suppose the graph of ( f ) is given. describe how the graph of each function can be obtained from the graph…

suppose the graph of ( f ) is given. describe how the graph of each function can be obtained from the graph of ( f ).\n(a) ( y=-f(x)+4 )\nreflect about the ( y )-axis, then shift down 4 units\nreflect about the ( y )-axis, then shift left 4 units\nreflect about the ( x )-axis, then shift down 4 units\nreflect about the ( x )-axis, then shift up 4 units\nreflect about the ( y )-axis, then shift up 4 units\n(b) ( y = 5 f(x)-4 )\nstretch vertically by a factor of 5, then shift down 4 units\nstretch vertically by a factor of 5, then shift up 4 units\nshift down 4 units, then stretch vertically by a factor of 5\nshrink vertically by a factor of 5, then shift down 4 units\nstretch vertically by a factor of 4, then shift down 5 units

suppose the graph of ( f ) is given. describe how the graph of each function can be obtained from the graph of ( f ).\n(a) ( y=-f(x)+4 )\nreflect about the ( y )-axis, then shift down 4 units\nreflect about the ( y )-axis, then shift left 4 units\nreflect about the ( x )-axis, then shift down 4 units\nreflect about the ( x )-axis, then shift up 4 units\nreflect about the ( y )-axis, then shift up 4 units\n(b) ( y = 5 f(x)-4 )\nstretch vertically by a factor of 5, then shift down 4 units\nstretch vertically by a factor of 5, then shift up 4 units\nshift down 4 units, then stretch vertically by a factor of 5\nshrink vertically by a factor of 5, then shift down 4 units\nstretch vertically by a factor of 4, then shift down 5 units

Answer

Brief Explanations:

Part (a)

  • For the transformation (y = -f(x)+4):
    • The negative sign in front of (f(x)) (i.e., (y=-f(x))) reflects the graph of (y = f(x)) about the (x) - axis. The rule for reflecting (y = f(x)) about the (x) - axis is ((x,y)\to(x, - y)).
    • The (+4) (i.e., (y=-f(x)+4)) shifts the graph of (y=-f(x)) vertically. The rule for vertical shift is: if (y = g(x)+k), when (k>0), the graph of (y = g(x)) is shifted up by (k) units. Here (g(x)=-f(x)) and (k = 4), so it shifts up 4 units.

Part (b)

  • For the transformation (y=5f(x)-4):
    • The coefficient (5) in front of (f(x)) (i.e., (y = 5f(x))) causes a vertical stretch. The rule for vertical stretch is: if (y=af(x)) with (|a|>1), the graph of (y = f(x)) is stretched vertically by a factor of (|a|). Here (a = 5), so it is stretched vertically by a factor of 5.
    • The (-4) (i.e., (y=5f(x)-4)) shifts the graph of (y = 5f(x)) vertically. Using the rule (y = g(x)+k) (where (g(x)=5f(x)) and (k=-4)), the graph is shifted down 4 units since (k<0).

Answer:

(a) reflect about the (x) - axis, then shift up 4 units
(b) stretch vertically by a factor of 5, then shift down 4 units