Graph Quest Worksheet GRAPH QUEST
Logarithmic Transformations & Visual Analysis
10th Grade Advanced Math
Student:
Date:
Warm-Up: Parameter Match (5 min)
Match the variable from the standard transformation formula \( f(x) = a \cdot \log_b(x - h) + k \) to its graphical effect.
a
h
k
Horizontal Shift
Vertical Stretch / Reflection
Vertical Shift
Video Quest (10 min)
1. The Difference
According to Randy, what is the primary difference between the asymptotes of exponential functions and logarithmic functions?
2. Inverse Insight
If you reflect \( y = \log_2(x) \) over the diagonal line \( y = x \), what function do you get?
3. The Reference Point Strategy
Why is it critical to choose the point exactly 1 unit horizontally from the asymptote when checking for vertical shifts or reflections?
Desmos Polygraph Log (25 min)
Use this space to track your questions and deductions. Remember: your goal is to identify your partner's hidden function by asking mathematical Yes/No questions about transformations!
Round My Questions (e.g., "Is there a vertical asymptote at x = 2?") Eliminated Parameters 1 2
Closure & Reflection (5 min)
Which transformation (a, h, or k) was the most difficult to "guess" during the Desmos activity? Why?
"One step at a time, and remember the meaning of each variable." — Randy
Logarithm Landscapes Slides LOGARITHM
LANDSCAPES
Translating Algebra to Art
10th Grade Advanced Math
Transformations & Graphical Analysis
Warm-Up: Parameter Match
In the equation \( f(x) = a \cdot \log_b(x - h) + k \), identify the role of each variable:
a
h
k
Video Insight: Graphs of Logs
Embedded media
Watch For:
Vertical Asymptotes vs. Horizontal Asymptotes
The "Reference Point" Strategy (1 unit away)
Direction of horizontal shifts (\(x-h\))
The Reference Point
"Find the point that is exactly one unit from the asymptote on both curves."
Why?
Since \( \log_b(1) = 0 \), identifying the point where the argument is 1 isolates the vertical shift (\(k\)) perfectly.
Asymptote
1 Unit Away
Desmos Polygraph
TIME: 25 MIN
How to Play:
1 Log in to your laptop and join the class session.
2 One partner picks a "hidden" log function.
3 Ask Yes/No questions to eliminate graphs!
Use Math Vocabulary!
Vertical Asymptote Vertical Stretch Reflection Horizontal Shift Range / Domain Intercepts
Class Reflection
Which hidden function was the most difficult to guess? Why?
Log Logic Teacher Guide TEACHER GUIDE
Logarithm Landscapes
Objective
Students will translate between algebraic formulas and graphical representations of logarithmic functions, identifying specific transformations including scaling, reflecting, and shifting.
Materials Needed
Laptops/Tablets (1 per student)
Desmos Classroom Access
Graph Quest Worksheets
Lesson Flow
5 min Warm-Up
Parameter Match
Have students complete the matching section on their worksheet. Display the slide for visual reference. Circulate to check for misconceptions about \(h\) (horizontal) vs. \(k\) (vertical).
10 min Instruction
Video Viewing & Guided Notes
Watch Randy's video. Teaching Moment: Pause at 1:27 and 7:07 to allow students to record the "Reference Point Strategy".
Discussion Anchor (7:15):
"Why is the point 1 unit away from the asymptote so important? (Hint: \(\log_b(1) = 0\)). How does this simplify finding the vertical shift 'k'?"
25 min Activity
Desmos Polygraph
Students pair up on Desmos Classroom. One student selects a function; the other asks Yes/No questions. Encourage high-level vocabulary.
Pro-Tip: Monitor the teacher dashboard to highlight effective questions like "Is your asymptote to the right of the y-axis?"
5 min Closure
Class Reflection
Discuss the most difficult functions. Often, functions with both reflections and shifts are hardest. Remind students of the "one unit away" rule for these cases.
Differentiation Strategies
Scaffolding (Struggling Learners)
Provide a "Cheat Sheet" showing the parent graph of \( \log_2(x) \) next to basic shifts. Encourage them to only vary one parameter at a time during Desmos.
Extension (Advanced Learners)
Challenge students to use base values other than 2 or 3 (e.g., fractional bases like 1/2) and observe how the "direction" of the curve changes.
Solution Key Teacher Resource ANSWER KEY & SOLUTIONS
Lesson: Logarithm Landscapes
Warm-Up: Parameter Match
a Vertical Stretch / Reflection
h Horizontal Shift
k Vertical Shift
Video Quest: Sample Responses
1. The Difference
Logarithmic functions have vertical asymptotes , whereas exponential functions have horizontal asymptotes.
2. Inverse Insight
You would get the exponential function \( y = 2^x \).
3. The Reference Point Strategy
Since \( \log_b(1) = 0 \), picking the point exactly 1 unit from the asymptote means the log term evaluates to zero. This leaves only the value of \( k \), making it easy to identify the exact vertical shift without being confused by reflections or stretches.
Video Example Solutions
Example 1 (Shifts)
Answer: \( g(x) = \log_2(x-3) - 2 \)
Asymptote moved from x=0 to x=3 (h=3).
Reference point moved down 2 units (k=-2).
Example 2 (Reflection)
Answer: \( g(x) = -\log_3(x+2) - 1 \)
Asymptote moved left 2 units (h=-2).
Graph is flipped vertically (a is negative).
Ref point is down 1 from green ref point (k=-1).
