Question 119·200 Super-Hard SAT Math Questions·Advanced Math

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The function $h$ is defined by $h(x)=72(16)^{x/6}$ . A new variable is defined by $u=2x-3$ . Which choice is an equivalent expression for $h$ in terms of $u$ that displays, as the base of its exponential factor, the factor by which $h$ increases for every increase of $1$ in $u$ ?

A new input in an exponential function can change both its starting value and its per-unit growth factor. Solve for the old input, use Desmos function evaluation to find the output when the new input is $0$ , then find how much the exponent changes per unit. An equivalent expression can still have a base that represents the wrong interval.

Hints

Hint 1
To rewrite a function in terms of a new variable, first solve the variable equation for the old input. What value of $x$ corresponds to $u=0$ ?
Hint 2
In the exponential form $A B^u$ , the value at $u=0$ is $A$ because $B^0=1$ . Evaluate the original function at the $x$ -value you found, rather than assuming its original coefficient is still the starting value.
Hint 3
The one-unit factor depends on how much the old exponent changes when $u$ increases by $1$ . Use your expression for $x$ to find that change, then rewrite the resulting power as a cube root.

Step-by-step

Find the new start and one-unit factor
Step 1
Write the old input in terms of the new one
To use $u$ as the input, solve for $x$ . Add $3$ : $u+3=2x$ . Divide by $2$ : $x=\frac{u+3}{2}$ . So a one-unit increase in $u$ increases $x$ by only $\frac12$ .
Step 2
Find the value at the new input zero
At $u=0$ , the old input is $x=\frac{0+3}{2}=\frac32$ . Type $h(x)=72(16)^{x/6}$ and then $h(3/2)$ in Desmos. It shows $144$ , the starting value when $u=0$ . The $72$ in the given formula belongs to $x=0$ , not $u=0$ .
Step 3
Find how much the exponent changes
Each increase of $1$ in $u$ increases $x$ by $\frac12$ . The exponent is $x/6$ , so divide that change by $6$ : $\frac{1/2}{6}=\frac1{12}$ . The exponent rises by $\frac1{12}$ , not by $1$ , for each new-input step.
Step 4
Calculate the factor for one step in u
Raising an exponent by $\frac1{12}$ multiplies the output by $16^{1/12}$ . Type $16^{1/12}$ below the earlier Desmos lines. It shows about $1.259921$ : this is the factor for each increase of $1$ in $u$ .
Step 5
Write that factor exactly
The choices use an exact root, so use $16=2^4$ . Rewrite the base: $16^{1/12}=(2^4)^{1/12}$ . Apply the power-of-a-power rule, which multiplies exponents: $(2^4)^{1/12}=2^{4/12}$ . Reduce the fraction: $2^{4/12}=2^{1/3}$ . A power of $\frac13$ is a cube root: $2^{1/3}=\sqrt[3]{2}$ . This root is the base the question wants displayed.
Step 6
Build the requested expression
In $A B^u$ , $A$ is the value at $u=0$ and $B$ multiplies the value whenever $u$ increases by $1$ . Use the new start and the one-unit factor: $144\left(\sqrt[3]{2}\right)^u$ . This expression has the requested base. Choice C.

Question 119·200 Super-Hard SAT Math Questions·Advanced Math

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