Question 26·SAT Desmos Regression Practice·Advanced Math

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A function $f$ is defined by $f(x)=a(b)^x$ , where $a$ and $b$ are positive constants. If $f(2)=72$ and $f(5)=1944$ , which choice is the value of $f(0)$ ?

In an exponential model $f(x)=ab^x$ , the coefficient $a$ is the starting value: at input $0$ , the power equals $1$ . If two outputs are given, turn them into input-output pairs and fit the stated model with a Desmos table regression. Read the fitted starting value, not the growth factor or one of the later outputs.

Hints

Hint 1
Use the zero-exponent rule: because $b$ is positive, $b^0=1$ . What does the given function become at $x=0$ , and which constant do you need to find?
Hint 2
Each statement $f(k)=v$ gives an input-output pair $(k,v)$ . Put the two pairs in a Desmos table, then fit the given exponential form rather than treating either known output as the starting value.

Step-by-step

Fit the given model in Desmos
Step 1
Identify the value you need
Because $b$ is positive, the zero-exponent rule gives $b^0=1$ . Put $0$ into the function: $f(0)=a(b)^0=a$ . The starting value is $a$ , not the base $b$ .
Step 2
Turn the known outputs into points
Putting $2$ and $5$ into the given rule gives $a(b)^2=72,\qquad a(b)^5=1944$ . So the two input-output pairs are $(2,72)$ and $(5,1944)$ . Both points must fit the same $a$ and $b$ .
Step 3
Fit both points and read the starting value
Enter those pairs in a Desmos table, then type $y_1\sim a(b)^{x_1}$ . The regression symbol $\sim$ asks Desmos to find constants that fit both rows. Under PARAMETERS, Desmos shows $a=8$ and $b=3$ . Since $f(0)=a$ , the requested value is $8$ . Choice B.

Question 26·SAT Desmos Regression Practice·Advanced Math

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