Question

1. (10 points) write the equation of the function where (g(x) = sqrt{x}) has been stretched vertically by a factor of two, shifted to the left by 3 units and shifted down by 5 units.

Explanation:

Step1: Apply vertical stretch

Multiply the parent function by the vertical stretch factor of \(2\).

$$g_1(x) = 2\sqrt{x}$$

Step2: Apply horizontal shift

Replace \(x\) with \(x + 3\) to shift left by \(3\) units.

$$g_2(x) = 2\sqrt{x + 3}$$

Step3: Apply vertical shift

Subtract \(5\) from the function to shift down by \(5\) units.

$$f(x) = 2\sqrt{x + 3} - 5$$

Answer:

\(f(x) = 2\sqrt{x + 3} - 5\)