Algebraic Foundations

Difference Quotient & Conjugates

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Warm-up: The Conjugate Trick

Recall: Multiplying conjugates results in the "Difference of Squares": \( (A-B)(A+B) = A^2 - B^2 \)

1. Simplify: \( (\sqrt{x} - 3)(\sqrt{x} + 3) \)

2. Simplify: \( (\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b}) \)

3. Rationalize the Numerator: \( \frac{\sqrt{x+h} - \sqrt{x}}{1} \cdot \frac{\sqrt{x+h} + \sqrt{x}}{\sqrt{x+h} + \sqrt{x}} \)

The Difference Quotient

The Difference Quotient measures the average rate of change of a function \( f \) over an interval of length \( h \).

\[ \frac{f(x+h) - f(x)}{h} \]

Guided Practice: Problem 11

Follow along with the video. Function: \( f(x) = \sqrt{x+2} \)

Step 1: Setup

Substitute \( (x+h) \) and set up the quotient.

Step 2: Conjugate Multiplication

Multiply numerator and denominator by the conjugate.

Step 3: Simplify and Final Form

Cancel terms in the numerator, then cancel the \( h \).

Difference Quotient Relay

GROUP PASSPORT

Relay Rules:

Each student in the group completes one step then passes the paper.
The next student must check the previous step before starting theirs.
Final student simplifies the expression to its most reduced form.

Problem 1: Linear f(x) = 4x + 7

STEP 1: FIND f(x+h)

STEP 2: SUBTRACT f(x)

STEP 3: DIVIDE BY h

STEP 4: SIMPLIFY

Problem 2: Quadratic f(x) = x² - 3x

STEP 1: FIND f(x+h)

STEP 2: SUBTRACT f(x)

STEP 3: DIVIDE BY h

STEP 4: SIMPLIFY

Problem 3: The Radical Challenge f(x) = \sqrt{x - 1}

Step 1: Difference Quotient Setup

Evaluate \( f(x+h) \) and write the initial expression.

Step 2: The Conjugate Move

Multiply by the conjugate and FOIL the numerator. Keep the denominator factored!

Step 3: Elimination & Final Result

Cancel the terms in the numerator and eliminate the \( h \) term.

Closure Thinking Question:

In Problem 3, as the value of \( h \) gets closer and closer to zero, what happens to your final expression? What specific value does it approach?

Radical Difference Quotient

Relay and Review Packet

Radical Quotient Slides

Relay Answer Key

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