Attached is a very useful chart for translations. [
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A number added or subtracted outside of parentheses shifts the graph vertically up (if number is added) or down (if number is subtracted).
As you can see in the chart,
f(x) + k shifts the graph up k units, and
f(x) - k shifts the graph down k units.
A number that is multiplied to x outside parentheses stretches or shrinks the graph vertically. If that number is greater than 1, then it stretches the graph. If the number is a fraction or between 0 and 1, then it shrinks the graph vertically.
As you can see in the chart,
a * f(x), where a>1 stretches the graph f(x) vertically by a factor a.
a * f(x), where a is between 0 and 1 shrinks the graph f(x) vertically by a factor of a.
When h is added or subtracated from x inside the parentheses, the graph is shifted horizontally right or left. Remember that this is opposite of what the sign is:
If h is positive, f(x + h), then the graph shifts left h units.
If h is negative, f(x - h), the graph shifts right h units.
Note that the attached chart uses c instead of h.
Since the function will be quadratic (degree is 2), the domain will be all real numbers, from -infinity to +infinity (-∞, +
∞).
Depending on whether the value of a is positive or negative, the graph will open up like a U (if a is positive) or open down like an upside down U (if a is negative).
Since the equation is in vertex form f(x) = a(x - h)^2 = k, the vertex is (h, k). The range (set of y-values) will be from y-coordinate of the vertex (k) to positive or negative infinity depending on whether the graph opens up or down.