Function Patterns Worksheet
Algebraic Reasoning TEKS AR.2.A - AR.2.D
Function Patterns & Differences
Analyze tables, graphs, and rates of change to classify and formulate mathematical models.
Name:
Date: Period:
Linear
Constant 1st Differences:
\(\Delta y = m \cdot \Delta x\)
Quadratic
Constant 2nd Differences:
\(\Delta^2 y = 2a\)
Cubic
Constant 3rd Differences:
\(\Delta^3 y = 6a\)
Exponential
Constant Successive Ratio:
\(\frac{y_n}{y_{n-1}} = b\)
1 Part I: Finite Difference Analysis & Function Family Identification
Compute differences or ratios, then classify function family
Set A: Data Sequence
| \(x\) | \(y\) | \(\Delta y\) | \(\Delta^2 y\) |
|---|
| 0 | 3 | | |
| 1 | 6 | | |
| 2 | 13 | | |
| 3 | 24 | | |
| 4 | 39 | | |
Constant Value:
Classification:
Set B: Data Sequence
| \(x\) | \(y\) | \(\Delta y\) | Ratio |
|---|
| 0 | 2 | | |
| 1 | 6 | | |
| 2 | 18 | | |
| 3 | 54 | | |
| 4 | 162 | | |
Constant Value:
Classification:
Set C: Data Sequence
| \(x\) | \(y\) | \(\Delta y\) | \(\Delta^2 y\) |
|---|
| 0 | 19 | | |
| 1 | 15 | | |
| 2 | 11 | | |
| 3 | 7 | | |
| 4 | 3 | | |
Constant Value:
Classification:
2 Part II: Classifying Across Representations
Label as Linear, Quadratic, Cubic, or Exponential & justify
1. Symbolic Representation f(x) Form
\(g(x) = 5(1.45)^x - 3\)
Classification:
Key Feature/Justification:
2. Graphical Feature Description Curvature
A continuous curve passes through the origin, has rotational symmetry about \((0,0)\), and has two turning points (a local maximum and local minimum).
Classification:
Key Feature/Justification:
3. Average Rate of Change Analysis \(\frac{\Delta y}{\Delta x}\)
The average rate of change over consecutive unit intervals increases by a constant value of \(+6\) units per interval as \(x\) increases.
Classification:
Key Feature/Justification:
4. Contextual Behavior Real-World
A sample of medicine in the bloodstream decreases such that exactly \(40\%\) of the remaining drug quantity breaks down every elapsed hour.
Classification:
Key Feature/Justification:
Algebraic Reasoning • Section 2: Patterns & Structure Page 1 of 2
Part III & IV: Mathematical Modeling
Constructing Function Models from Differences & Rates
Name:
5 Quadratic Modeling from Second Differences: \(f(x) = ax^2 + bx + c\)
AR.2.C, AR.2.D
The table below gives values for a quadratic function \(f(x)\). Determine the values of finite differences to solve for parameters \(a\), \(b\), and \(c\), then state the function model.
| \(x\) | \(f(x)\) | \(\Delta y\) | \(\Delta^2 y\) |
|---|
| 0 | 5 | | |
| 1 | 8 | | |
| 2 | 15 | | |
| 3 | 26 | | |
| 4 | 41 | | |
Step A: Parameter \(a\) \(\Delta^2 y = 2a\)
Step B: Parameter \(c\) \(c = f(0)\)
Step C: Parameter \(b\) \(a(1)^2 + b(1) + c = f(1)\)
Final Quadratic Function Model: \(f(x) =\)
6 Exponential Modeling via Common Ratio: \(g(x) = a \cdot b^x\)
AR.2.C, AR.2.D
A laboratory culture of microbes begins at count \(a\) and multiplies across uniform time periods \(x\).
| \(x\) (hrs) | \(N(x)\) | Ratio \(r\) |
|---|
| 0 | 250 | -- |
| 1 | 750 | |
| 2 | 2,250 | |
| 3 | 6,750 | |
Initial Value \(a = N(0)\):
Base / Common Ratio \(b = \frac{N(x)}{N(x-1)}\):
Exponential Model: \(N(x) =\)
7 Real-World Physics Application: Vehicle Braking Distance
AR.2.A - AR.2.D
Engineers recorded the total stopping distance \(d(s)\) in feet for a sedan traveling at speed \(s\) in miles per hour:
| Speed \(s\) (mph) | 10 | 20 | 30 | 40 | 50 |
|---|
| Distance \(d\) (ft) | 5.0 | 20.0 | 45.0 | 80.0 | 125.0 |
a) Compute \(\Delta d\) and \(\Delta^2 d\) for \(\Delta s = 10\). Identify the model family:
b) Determine equation \(d(s)\) and predict stopping distance at \(60\text{ mph}\):
Algebraic Reasoning • Section 2: Patterns & Structure Page 2 of 2
