To analyze the statements related to the functions Clara is studying, let's first clarify what each function represents:
- p(x) = 1.3x: This is a linear function, specifically it represents a line with a slope of 1.3.
- q(x) = 1.3x + 10: This is also a linear function, but it has the same slope as p(x) and is simply shifted upwards by 10 units.
- t(x) = 1.3^x: This is an exponential function. Exponential functions grow faster than linear functions as x increases.
Now, let's analyze each statement:
A. For small values of x, the growth rate of t(x) = 1.3^x is less than the growth rate of p(x) = 1.3x and the growth rate of q(x) = 1.3x + 10.
- For small values of \( x \) (like \( x = 0 \) or \( x = 1 \)), \( t(x) \) starts close to 1 (because \( 1.3^0 = 1 \)) and grows slowly compared to the linear functions, which would produce much larger values. This statement is true, at least for sufficiently small \( x \).
B. For small values of x, the growth rate of t(x) = 1.3^x is less than the growth rate of p(x) = 1.3x and the growth rate of q(x) = 1.3x + 10.
- This is similar to statement A. However, it incorrectly claims that \( t(x) \) is less than both linear functions when in fact it is the value that is less, but as \( x \) increases, exponential growth will outstrip linear growth. Therefore, this statement is somewhat misleading but depends on interpretation.
C. The growth rate of q(x) = 1.3x + 10 is always greater than the growth rate of t(x) = 1.3^x and the growth rate of p(x) = 1.3x.
- This statement implies that the linear function \( q(x) \) grows faster than both p(x) and t(x) for all \( x \). While it is certainly true for small values, as \( x \) increases, \( t(x) \) grows faster because exponential functions eventually surpass linear functions.
D. The growth rate of t(x) = 1.3^x is always greater than the growth rate of p(x) = 1.3x and the growth rate of q(x) = 1.3x + 10.
- This is incorrect for small values of \( x \), as initially \( t(x) \) grows slower than the linear functions.
The most accurate statement about the growth rates of these functions at small values of x is: A. For small values of x, the growth rate of t(x)=1.3^x is less than the growth rate of p(x)=1.3x and the growth rate of q(x)=1.3x + 10.