HSF.LE.A.1.a Tutor Intake — Linear and Exponential Growth Properties

Which statement best describes what it means to prove that linear functions
grow by equal differences over equal intervals?

For the exponential function $g(x) = a \cdot b^x$, which algebraic property
is the key tool for simplifying $g(x + d)$?

A student wants to test whether $h(x) = 5 \cdot 3^x$ is exponential.
Which expression should the student simplify?

The exponent rule used in the exponential-growth proof states
$b^{x+d} = b^x \cdot b^d$. Using this rule, simplify $2^7 \div 2^4$.
Enter the numerical value.

Let $f(x) = 4x + 9$. Compute $f(x + 3) - f(x)$ and simplify completely.
Enter the numerical value of the constant difference.

When computing $f(x + d) - f(x)$ for $f(x) = 7x - 2$, a student writes:
$f(x + d) = 7x + d - 2$
What error did the student make?

Let $g(x) = 3 \cdot 2^x$. Compute $g(x + 4) \div g(x)$ and simplify.
Enter the numerical value of the constant ratio.

For $f(x) = 6x + 1$, the proof shows that $f(x + d) - f(x) = md$.
If the constant difference over an interval of width $d = 5$ is $30$,
what is the slope $m$?

For $g(x) = 5 \cdot b^x$, the ratio $g(x + 1) \div g(x) = 3$.
What is the base $b$?