Question

1. the scatterplot and table show the height, in meters, of a certain tree, y, and the number of years since it was planted, x. the relationship is best modeled by an exponential function. which of the following would result in a relationship that appears linear?

years since planting (x) | height of tree, meters (y)
0 | 0.51
1 | 0.82
2 | 1.33
3 | 2.14
4 | 3.45
5 | 5.56
6 | 8.97
7 | 14.38
8 | 23.19
9 | 37.21
10 | 60
a) graphing x on the vertical axis and y on the horizontal axis
b) graphing log x on the horizontal axis and y on the vertical axis
c) graphing x on the horizontal axis and log y on the vertical axis
d) graphing x on the horizontal axis and y² on the vertical axis
e) graphing √x on the horizontal axis and y on the vertical axis

Explanation:

Step1: Recall exponential - linear conversion

An exponential function is of the form $y = ab^{x}$. Taking the logarithm of both sides gives $\log y=\log a + x\log b$. This is in the form of a linear equation $Y = mx + c$, where $Y=\log y$, $m = \log b$, and $c=\log a$.

Step2: Analyze each option

• Option A: Just swapping the axes ($x$ on vertical and $y$ on horizontal) does not linearize an exponential relationship.
• Option B: Graphing $\log x$ and $y$ does not transform the exponential relationship $y = ab^{x}$ into a linear one.
• Option C: Since $\log y=\log a + x\log b$, graphing $x$ on the horizontal axis and $\log y$ on the vertical axis will result in a linear - looking graph.
• Option D: Graphing $x$ and $y^{2}$ for an exponential function $y = ab^{x}$ does not linearize the relationship.
• Option E: Graphing $\sqrt{x}$ and $y$ for an exponential function $y = ab^{x}$ does not linearize the relationship.

Answer:

C. Graphing $x$ on the horizontal axis and $\log y$ on the vertical axis