3. find the average rate of change from x=3 to x=5 (unit 1.2)\nf(x)= -12x² - 2x + 1\nfind the average rate…

3. find the average rate of change from x=3 to x=5 (unit 1.2)\nf(x)= -12x² - 2x + 1\nfind the average rate of change from x=2 to x=4? (unit 1.2)\nf(x)= -3x³ -x + 3\n4. a boy drops his teddy bear off a 70 foot building. the height of the bear over time can be modeled by the function below, where t is time in seconds and h is the height in feet. what is the average rate of change from 1 second to 3 seconds? (unit 1.2)\nh(t)= -16t²+4t+70\na bratty kid spits off a 60 foot balcony with an initial speed of 13 feet per second. what is the average rate of change from 2 seconds to 4 seconds? where t is time in seconds and h is the height in feet. (unit 1.2)\nh(t)= -16t²+13t+60\n5. describe the transformations to the following function: (unit 1.3)\ny= -f(x+11)^2 - 7\npoint (-1, 2) lies on the graph of the function f(x)\nsuppose the function g represents a transformation of function f, g(x) = f(x) + 3 (unit 1.3)\nwrite the new ordered pair after the transformation.

3. find the average rate of change from x=3 to x=5 (unit 1.2)\nf(x)= -12x² - 2x + 1\nfind the average rate of change from x=2 to x=4? (unit 1.2)\nf(x)= -3x³ -x + 3\n4. a boy drops his teddy bear off a 70 foot building. the height of the bear over time can be modeled by the function below, where t is time in seconds and h is the height in feet. what is the average rate of change from 1 second to 3 seconds? (unit 1.2)\nh(t)= -16t²+4t+70\na bratty kid spits off a 60 foot balcony with an initial speed of 13 feet per second. what is the average rate of change from 2 seconds to 4 seconds? where t is time in seconds and h is the height in feet. (unit 1.2)\nh(t)= -16t²+13t+60\n5. describe the transformations to the following function: (unit 1.3)\ny= -f(x+11)^2 - 7\npoint (-1, 2) lies on the graph of the function f(x)\nsuppose the function g represents a transformation of function f, g(x) = f(x) + 3 (unit 1.3)\nwrite the new ordered pair after the transformation.

Answer

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function (y = f(x)) from (x=a) to (x = b) is given by (\frac{f(b)-f(a)}{b - a}).

Step2: Calculate (f(3)) and (f(5)) for (f(x)=-12x^{2}-2x + 1)

  • For (x = 3): (f(3)=-12\times(3)^{2}-2\times3 + 1=-12\times9-6 + 1=-108-6 + 1=-113)
  • For (x = 5): (f(5)=-12\times(5)^{2}-2\times5 + 1=-12\times25-10 + 1=-300-10 + 1=-309)

Step3: Apply the average - rate - of - change formula

(\frac{f(5)-f(3)}{5 - 3}=\frac{-309-(-113)}{2}=\frac{-309 + 113}{2}=\frac{-196}{2}=-98)

Step4: Calculate (f(2)) and (f(4)) for (f(x)=-3x^{3}-x + 3)

  • For (x = 2): (f(2)=-3\times(2)^{3}-2 + 3=-3\times8-2 + 3=-24-2 + 3=-23)
  • For (x = 4): (f(4)=-3\times(4)^{3}-4 + 3=-3\times64-4 + 3=-192-4 + 3=-193)

Step5: Apply the average - rate - of change formula

(\frac{f(4)-f(2)}{4 - 2}=\frac{-193-(-23)}{2}=\frac{-193 + 23}{2}=\frac{-170}{2}=-85)

Step6: Calculate (h(1)) and (h(3)) for (h(t)=-16t^{2}+4t + 70)

  • For (t = 1): (h(1)=-16\times(1)^{2}+4\times1 + 70=-16 + 4+70=58)
  • For (t = 3): (h(3)=-16\times(3)^{2}+4\times3 + 70=-16\times9+12 + 70=-144+12 + 70=-62)

Step7: Apply the average - rate - of change formula

(\frac{h(3)-h(1)}{3 - 1}=\frac{-62 - 58}{2}=\frac{-120}{2}=-60)

Step8: Calculate (h(2)) and (h(4)) for (h(t)=-16t^{2}+13t + 60)

  • For (t = 2): (h(2)=-16\times(2)^{2}+13\times2 + 60=-16\times4+26 + 60=-64+26 + 60=22)
  • For (t = 4): (h(4)=-16\times(4)^{2}+13\times4 + 60=-16\times16+52 + 60=-256+52 + 60=-144)

Step9: Apply the average - rate - of change formula

(\frac{h(4)-h(2)}{4 - 2}=\frac{-144-22}{2}=\frac{-166}{2}=-83)

Step10: Analyze the transformation (y=-f(x + 11)^{2}-7)

  • Horizontal shift: The graph of (y = f(x)) is shifted left by (11) units (because of (x+11)).
  • Reflection: The graph is reflected about the (x) - axis (because of the negative sign in front of (f)).
  • Vertical stretch/squeeze: Since it's (f(x + 11)^{2}), if (|a|>1) (here (a = 1) for the squaring of the function value, but if we consider the general form (y = af(x - h)+k), the squaring is a non - linear transformation. In terms of basic transformations:
  • Vertical shift: The graph is shifted down by (7) units (because of (-7)).

Step11: Find the new point for (g(x)=f(x)+3)

If ((x,y)) lies on (y = f(x)), then for (y = g(x)=f(x)+3), when (x=-1), (y=f(-1)+3). Since (f(-1) = 2), then (g(-1)=2 + 3=5). The new ordered pair is ((-1,5))

Answer:

  • The average rate of change of (f(x)=-12x^{2}-2x + 1) from (x = 3) to (x = 5) is (-98).
  • The average rate of change of (f(x)=-3x^{3}-x + 3) from (x = 2) to (x = 4) is (-85).
  • The average rate of change of (h(t)=-16t^{2}+4t + 70) from (t = 1) to (t = 3) is (-60).
  • The average rate of change of (h(t)=-16t^{2}+13t + 60) from (t = 2) to (t = 4) is (-83).
  • The transformation of (y=-f(x + 11)^{2}-7) is a left shift of (11) units, reflection about the (x) - axis, and a down shift of (7) units.
  • The new ordered pair for (g(x)=f(x)+3) with the point ((-1,2)) on (f(x)) is ((-1,5)).