Question

find the function that is finally graphed after the following transformations are applied to the graph of \\(y = \sqrt{x}\\) in the order listed.

(1) reflect about the x-axis
(2) shift up 7 units
(3) shift left 9 units

Explanation:

Identify the base function

Using the Function Transformation knowledge point

$$ f(x) = \sqrt{x} $$

Apply reflection about the x-axis

Using the Function Transformation knowledge point

$$ g(x) = -f(x) = -\sqrt{x} $$

Apply vertical shift up

Using the Function Transformation knowledge point

$$ h(x) = g(x) + 7 = -\sqrt{x} + 7 $$

Apply horizontal shift left

Using the Function Transformation knowledge point

$$ y = h(x + 9) = -\sqrt{x + 9} + 7 $$

Answer:

\(y = -\sqrt{x + 9} + 7\)