Answer to Question #236711 in Algebra for Lara

Question #236711
Let g(x) be the transformation of f(x)= x2. Write the rule for g(x) using the change described.

1. horizontal compression by a factor of 1/5 followed by a vertical shift down 7 units and reflection across y-axis. ______________

2. reflection across the x-axis followed by a vertical stretch by a factor of 3, a horizontal shift 6 units left, and a vertical shift 4 units down. _____________
1
Expert's answer
2021-09-16T05:15:02-0400

1)

Parent Function: "f(x) = x\\\\^2"

There are:

  • A horizontal compression by a factor of "\\frac{1}{5}"
  • A vertical shift down 7 units
  • A reflection across y-axis, means that b is negative

The general form for the rule is: "f(x) = a \\times f([b(x-h]) + k"

Therefore, "b = -5" because of horizontal compression by a factor of "\\frac{1}{5}" and reflection across y-axis

k= - 7

Replacing b and k in the general form of the rule "f(x) = a f[ b(x \u2212 h)] + k" where "f(x) = x\\\\^2" , we find, "f(x) = (-5(x))\\\\^2 - 7"

"= f(x) = (-5(x))\\\\^2 - 7"


2)

Parent Function: "f(x) = x\\\\^2"

The general form for the rule is: "f(x) = a \\times f([b(x-h]) + k"

There are:

Reflection across the x-axis, indicating that "a" is negative

Vertical stretch by a factor of 3, means that "a= -3"

A horizontal shift 6 units left, means that "h=-6"

 A vertical shift 4 units down, means that "k= -4"

Replacing a, k, and h in the general form of the rule "f(x) = a f[ b(x \u2212 h)] + k" where "f(x) = x\\\\^2" , we find "f(x) = -3((x + 6)\\\\^2) - 4"

"= -3((x + 6)\\\\^2) - 4"





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