Relay and Review Packet 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 Quotient Slides \[ \frac{f(x+h)-f(x)}{h} \]
\[ \lim_{h \to 0} \]
Calculus Prep: Unit 1
THE DIFFERENCE
QUOTIENT
Mastering Radical Functions with the Conjugate Method
Today's Mission
Objective
Calculate the difference quotient for radical functions using the conjugate method.
Key Skill
Multiplying by the Conjugate
Agenda
01 Algebra Warm-up
02 Video: Radical Analysis
03 Difference Quotient Relay
04 Limits Preview
Warm-up
Multiply the conjugates. What happens to the radicals?
Problem A
\[ (\sqrt{5} - 2)(\sqrt{5} + 2) \]
Problem B
\[ (\sqrt{x} + 4)(\sqrt{x} - 4) \]
The Golden Rule
The product of conjugates always eliminates the middle terms!
\[ (A-B)(A+B) = A^2 - B^2 \]
Radical Case Study
Watch: Problem 11 (11:06 - End)
Take notes in Section 3 of your packet
Embedded media
Watch For:
How the numerator is rationalized to "free" the variables from under the radical sign.
Pro-Tip:
Don't distribute the \( h \) in the denominator. Keep it factored!
DIFFERENCE QUOTIENT
RELAY
Round 1
LINEAR
Warm up those algebraic muscles.
Round 2
QUADRATIC
Expansion and simplification is key.
Round 3
RADICAL
The conjugate method showdown.
Rules: 1 Step per Student | Check Previous Work | High Speed / High Accuracy
The Calculus Connection
In our final problem, what happens as \( h \to 0 \)?
\[ \frac{1}{\sqrt{x+h-1} + \sqrt{x-1}} \xrightarrow{h \to 0} ? \]
This is the instantaneous rate of change.derivative !
Ready for Derivatives?
Relay Answer Key Teacher Guide & Answer Key
Radical Difference Quotient Mastery
1. Warm-up: Conjugates
Q1: \( (\sqrt{x}-3)(\sqrt{x}+3) \)
Answer: \( x - 9 \)
Q2: \( (\sqrt{a}+\sqrt{b})(\sqrt{a}-\sqrt{b}) \)
Answer: \( a - b \)
Q3: Rationalizing Numerator
Answer: \( \frac{(x+h) - x}{\sqrt{x+h} + \sqrt{x}} = \frac{h}{\sqrt{x+h} + \sqrt{x}} \)
2. Video Case: \( f(x) = \sqrt{x+2} \)
Full Solution Steps:
Setup: \( \frac{\sqrt{x+h+2} - \sqrt{x+2}}{h} \)
Conjugate: \( \frac{\sqrt{x+h+2} - \sqrt{x+2}}{h} \cdot \frac{\sqrt{x+h+2} + \sqrt{x+2}}{\sqrt{x+h+2} + \sqrt{x+2}} \)
Foil Numerator: \( \frac{(x+h+2) - (x+2)}{h(\sqrt{x+h+2} + \sqrt{x+2})} \)
Simplify Numerator: \( \frac{h}{h(\sqrt{x+h+2} + \sqrt{x+2})} \)
Final Answer: \( \frac{1}{\sqrt{x+h+2} + \sqrt{x+2}} \)
3. Relay Solutions
P1 (Linear): \( f(x) = 4x + 7 \)
Step 1: \( 4(x+h)+7 \); Step 2: \( 4x+4h+7-4x-7 \); Step 3: \( 4h/h \)
Final Answer: 4
P2 (Quadratic): \( f(x) = x^2 - 3x \)
Step 1: \( (x+h)^2 - 3(x+h) \); Step 2: \( x^2+2xh+h^2-3x-3h - (x^2-3x) \); Step 3: \( 2xh+h^2-3h \)
Final Answer: 2x + h - 3
P3 (Radical): \( f(x) = \sqrt{x-1} \)
Final Answer: \( \frac{1}{\sqrt{x+h-1} + \sqrt{x-1}} \)
"As h approaches 0, the radical result becomes \( \frac{1}{2\sqrt{x-1}} \)."
