Question

level k 〉 aa.7 write linear, quadratic, and exponential functi you have prizes to reveal! go to your game board, x video questio answer look at this table: x y 1 12 2 36 3 108 4 324 5 972 write a linear (y = mx + b), quadratic (y = ax²), or exponential (y = a(b)ˣ) function that models the data. y =

Explanation:

Step1: Check linearity (constant slope)

Slope between (1,12) & (2,36): $\frac{36-12}{2-1}=24$
Slope between (2,36) & (3,108): $\frac{108-36}{3-2}=72$
Slopes are not equal, so not linear.

Step2: Check quadratic (constant second difference)

First differences: $36-12=24$, $108-36=72$, $324-108=216$, $972-324=648$
Second differences: $72-24=48$, $216-72=144$, $648-216=432$
Second differences are not equal, so not quadratic.

Step3: Check exponential (constant ratio)

Ratio of consecutive y-values: $\frac{36}{12}=3$, $\frac{108}{36}=3$, $\frac{324}{108}=3$, $\frac{972}{324}=3$
Constant ratio $b=3$, so it is exponential.

Step4: Find a in $y=a(b)^x$

Use $x=1, y=12$: $12=a(3)^1$
Solve for a: $a=\frac{12}{3}=4$

Step5: Write the function

Substitute $a=4, b=3$ into $y=a(b)^x$

Answer:

$y=4(3)^x$