Question 19·SAT Desmos Regression Practice·Advanced Math

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The function $f$ is defined by

f(x)=a\cdot 3^{x-1}-5

where $a$ is a constant. If $f(2)=4$ , which choice is the value of $f(0)$ ?

In an exponential function, the input changes the exponent while a constant in front stays fixed. If you're given one output and asked for another, use the known input to find the constant, then evaluate the function at the requested input in Desmos. Watch the shift inside the exponent: $3^{x-1}$ is not $3^x$ .

Hints

Hint 1
In function notation, $f(2)=4$ means input $2$ gives output $4$ . Replace $x$ with $2$ in the rule and set the result equal to $4$ . What does the exponent become?
Hint 2
The constant $a$ keeps the same value when the input changes. After finding it, use input $0$ in the rule. A negative exponent means a reciprocal: $3^{-1}=\frac13$ , not $-3$ .

Step-by-step

Find the constant, then evaluate
Step 1
Turn the known output into an equation
$f(2)=4$ means that putting $2$ in for $x$ gives $4$ . Substitute $2$ into the entire exponent $x-1$ :
$4=a\cdot3^{2-1}-5.$
Step 2
Find the constant in Desmos
Type $4\sim a\cdot3^{2-1}-5$ . The tilde asks Desmos to find the unknown rather than graph an equation. Under PARAMETERS, Desmos shows $a=3$ . Use the known input to find the constant before switching to the requested input.
Step 3
Define the function with that constant
Type $f(x)=a\cdot3^{x-1}-5$ on the next line. Desmos accepts the function definition and uses the $a$ it found. The constant stays $3$ when you change the input.
Step 4
Read the output at zero
Type $f(0)$ . Desmos shows $-4$ . At input $0$ , the exponent is $0-1=-1$ , and $3^{-1}=\frac13$ : a negative exponent gives a reciprocal, not a negative number. So the requested value is $f(0)=-4$ . Choice D.

Question 19·SAT Desmos Regression Practice·Advanced Math

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