Question

level k > aa.7 write linear, quadratic, and exponential functi you have prizes to reveal! go to your game board, look at this table: x | y 3 | -90 4 | -160 5 | -250 6 | -360 7 | -490 write a linear ($y = mx + b$), quadratic ($y = ax^2$), or exponential ($y = a(b)^x$) function that models the data. $y = $

Explanation:

Step1: Check first differences

Calculate $\Delta y$ for consecutive $x$:
$\Delta y_1 = -160 - (-90) = -70$
$\Delta y_2 = -250 - (-160) = -90$
$\Delta y_3 = -360 - (-250) = -110$
$\Delta y_4 = -490 - (-360) = -130$
First differences are not constant, so not linear.

Step2: Check second differences

Calculate $\Delta^2 y$ (difference of first differences):
$\Delta^2 y_1 = -90 - (-70) = -20$
$\Delta^2 y_2 = -110 - (-90) = -20$
$\Delta^2 y_3 = -130 - (-110) = -20$
Second differences are constant, so quadratic.

Step3: Find $a$ in $y=ax^2$

Use $x=3, y=-90$:
$-90 = a(3)^2$
$-90 = 9a$
$a = \frac{-90}{9} = -10$

Step4: Verify the function

Test $x=4$: $y=-10(4)^2=-160$ (matches)
Test $x=5$: $y=-10(5)^2=-250$ (matches)
All points fit.

Answer:

$y = -10x^2$