Question

Describe the role of 'a', 'k', 'd', and 'c' in transforming functions under different conditions:

(A) a > 0 and 0 < a < 1
(B) a is negative
(C) k > 0 and 0 < k < 1
(D) k is negative
(E) d is positive (1 - d) or negative (1 - (-d))
(F) c is positive or negative

Answer 1

(A) For
`\(a > 0\)` and
`\(0 < a < 1\)`, the parameter 'a' in a function transforms it by vertically compressing it without changing its direction. (B) When 'a' is negative, the function is reflected across the x-axis. (C) For
`\(k > 0\)` and
`\(0 < k < 1\)`, the parameter 'k' vertically compresses the function. (D) When 'k' is negative, the function is reflected across the x-axis and vertically compressed. (E) For (d) positive, a horizontal shift occurs to the right, and for (1 - d) negative, a horizontal shift occurs to the left. (F) The parameter 'c' affects the vertical position of the function, shifting it up or down based on whether 'c' is positive or negative.

Step-by-step explanation:

(A) When
`\(a > 0\)` and
`\(0 < a < 1\)`, the parameter 'a' in a function of the form
`\(f(x) = a \cdot g(x)\)` causes a vertical compression by a factor of \(a\) without changing the direction of the function. For example, if
`\(a = (1)/(2)\)`, the function's amplitude is halved, leading to a compression.

(B) If 'a' is negative, the function
`\(f(x) = a \cdot g(x)\)` is reflected across the x-axis. This reflection changes the sign of the function values, effectively flipping it upside down.

(C) For
`\(k > 0\)` and
`\(0 < k < 1\)`, the parameter 'k' in a function of the form
`\(f(x) = g(kx)\)` results in a vertical compression by a factor of (k). The graph is squeezed vertically, making it narrower.

(D) When 'k' is negative, the function is reflected across the x-axis and vertically compressed by the absolute value of 'k'. This leads to an upside-down reflection and a compression or stretching depending on the absolute value of 'k'.

(E) For (d) positive, a positive value of 'd' in
`\(f(x) = g(x - d)\)` causes a horizontal shift to the right. For (1 - d) negative, a negative value of 'd' in
`\(f(x) = g(x - d)\)` results in a horizontal shift to the left.

(F) The parameter 'c' in
`\(f(x) = g(x) + c\)` shifts the entire function vertically by 'c' units. If 'c' is positive, the graph moves up; if 'c' is negative, the graph moves down. This parameter controls the vertical position of the function.