Question

for the set of points to the right, determine whether an exponential function, a logarithmic function, a linear function, or a quadratic function is the best choice for modeling the data. choose the correct answer below. a. exponential function b. quadratic function c. linear function d. logarithmic function

Explanation:

• Exponential function: Its graph has a characteristic rapid growth (or decay) curve. For example, \(y = a\cdot b^{x}\) (\(b> 1\) for growth). But the given data points do not show the typical exponential - like curvature (where the rate of change of \(y\) with respect to \(x\) is proportional to \(y\) itself).
• Quadratic function: A quadratic function \(y=ax^{2}+bx + c\) (\(a

eq0\)) has a parabolic graph. A parabola is symmetric about its axis of symmetry \(x =-\frac{b}{2a}\). The given data points do not show the U - shaped (if \(a>0\)) or inverted U - shaped (if \(a < 0\)) symmetry of a parabola.

• Linear function: A linear function has the form \(y=mx + b\), and its graph is a straight line. The rate of change (slope \(m=\frac{\Delta y}{\Delta x}\)) is constant. If we assume two - point differences (a rough estimate of the slope between consecutive points), we can see that the change in \(y\) relative to the change in \(x\) is approximately constant. For example, if we consider the general trend of the points moving from left - to - right, the increase in \(y\) for a relatively constant increase in \(x\) is in line with the behavior of a linear function \(y=mx + b\) where \(m>0\) (since \(y\) increases as \(x\) increases).
• Logarithmic function: A logarithmic function \(y = a\ln(x)+b\) (or \(y=a\log_{10}(x)+b\)) has a graph that has a vertical asymptote (for \(y = a\ln(x)+b\), the vertical asymptote is \(x = 0\)) and the rate of change of \(y\) with respect to \(x\) decreases as \(x\) increases. The given data points do not show the characteristic slow - down in the rate of increase (as \(x\) gets larger) that is typical of a logarithmic function.

Answer:

C. linear function