y=3(0)+5. Step 3: The vertex y-coordinate is 5.

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Quadratic Shift

Objective: Explain quadratic shift, apply the rule, and justify relevant restrictions.

Formulas

$y=A(x−h)²+k; vertex=(h,k)$

Worked examples

Try itFor y=3(x−7)²+5, find the vertex y-coordinate.Show answer

Step 1: The squared term is zero at x=7. Step 2: Then y=3(0)+5. Step 3: The vertex y-coordinate is 5.

Feasible Vertex

Objective: Explain feasible vertex, apply the rule, and justify relevant restrictions.

The algebraic turning point is a contextual maximum or minimum only if its input lies inside the permitted domain. Otherwise check domain endpoints. A restricted interval may place its optimum at an edge. 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

$For y=(x−h)²+k: y≥k$

Worked examples

Try itFor y=(x−7)²+5, find the minimum output.Show answer

Step 1: A square is at least zero. Step 2: At x=7, the square is zero. Step 3: The minimum output is 5.

Units in a Model

Objective: Explain units in a model, apply the rule, and justify relevant restrictions.

A vertex (time, height) has different units on each coordinate. Report the physical meaning and avoid extending a quadratic model past plausible conditions. Not when time starts at the launch event. 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

$y=A(x−h)²+k; vertex=(h,k)$

Worked examples

Try itFor y=3(x−7)²+5, find the vertex y-coordinate.Show answer

Step 1: The squared term is zero at x=7. Step 2: Then y=3(0)+5. Step 3: The vertex y-coordinate is 5.

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