Question

date: feb 6, 2026period: 3average rate of change (given a function) mazedirections: find the average rate of change of the functions over the interval specified in each box. start with the given information the start box. use your solution to move to the next box. continue using each solution until you reach the finished! box. show your work for each question. some boxes may not be usedstart:$f(x)=-\frac{1}{2}x^{2}+2x-1;$$-8,-6$$roc=\frac{31-(49)}{-6-(-8)}=9$$f(x)=\frac{1}{2}x^{2}+2x-1$$-2,4$$roc=\frac{4-(-2)}{4-(-2)}=3$$f(x)=x^{2}-2x-3$$5,8$$roc=\frac{45-12}{8-5}=11$$f(x)=x^{2}+8x+18$$-7,-3$$roc=______=-2$$f(x)=2x^{2}+4x+3$$-2,0$$roc=____=0$$f(x)=x^{2}+8x+12$$-7,-4$$roc=______=-3$$f(x)=-2x^{2}+16x-31$$4,7$$roc=__=-6$$f(x)=-x^{2}+6x-5$$5,9$$roc=______=-8$$f(x)=-x^{2}+8x-18$$2,4$$f(x)=x^{2}-4x+8$$-2,1$$roc=__=-5$$f(x)=-2x^{2}-4x-5$$0,3$$roc=____=-10$finished!© chelsea se

Explanation:

The average rate of change of a function $f(x)$ over interval $[a,b]$ is $\frac{f(b)-f(a)}{b-a}$. We solve each box in the maze path starting from the START box.

Step1: Solve START box

Function: $f(x)=-\frac{1}{2}x^2+2x-1$, interval $[-8,-6]$
Calculate $f(-8)$:
$f(-8)=-\frac{1}{2}(-8)^2+2(-8)-1=-\frac{1}{2}(64)-16-1=-32-16-1=-49$
Calculate $f(-6)$:
$f(-6)=-\frac{1}{2}(-6)^2+2(-6)-1=-\frac{1}{2}(36)-12-1=-18-12-1=-31$
Rate of change:
$\frac{f(-6)-f(-8)}{-6-(-8)}=\frac{-31-(-49)}{-6+8}=\frac{18}{2}=9$
Note: The handwritten calculation had a sign error; correct value is 9, which connects to the box with path 9 (the $f(x)=x^2+8x+18$ box)

Step2: Solve $f(x)=x^2+8x+18$, $[-7,-3]$

Calculate $f(-7)$:
$f(-7)=(-7)^2+8(-7)+18=49-56+18=11$
Calculate $f(-3)$:
$f(-3)=(-3)^2+8(-3)+18=9-24+18=3$
Rate of change:
$\frac{f(-3)-f(-7)}{-3-(-7)}=\frac{3-11}{-3+7}=\frac{-8}{4}=-2$
This connects to the box with path -2 (the $f(x)=\frac{1}{2}x^2+2x-1$ box)

Step3: Solve $f(x)=\frac{1}{2}x^2+2x-1$, $[-2,4]$

Calculate $f(-2)$:
$f(-2)=\frac{1}{2}(-2)^2+2(-2)-1=\frac{1}{2}(4)-4-1=2-4-1=-3$
Calculate $f(4)$:
$f(4)=\frac{1}{2}(4)^2+2(4)-1=\frac{1}{2}(16)+8-1=8+8-1=15$
Rate of change:
$\frac{f(4)-f(-2)}{4-(-2)}=\frac{15-(-3)}{4+2}=\frac{18}{6}=3$
This connects to the box with path 3 (the $f(x)=x^2+8x+12$ box)

Step4: Solve $f(x)=x^2+8x+12$, $[-7,-4]$

Calculate $f(-7)$:
$f(-7)=(-7)^2+8(-7)+12=49-56+12=5$
Calculate $f(-4)$:
$f(-4)=(-4)^2+8(-4)+12=16-32+12=-4$
Rate of change:
$\frac{f(-4)-f(-7)}{-4-(-7)}=\frac{-4-5}{-4+7}=\frac{-9}{3}=-3$
This connects to the box with path -3 (the $f(x)=-2x^2-4x-5$ box)

Step5: Solve $f(x)=-2x^2-4x-5$, $[0,3]$

Calculate $f(0)$:
$f(0)=-2(0)^2-4(0)-5=-5$
Calculate $f(3)$:
$f(3)=-2(3)^2-4(3)-5=-2(9)-12-5=-18-12-5=-35$
Rate of change:
$\frac{f(3)-f(0)}{3-0}=\frac{-35-(-5)}{3}=\frac{-30}{3}=-10$
This connects to the FINISHED! box via path -10

Answer:

The correct maze path and calculations are:

1. START box: Average rate of change = $9$
2. $f(x)=x^2+8x+18$, $[-7,-3]$: Average rate of change = $-2$
3. $f(x)=\frac{1}{2}x^2+2x-1$, $[-2,4]$: Average rate of change = $3$
4. $f(x)=x^2+8x+12$, $[-7,-4]$: Average rate of change = $-3$
5. $f(x)=-2x^2-4x-5$, $[0,3]$: Average rate of change = $-10$
6. Reaches FINISHED! box