πŸƒ Flashcards Β· Topic 1.3

Rates of Change:
Linear & Quadratic

10 cards across 2 print sheets. Print double-sided, cut along dashed lines.

πŸ“‹ Notes πŸƒ Flashcards 🧠 Quiz
Print double-sided (flip on long edge) β†’ cut along dashed lines.
10 cards Β· 2 print sheets
πŸ“„ Page 1 β€” Questions FRONT Β· Sheet 1/2
1/10
Topic 1.3 Β· Linear β€” AROC
What is true about the AROC of a linear function over any equal-length interval?
AROC = Ξ”y/Ξ”x
Think: slope
2/10
Topic 1.3 Β· Quadratic β€” AROC
What is true about the AROC of a quadratic function over consecutive equal-length intervals?
The AROCs themselves form what kind of pattern?
3/10
Topic 1.3 Β· 3-Step Method
What are the 3 steps to classify a function as linear, quadratic, or neither from a table?
Step 1: check Ξ”x Β· Step 2: compute AROC Β· Step 3: check Ξ”(AROC)
4/10
Topic 1.3 Β· Concave Up
A function is concave up when the AROC over equal-length intervals is ___?
The slope is getting...
5/10
Topic 1.3 Β· Concave Down
A function is concave down when the AROC over equal-length intervals is ___?
The slope is getting...
6/10
Topic 1.3 Β· Example 2: g(x)=xΒ²
g(x)=xΒ². Over Ξ”x=2 intervals, AROCs are βˆ’4, 0, 4, 8. What type of function and what is the concavity?
Check if AROCs change at a constant rate
7/10
Topic 1.3 Β· Example 4a: k(x)
k(x) has AROCs of βˆ’30, βˆ’20, βˆ’10. What is the concavity? Is k increasing or decreasing?
k is going down but the slope is getting bigger
8/10
Topic 1.3 Β· Concavity vs. Direction
Can a function be DECREASING and CONCAVE UP at the same time? Explain.
Think Example 4a
πŸ“„ Page 2 β€” Answers BACK Β· columns swapped
2/10
βœ“ Quadratic β€” AROC Pattern
AROC changes at a constant rate
The AROCs form a linear pattern β€” they increase or decrease by the same constant amount each step.
Ξ”(AROC) = constant β‰  0.
1/10
βœ“ Linear β€” AROC
AROC is CONSTANT (same every interval)
A linear function has constant slope everywhere. AROC = slope of the line, regardless of which interval you pick.
4/10
βœ“ Concave Up
AROC is INCREASING
Concave up (βˆͺ) = slope is getting larger (more positive or less negative).
This does NOT mean the function is increasing!
3/10
βœ“ 3-Step Method
1) Verify equal Ξ”x 2) Compute AROC=Ξ”y/Ξ”x 3) Check Ξ”(AROC)
Ξ”(AROC)=0 β†’ Linear
Δ(AROC)=constant≠0 → Quadratic
Ξ”(AROC)=not constant β†’ Neither
6/10
βœ“ Example 2: g(x)=xΒ²
Quadratic Β· Concave Up
AROCs βˆ’4, 0, 4, 8 increase by 4 each step β†’ quadratic.
AROC is increasing β†’ concave up.
5/10
βœ“ Concave Down
AROC is DECREASING
Concave down (∩) = slope is getting smaller (less positive or more negative).
The function can still be increasing while concave down!
8/10
βœ“ Decreasing AND Concave Up
YES β€” these are independent properties
Direction: is the function going up or down? (sign of AROC)
Concavity: is the slope increasing or decreasing?
Example 4a: k has negative AROCs (decreasing) that are increasing (βˆ’30β†’βˆ’20β†’βˆ’10) β†’ decreasing + concave up.
7/10
βœ“ Example 4a: k(x)
Concave Up Β· Decreasing
AROCs: βˆ’30, βˆ’20, βˆ’10 are negative β†’ k is decreasing.
AROCs are increasing (βˆ’30 < βˆ’20 < βˆ’10) β†’ concave up. Both simultaneously!
πŸ“„ Page 3 β€” Questions FRONT Β· Sheet 2/2
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Topic 1.3 Β· Example 3b
g(x): (1,0),(2,1),(5,4),(10,9) with unequal Ξ”x. AROCs = 1/1, 3/3, 5/5. What type?
Even though intervals are unequal β€” is AROC constant?
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Topic 1.3 Β· Neither (linear concavity)
A function has constant AROC. What is its concavity β€” concave up, concave down, or neither?
Constant AROC = what type of function?
πŸ“„ Page 4 β€” Answers BACK Β· columns swapped
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βœ“ Neither (Linear) β€” Concavity
Neither β€” linear functions have no concavity
Constant AROC β†’ linear function β†’ slope never changes β†’ neither concave up nor concave down. "Neither" in concavity context = linear.
9/10
βœ“ Example 3b β€” Linear
Linear β€” AROC is constant = 1
Despite unequal Ξ”x (1, 3, 5), AROC = Ξ”y/Ξ”x = 1/1 = 3/3 = 5/5 = 1. Constant AROC β†’ Linear. Always compute Ξ”y/Ξ”x, not just Ξ”y.