Question

which describes the effect of the transformations on the graph of $f(x)=x^2$ when changed to $f(x)=\frac{1}{2}(x - 5)^2+7$?
a stretched vertically, shifted left 5 units, and shifted down 7 units
b stretched vertically, shifted right 5 units, and shifted up 7 units
c compressed vertically, shifted left 5 units, and shifted down 7 units
d compressed vertically, shifted right 5 units, and shifted up 7 units

Explanation:

Step1: Analyze vertical scaling factor

For $f(x)=a(x-h)^2+k$, $|a|<1$ means vertical compression. Here $a=\frac{1}{2}$, so vertical compression.

Step2: Analyze horizontal shift

$h=5$: $x\to x-5$ shifts graph right 5 units.

Step3: Analyze vertical shift

$k=7$: adding 7 shifts graph up 7 units.

Answer:

D. compressed vertically, shifted right 5 units, and shifted up 7 units