Question

1. the tables below show the input and output values of four different functions. which table represents a linear function?

a. f(x)
b. g(x)
c. h(x)
d. j(x)

1. a census of all the 12,475 current students at the local community college showed that 56.5% want to transfer to a university, their average age is 21.7 years, and 60 of them are pursuing degrees in mathematics. in order to attract more students, the math department organized an \opportunities in mathematics\ fair for current students of the college. a total of 267 students from those who attended responded to a poll. the poll showed that 83% of those who answered want to transfer to a university, the average poll respondent was 22.3 years old, and 77 of them were interested in pursuing a degree in mathematics.

(a) for the math fair poll, identify the population and the sample.
population:
○the 267 students who responded to the poll
○all the current students at the community college
sample:
○the 267 students who responded to the poll
○all the current students at the community college
(b) choose whether each number described below is a parameter or a statistic for the math fair poll.
number | parameter | statistic
the 60 students pursuing a math degree among all the current students at the community college | |
the average age of 22.3 years of the students who responded to the poll | |
the 83% of students who responded to the poll who want to transfer to a university | |

Explanation:

Question 10 (Mathematics - Subfield: Algebra (specifically Linear Functions))

Step 1: Recall the definition of a linear function

A linear function has a constant rate of change, meaning the difference in \( f(x) \) (or \( y \)) values is constant for equal differences in \( x \) values. Mathematically, for a function \( y = f(x) \), the slope \( m=\frac{\Delta y}{\Delta x} \) should be constant for all pairs of points.

Step 2: Analyze Table A (f(x))

• \( x \): -2, -1, 0, 1, 2, 3
• \( f(x) \): 6, 3, 1, -2, -5, 1
• Calculate \( \Delta y \) (change in \( f(x) \)) for \( \Delta x = 1 \) (since \( x \) increases by 1 each time):
• From \( x=-2 \) to \( x=-1 \): \( 3 - 6=-3 \)
• From \( x=-1 \) to \( x=0 \): \( 1 - 3=-2 \)
• The changes are not constant (-3, -2,...), so not linear.

Step 3: Analyze Table B (g(x))

• \( x \): -4, -3, -2, -1, 0, 1
• \( g(x) \): 3, 2, 1, 0, -1, -2
• Calculate \( \Delta y \) for \( \Delta x = 1 \):
• From \( x=-4 \) to \( x=-3 \): \( 2 - 3=-1 \)
• From \( x=-3 \) to \( x=-2 \): \( 1 - 2=-1 \)
• From \( x=-2 \) to \( x=-1 \): \( 0 - 1=-1 \)
• From \( x=-1 \) to \( x=0 \): \( -1 - 0=-1 \)
• From \( x=0 \) to \( x=1 \): \( -2 - (-1)=-1 \)
• The change in \( y \) is constant (-1) for \( \Delta x = 1 \), so this is a linear function. (We can check the other tables for confirmation, but since we found a constant rate here, we can conclude Table B is linear.)

Step 4: (Optional) Verify other tables (for thoroughness)

• Table C (h(x)): The \( y \)-values do not show a constant rate of change (e.g., from \( x=-5 \) to \( x=-3 \), the change is not consistent with linearity).
• Table D (j(x)): The \( y \)-values also do not show a constant rate of change (e.g., from \( x=-2 \) to \( x=0 \), the change is not consistent).

• Population: The population is the entire group we are interested in studying. In this case, the math fair is for all current students at the community college, so the population is all the current students at the community college.
• Sample: The sample is a subset of the population that is actually surveyed. Here, the 267 students who responded to the poll are the sample (since they are a part of the population of all current students).

• Parameter: A parameter is a numerical characteristic of a population.
• Statistic: A statistic is a numerical characteristic of a sample.

1. "The 60 students pursuing a math degree among all the current students at the community college": This is about the entire population (all current students), so it is a parameter.
2. "The average age of 22.3 years of the students who responded to the poll": This is about the sample (the 267 respondents), so it is a statistic.
3. "The 83% of students who responded to the poll who want to transfer to a university": This is about the sample (the 267 respondents), so it is a statistic.

Answer:

B (g(x))

Question 11 (Mathematics - Subfield: Statistics (Population and Sample, Parameters and Statistics))

Part (a)