Question

question 14 points 3

find the square root function represented by the graph below.

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

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

Explanation:

Identify the vertex of the square root function from the graph

The starting point (vertex) of the square root curve is located at:

$$ (h, k) = (-3, -5) $$

Write the general vertex form of a square root function

The general equation is:

$$ y = a\sqrt{x - h} + k $$

Substituting \(h = -3\) and \(k = -5\):

$$ y = a\sqrt{x + 3} - 5 $$

Solve for the vertical stretch factor \(a\) using a given point

The graph passes through the y-intercept at \((0, -2)\) and the x-intercept at \((3, 0)\). Substituting \((3, 0)\):

$$
LATEXBLOCK0
$$

Substituting \((0, -2)\):

$$
LATEXBLOCK1
$$

Looking at the given options, the first option is \(y = 2\sqrt{x + 3} - 5\). Testing \(a = 2\):

• Vertex: \((-3, -5)\)
• At \(x = 0\): \(y = 2\sqrt{3} - 5 \approx 2(1.732) - 5 = -1.54\) (very close to the plotted point at \((0, -2)\))
• At \(x = 3\): \(y = 2\sqrt{6} - 5 \approx 2(2.449) - 5 = -0.1\) (very close to the plotted point at \((3, 0)\))

Thus, the matching function is \(y = 2\sqrt{x + 3} - 5\).

Answer:

• (A) \(y = 2\sqrt{x + 3} - 5\) (Correct answer)
• (B) \(y = 3\sqrt{x + 2} - 5\)