Question

question 17 (3 points)
assume that the growth of the membership of a country club was linear from 1996 to 2000 with a membership of 742 in 1996 and a rate of growth of 236 per year. write an equation for the membership p of this country club as a function of the number of years x after 1996.

\\( p = 742x + 236 \\)
\\( p = 236x + 2000 \\)
\\( p = 2000x + 236 \\)
\\( p = 236x + 742 \\)

question 18 (3 points)
select the best type of modeling function for the data shown in the graph.

Explanation:

Question 17

Step1: Recall linear function form

A linear function is in the form \( y = mx + b \), where \( m \) is the slope (rate of change) and \( b \) is the y - intercept (initial value). In this problem, \( P \) is the membership (dependent variable), \( x \) is the number of years after 1996 (independent variable).

Step2: Identify slope and y - intercept

The rate of growth (slope \( m \)) is 236 per year. The initial membership (in 1996, when \( x = 0 \)) is 742, so the y - intercept \( b = 742 \).

Step3: Form the equation

Using the linear function form \( P=mx + b \), substituting \( m = 236 \) and \( b = 742 \), we get \( P = 236x+742 \).

Answer:

\( P = 236x + 742 \) (the last option among the given choices)

Question 18

Since the graph is not fully visible, but from the description of the data points (they seem to follow a curve that is increasing at an increasing rate, or a curve that is concave up), if the data points show a pattern where the rate of increase is increasing (e.g., the slope between consecutive points is increasing), an exponential function or a quadratic function (if it's a parabola - shaped curve) could be a candidate. But if we assume the points are following a curve that is best modeled by an exponential function (common for growth - like data) or a quadratic function. However, since the graph is not clear, but from the general shape of points that are rising and the distance between them is increasing, a possible best - fit function could be an exponential function or a quadratic function. But if we consider the typical cases, if the data has a curve that is concave up (increasing slope), a quadratic function \( y = ax^{2}+bx + c\) (where \( a>0 \)) or an exponential function \( y = ab^{x}\) (\( b > 1 \)) can be used. But without the full graph, we can infer based on the point pattern. If the points are forming a curve that is not linear (since the spacing between points is changing), and if it's a curve that is growing faster, exponential or quadratic. But if we have to choose from common models, and the points are showing a curve that is concave up, a quadratic function or exponential function. But since the problem is about modeling, and if the data is increasing with an increasing rate, an exponential model or a quadratic model. However, if we assume the graph is of a curve that is best modeled by a quadratic function (for example, if it's a parabola - like curve) or an exponential function. But as the graph is not clear, we can't give a definite answer without more details. But if we consider the points given (the dots are increasing and the vertical distance between them is increasing), a possible best - fit is an exponential function or a quadratic function. If we have to choose from linear, quadratic, exponential, etc., and the data is not linear (since the slope between points is changing), then a non - linear model like quadratic or exponential. But since the problem is presented, and if we assume the graph is of a curve that is best modeled by a quadratic function (for example, if it's a parabola opening upwards) or an exponential function. But without the full graph, we can't be certain.