FRQs 1-2 – Graphing Calculator Required

Overview

• FRQ 1 and 2 together make up 18.75% of your total exam score
• 30 minutes total for both questions (15 minutes per question suggested)
• Graphing calculator required - must be in radian mode
• Each question is worth 6 points with specific point allocations by skill
• Store values in calculator to maintain precision for multi-part questions

FRQ 1: Function Concepts

• Tests multiple function representations (graphs, tables, equations)
• Common topics: composition, inverse functions, zeros, end behavior
• Point distribution typically includes procedural work, interpretation, and reasoning

FRQ 2: Modeling a Non-Periodic Context

• Real-world scenario requiring function construction
• Function types: polynomial, piecewise, exponential, or logarithmic
• Heavy emphasis on rate of change and model limitations

Strategy Deep Dive

These calculator questions aren't just about computation—they're carefully designed to test whether you can use technology as a tool for mathematical investigation. Understanding the intent behind each question type transforms your approach.

FRQ 1: The Function Investigation Framework

Function Concepts questions follow a predictable arc: they give you functions in various forms and ask you to explore relationships between them. The key insight is that each part builds on previous parts, even when it's not obvious.

When you first read the question, identify what representations you're given:

• Table of values for one function
• Equation for another function
• Perhaps a graph for context

Before diving into part (a), take 30 seconds to understand the big picture. What functions are in play? How might they relate? This investment pays off when later parts reference earlier work.

Composition and Inverse Mastery: These are FRQ 1 staples. For composition h(x) = (g ∘ f)(x) = g(f(x)):

1. Work from inside out
2. Use your calculator to store intermediate values
3. Show the substitution clearly: "Since f(5) = 34, we have h(5) = g(34)..."

For inverse functions, remember that f^(-1)(a) = b means f(b) = a. When working with tables, this becomes a simple lookup, but document your reasoning.

End Behavior Analysis: When asked about limits as x → ∞ or x → -∞:

1. For rational functions, compare degrees of numerator and denominator
2. For polynomial compositions, the highest degree term dominates
3. Write your conclusion using proper limit notation

The calculator helps verify your analytical work. Graph the function with an appropriate window to confirm your limit conclusion, but remember: the analytical reasoning earns the point, not the graph.

Function Type Identification: Part (c) often asks "Which type of function best models the data?" This isn't guesswork—it's pattern recognition:

• Linear: constant first differences
• Quadratic: constant second differences
• Exponential: constant ratios
• Logarithmic: linear pattern when you plot (ln(x), y)

Show the calculations that lead to your conclusion. A table of differences or ratios demonstrates clear thinking.

FRQ 2: The Modeling Method

Modeling questions test whether you can translate real-world information into mathematical language. The process is systematic, not creative.

Parameter Finding Strategy: You're typically given:

• A function form like R(t) = a + b·ln(t+1)
• Specific data points
• Asked to find parameters a and b

The approach is mechanical:

1. Substitute each data point to create equations
2. Solve the system (usually just 2 equations, 2 unknowns)
3. Store exact values in your calculator
4. Write the complete function with decimal approximations

But here's the key: keep exact values in your calculator memory for subsequent parts. If you round too early, errors compound.

Rate of Change Analysis: Part (b) invariably involves average rate of change: Average rate = [f(b) - f(a)]/(b - a)

The calculator computation is straightforward, but interpretation matters. Always:

• Include units in your answer
• Explain what the sign means in context
• Connect to the physical situation

When comparing rates of change over different intervals, consider the function type. For logarithmic models, rates decrease over time. For exponential models, rates increase. Your explanation should reference both the mathematical property and the real-world meaning.

Model Evaluation: Part (c) often asks about model appropriateness or limitations. This tests mathematical maturity—can you critique a model thoughtfully?

Common evaluation criteria:

• Does the model make sense outside the given data range?
• Are there physical constraints the model violates?
• How well does it match the described behavior?

Your response should blend mathematical analysis with contextual reasoning. "Since the logarithmic model grows without bound as t increases, it cannot accurately model test scores, which have a maximum of 100 points."

Rubric Breakdown

Understanding how points are awarded transforms your approach to these questions. Each point has specific criteria, and knowing these helps you include exactly what graders seek.

FRQ 1 Point Distribution

Computation Points (typically 3 points):

• Usually one point each for parts (a)(i), (a)(ii), and (b)(i)
• Must show substitution or method, not just answer
• Calculator work should be indicated: "Using the calculator to solve..."
• Decimal approximations must be accurate to 3 decimal places

Reasoning Points (typically 2 points):

• Often in parts (b)(ii) and (c)(i)
• Require mathematical justification
• "The limit is infinity" alone won't earn the point
• Need: "As x decreases without bound, the denominator approaches -2 while the numerator becomes increasingly negative, so the fraction grows without bound"

Communication Point (typically 1 point):

• Usually the explanation in part (c)(ii)
• Must reference specific calculations
• Connect mathematical pattern to function type
• "The second differences are constant (all equal 4), which indicates a quadratic model"

FRQ 2 Point Distribution

Setup Points (typically 2 points):

• Creating correct equations from given information
• Clear substitution shown
• Proper notation used

Calculation Points (typically 2 points):

• Solving for parameters correctly
• Computing rate of change accurately
• Including appropriate units

Interpretation Points (typically 2 points):

• Explaining meaning in context
• Evaluating model appropriateness
• Discussing limitations with specific reference to mathematics

Common Pitfalls and How to Avoid Them

Learning from common mistakes can boost your score significantly. These aren't just careless errors—they're systematic misunderstandings the exam exposes.

Technology Pitfalls

Rounding Too Early: The biggest calculator mistake is rounding intermediate values. If part (a) asks for parameter b = -3.001, and you store b = -3 in your calculator, part (b) will be slightly off, and by part (c), you're significantly wrong. Always store exact values and only round for final answers.

Window Woes: When graphing to verify behavior or find intersections, bad windows hide critical features. For logarithmic functions, remember the domain restriction. For exponentials, the y-values can grow rapidly. Start with ZoomFit, then adjust as needed.

Regression Confusion: When finding an exponential model y = ab^x, make sure your calculator is set to the right regression type. Some calculators offer y = ae^(bx) instead. Know which form the question wants and how to convert if needed.

Communication Pitfalls

Naked Numbers: Writing "Average rate = -1.387" without units or interpretation earns partial credit at best. Complete responses include: "The average rate of change is -1.387 points per month, meaning the group's retention score decreased by an average of 1.387 points each month during this period."

Vague Reasoning: "The function is increasing" isn't enough. Specify where and how you know: "Since R(t) is logarithmic with a negative coefficient on the logarithm term, R(t) is decreasing for all t > 0."

Missing Connections: In modeling questions, connect mathematics to context. Don't just say "The model is inappropriate for large t." Explain: "As t approaches infinity, the logarithmic model predicts negative test scores, which is impossible in this context."

Strategic Pitfalls

Part Independence Assumption: Students often think each part stands alone. In reality, part (b) often uses results from part (a), and part (c) synthesizes everything. Read ahead to see connections.

Time Imbalance: Spending 20 minutes perfecting FRQ 1 leaves you rushing FRQ 2. Aim for 15 minutes each. If you're stuck on a part, leave space and move on. Partial credit on both questions beats perfect on one.

Calculator Dependence: The calculator is a tool, not a crutch. You still need to show mathematical thinking. "By calculator" isn't sufficient justification. Explain what you asked the calculator to do and why.

Time Management Reality

Thirty minutes for two multi-part questions creates unique pressure. Here's the reality of managing this section effectively.

The First Read (2 minutes)

Don't skip this. Read both questions completely. This isn't procrastination—it's strategy. Your brain begins processing while you work on FRQ 1. Sometimes FRQ 2 is more straightforward, and starting there builds confidence.

FRQ 1 Pacing (13-15 minutes)

Part (a) usually has two computational sub-parts. These should take 3-4 minutes total. They're testing whether you can carefully execute procedures. Don't rush, but don't overthink.

Part (b) typically combines computation with explanation. Budget 5 minutes here. The computation might be quick, but the explanation needs thought. Write your mathematical reasoning first, then translate to words.

Part (c) is often the discriminator. It requires synthesis and clear communication. If you're running behind, at least attempt the identification part (c)(i)—it's often worth a point for recognizing a pattern.

FRQ 2 Pacing (13-15 minutes)

Part (a) is mechanical: substitute, solve, find parameters. This should be 4-5 minutes of careful algebra. Show every step. Graders can't give partial credit if they can't follow your work.

Part (b) involves rate of change. The calculation is quick (2 minutes), but interpretation and comparison take time. Don't write the first thing that comes to mind. Think about what the rate means in context.

Part (c) on model evaluation rewards mathematical maturity. Even if you're running out of time, attempt this. A thoughtful paragraph about model limitations can earn points even if your earlier calculations were off.

The Final Minutes

With 5 minutes left, stop calculating and start checking:

• Did I answer every part?
• Are units included where needed?
• Is my work followable?
• Did I store values correctly for multi-part calculations?

If you're missing a part, write something relevant. "To find when the model is no longer appropriate, I would check when the predicted values become unrealistic for the context." This shows understanding and might earn partial credit.

Specific Topic Strategies

Certain topics appear repeatedly in FRQs 1 and 2. Mastering these specific approaches gives you confidence when you recognize familiar patterns.

Working with Composite Functions

When you see (g ∘ f)(x) or similar:

1. Write out the definition: (g ∘ f)(x) = g(f(x))
2. Evaluate from inside out
3. If given tables, trace through carefully
4. If given equations, substitute algebraically
5. Store intermediate values in calculator

Common error: confusing (g ∘ f)(x) with (f ∘ g)(x). These are rarely equal. Always work inside-out.

Finding Inverse Function Values

From tables:

• If f(3) = 7, then f^(-1)(7) = 3
• State this relationship explicitly

For equations requiring inverse:

1. Replace f(x) with y
2. Swap x and y
3. Solve for y
4. Verify with a test point

Average Rate of Change in Context

The formula is simple, but interpretation requires thought:

• Always include units
• Explain what positive/negative means
• Compare to instantaneous rate when relevant
• Reference concavity when discussing changing rates

For increasing functions with decreasing rates of change (like logarithmic growth), explain: "While the score continues to increase, it increases more slowly over time, reflecting diminishing returns in learning retention."

Model Selection and Justification

When choosing between model types:

1. Calculate diagnostic values (differences, ratios)
2. Show these calculations in a table
3. State the pattern clearly
4. Connect pattern to function type
5. Consider context appropriateness

Never just guess based on the graph's shape. The mathematical justification earns points.

Final Thoughts

FRQs 1 and 2 test whether you can use functions as tools for understanding. They're not about complex calculations—your calculator handles those. They're about mathematical thinking: Can you translate between representations? Can you use models to make predictions? Can you evaluate whether a mathematical model captures real-world behavior?

Success comes from systematic approaches, not brilliance. When you see a composition, you know to work inside-out. When asked about rates of change, you know to include units and interpretation. When evaluating models, you know to consider both mathematical behavior and contextual constraints.

The calculator is your partner, not your replacement. Use it to:

• Store exact values for precision
• Verify analytical conclusions
• Explore when the algebra gets messy
• Find values the problem requires

But remember: the calculator can't explain your reasoning or interpret in context. Those human skills earn the reasoning and communication points that separate good scores from great ones.

Practice these questions with your calculator in radian mode. Build muscle memory for common operations. Learn to write explanations that blend mathematical precision with contextual understanding. Most importantly, see these questions as investigations, not tests. You're exploring mathematical relationships and their real-world implications.

The 18.75% of your score from these questions rewards careful thinking and clear communication as much as computational accuracy. Master the patterns, manage your time wisely, and approach each question as a mathematical conversation. You're not just solving—you're explaining your mathematical thinking to someone who wants to understand.