how is the graph of $y=(x - 1)^2-3$ transformed to produce the graph of $y=\frac{1}{2}(x + 4)^2$?\nthe graph…

how is the graph of $y=(x - 1)^2-3$ transformed to produce the graph of $y=\frac{1}{2}(x + 4)^2$?\nthe graph is translated left 5 units, compressed vertically by a factor of $\frac{1}{2}$, and translated up 3 units.\nthe graph is stretched vertically by a factor of $\frac{1}{2}$, translated left 5 units, and translated up 3 units.\nthe graph is translated left 5 units, compressed horizontally by a factor of $\frac{1}{2}$, and translated down 3 units.\nthe graph is stretched horizontally by a factor of $\frac{1}{2}$, translated left 5 units, and translated down 3 units.

how is the graph of $y=(x - 1)^2-3$ transformed to produce the graph of $y=\frac{1}{2}(x + 4)^2$?\nthe graph is translated left 5 units, compressed vertically by a factor of $\frac{1}{2}$, and translated up 3 units.\nthe graph is stretched vertically by a factor of $\frac{1}{2}$, translated left 5 units, and translated up 3 units.\nthe graph is translated left 5 units, compressed horizontally by a factor of $\frac{1}{2}$, and translated down 3 units.\nthe graph is stretched horizontally by a factor of $\frac{1}{2}$, translated left 5 units, and translated down 3 units.

Answer

Explanation:

Step1: Analyze horizontal translation

For the quadratic - function form (y = a(x - h)^2+k), the original function is (y=(x - 1)^2-3) with (h_1 = 1) and the new function is (y=\frac{1}{2}(x + 4)^2) with (h_2=-4). The change in (x) is (\Delta x=h_2 - h_1=-4 - 1=-5), which means a left - shift of 5 units.

Step2: Analyze vertical transformation

The coefficient (a) of the original function is (a_1 = 1) and for the new function (a_2=\frac{1}{2}). Since (0\lt a_2\lt a_1), the graph is compressed vertically by a factor of (\frac{1}{2}).

Step3: Analyze vertical translation

The original function has (k_1=-3) and the new function has (k_2 = 0). The change in (y) is (\Delta y=k_2 - k_1=0-(-3)=3), which means a translation up 3 units.

Answer:

The graph is translated left 5 units, compressed vertically by a factor of (\frac{1}{2}), and translated up 3 units.