linear vs. exponential
i can determine whether a sequence is linear, exponential, or neither.
match the vocabulary word with its synonym:
9.) linear
a.) geometric
10.) exponential
b.) arithmetic
11.) $f(x) = 7x + 2$
choice: geo metrio
reason:
12.) $f(x) = 2 \cdot 7^x$
choice: arithmetic
reason:
13.) $f(x) = 7x^2$
choice: geometric
reason:
14.)
| x | f(x) |
| 0 | 5 |
| 1 | 15 |
| 2 | 45 |
| 3 | 135 |
| 4 | 405 |
choice:
reason:
15.)
| x | f(x) |
| 0 | -20 |
| 1 | -14 |
| 2 | -8 |
| 3 | -2 |
| 4 | 4 |
choice:
reason:
16.)
| x | f(x) |
| 0 | 1 |
| 1 | 1 |
| 2 | 2 |
| 3 | 3 |
| 4 | 5 |
choice:
reason:
17.)
graph
choice: exponential
reason:
18.)
graph
choice: exponetina
reason:
19.)
graph
choice: linear
reason:

Question

Accepted Answer

### Problem 9:
# Explanation:
## Step1: Recall synonyms of linear
Linear sequences (arithmetic sequences) have a common difference, and "arithmetic" is a synonym for linear in sequence terms. "Geometric" is for exponential (common ratio).
## Step2: Match the term
"Linear" is synonymous with "Arithmetic" (since linear sequences are arithmetic, exponential are geometric). So Linear matches B. Arithmetic, Exponential matches A. Geometric.
# Answer:
9. B. Arithmetic
10. A. Geometric

### Problem 11: $ f(x) = 7x + 2 $
# Explanation:
## Step1: Identify function type
A linear function has the form $ f(x)=mx + b $ (degree 1). This function is $ 7x + 2 $, which is linear (arithmetic sequence related, since linear functions have constant rate of change like arithmetic sequences).
## Step2: Correct choice
Previously marked "Geometric" is wrong. Correct choice: Linear (Arithmetic - related). Reason: It is a linear function with form $ f(x)=mx + b $, degree 1, constant slope (rate of change), characteristic of linear/arithmetic.
# Answer:
Choice: Linear (Arithmetic)
Reason: It is a linear function ($ f(x) = mx + b $ form, degree 1) with constant slope, so linear (arithmetic - like rate of change).

### Problem 12: $ f(x) = 2 \cdot 7^x $
# Explanation:
## Step1: Identify function type
An exponential function has the form $ f(x)=ab^x $ (variable in exponent). This function is $ 2 \cdot 7^x $, which is exponential (geometric sequence related, common ratio).
## Step2: Correct choice
Previously marked "Arithmetic" is wrong. Correct choice: Exponential (Geometric - related). Reason: It is an exponential function with form $ f(x)=ab^x $, variable in exponent, characteristic of exponential (geometric - like growth/decay).
# Answer:
Choice: Exponential (Geometric)
Reason: It is an exponential function ($ f(x) = ab^x $ form, variable in exponent), so exponential (geometric - like rate of change).

### Problem 13: $ f(x) = 7x^2 $
# Explanation:
## Step1: Identify function type
This is a quadratic function (degree 2), neither linear (degree 1) nor exponential (variable in exponent).
## Step2: Correct choice
Previously marked "Geometric" is wrong. Correct choice: Neither. Reason: It is a quadratic function ($ x^2 $ term), not linear (degree 1) or exponential (variable in exponent), so neither.
# Answer:
Choice: Neither
Reason: It is a quadratic function ($ f(x) = 7x^2 $, degree 2), not linear (degree 1) or exponential (variable in exponent).

### Problem 14: Table $ \begin{array}{c|c} x & f(x) \ \hline 0 & 5 \ 1 & 15 \ 2 & 45 \ 3 & 135 \ 4 & 405 \end{array} $
# Explanation:
## Step1: Check ratios/differences
Calculate ratios: $ \frac{15}{5}=3 $, $ \frac{45}{15}=3 $, $ \frac{135}{45}=3 $, $ \frac{405}{135}=3 $. Constant ratio (3), so exponential (geometric sequence).
## Step2: Determine type
Since there is a common ratio, it is exponential (geometric).
# Answer:
Choice: Exponential (Geometric)
Reason: The ratio between consecutive $ f(x) $ values is constant ($ \frac{15}{5}=\frac{45}{15}=3 $, etc.), so it is a geometric (exponential) sequence with common ratio 3.

### Problem 15: Table $ \begin{array}{c|c} x & f(x) \ \hline 0 & -20 \ 1 & -14 \ 2 & -8 \ 3 & -2 \ 4 & 4 \end{array} $
# Explanation:
## Step1: Check differences/ratios
Calculate differences: $ -14 - (-20)=6 $, $ -8 - (-14)=6 $, $ -2 - (-8)=6 $, $ 4 - (-2)=6 $. Constant difference (6), so linear (arithmetic sequence).
## Step2: Determine type
Since there is a common difference, it is linear (arithmetic).
# Answer:
Choice: Linear (Arithmetic)
Reason: The difference between consecutive $ f(x) $ values is constant ($ -14 - (-20)=6 $, etc.), so it is an arithmetic (linear) sequence with common difference 6.

### Problem 16: Table $ \begin{array}{c|c} x & f(x) \ \hline 0 & 1 \ 1 & 1 \ 2 & 2 \ 3 & 3 \ 4 & 5 \end{array} $
# Explanation:
## Step1: Check differences/ratios
Differences: $ 1 - 1 = 0 $, $ 2 - 1 = 1 $, $ 3 - 2 = 1 $, $ 5 - 3 = 2 $. Ratios: $ \frac{1}{1}=1 $, $ \frac{2}{1}=2 $, $ \frac{3}{2}=1.5 $, $ \frac{5}{3}\approx1.67 $. No constant difference or ratio.
## Step2: Determine type
Since neither common difference (linear) nor common ratio (exponential) exists, it is neither.
# Answer:
Choice: Neither
Reason: There is no constant difference (for linear) or constant ratio (for exponential) between consecutive $ f(x) $ values.

### Problem 17: Graph (wavy, polynomial - like)
# Explanation:
## Step1: Analyze graph shape
The graph is a polynomial (not linear or exponential). Linear graphs are straight lines, exponential are curves (increasing/decreasing with constant ratio, smooth curve). This graph has multiple peaks/valleys, so it is a polynomial (neither linear nor exponential).
## Step2: Determine type
Previously marked "Exponential" is wrong. Correct choice: Neither. Reason: It is a polynomial graph (wavy, multiple turning points), not linear (straight) or exponential (smooth curve with constant growth/decay rate).
# Answer:
Choice: Neither
Reason: The graph has multiple turning points (not straight like linear, not a smooth exponential - like curve), so it is neither linear nor exponential.

### Problem 18: Graph (exponential - like curve)
# Explanation:
## Step1: Analyze graph shape
The graph is a smooth curve increasing with a constant ratio (exponential growth - like, starts near zero and rises rapidly, characteristic of $ f(x)=ab^x $ with $ b > 1 $).
## Step2: Determine type
This is exponential. Reason: It is a smooth curve with exponential growth (constant ratio of change, curve shape typical of exponential functions).
# Answer:
Choice: Exponential (Geometric)
Reason: The graph is a smooth, increasing curve with the characteristic shape of an exponential function (constant ratio of change, typical of $ f(x)=ab^x $ with $ b > 1 $).

### Problem 19: Graph (straight line)
# Explanation:
## Step1: Analyze graph shape
The graph is a straight line, which is linear (constant slope, characteristic of linear functions/arithmetic sequences).
## Step2: Determine type
This is linear. Reason: The graph is a straight line, indicating a constant slope (rate of change), characteristic of linear functions.
# Answer:
Choice: Linear (Arithmetic)
Reason: The graph is a straight line, so it has a constant slope (rate of change), characteristic of linear functions.

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QUESTION IMAGE

Question

Explanation:

Problem 9:

Step1: Recall synonyms of linear

Step2: Match the term

Step1: Identify function type

Step2: Correct choice

Step1: Identify function type

Step2: Correct choice

Answer:

Problem 11: \( f(x) = 7x + 2 \)