Question 27·SAT Desmos Regression Practice·Advanced Math

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The function $h$ is defined by

h(t)=a\cdot 3^{t/2}+b

where $t$ is the number of hours after a culture is started.

If $h(0)=20$ and $h(4)=92$ , which choice is the value of $h(2)$ ?

An exponential function with an added constant doesn't change by the same amount over equal time intervals. Two known outputs can determine the two unknown constants. Use a matched-list regression in Desmos to find them together, then evaluate the function. Don't average the given outputs merely because the requested time is halfway between them.

Hints

Hint 1
In function notation, $h(0)$ means replace $t$ with $0$ , not multiply $h$ by $0$ . Do the same with $t=4$ to make a second equation for the same $a$ and $b$ .
Hint 2
Put the two expressions you get from the function in one list and their known outputs in another. A single regression in Desmos finds values of $a$ and $b$ that satisfy both conditions.
Hint 3
At two hours, the exponent is $2/2=1$ , so the power contributes a factor of $3$ . Once you've multiplied by $a$ , what does the function rule still tell you to add?

Step-by-step

Fit the constants, then evaluate
Step 1
Turn the known outputs into equations
In function notation, $h(0)=20$ says to put $0$ in for $t$ , and $h(4)=92$ says to put $4$ in. Both use the same $a$ and $b$ , so the given rule becomes
$a\cdot3^{0/2}+b=20,\qquad a\cdot3^{4/2}+b=92.$
Step 2
Find both constants in Desmos
Type $[a\cdot3^{0/2}+b,a\cdot3^{4/2}+b]\sim[20,92]$ . This regression pairs each expression with its known output and fits one pair of constants to both. Desmos shows $a=9$ and $b=11$ under PARAMETERS.
Step 3
Define the fitted function
Below the regression, type $h(t)=a\cdot3^{t/2}+b$ . Desmos uses the fitted $a=9$ and $b=11$ , so this is now the function you can evaluate. Keep $+b$ outside the power, as the given rule does.
Step 4
Evaluate after two hours
Type $h(2)$ ; Desmos prints $38$ . At $t=2$ , the exponent is $1$ , so the power supplies a factor of $3$ ; the rule then adds $b$ . Halfway in time is not halfway in output for an exponential function. The value of $h$ after $2$ hours is $38$ . Choice C.

Question 27·SAT Desmos Regression Practice·Advanced Math

Strategy

Hints

Step-by-step

Lessons that teach this

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Strategy

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Step-by-step

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