Question

transformations of functions homework
score: 9.5/13 answered: 12/13
question 11
the graph above is a transformation of the function \\(\sqrt{x}\\)
write an equation for the function graphed above
\\(g(x) =\\)

Explanation:

Identify the starting point of the graph

Using the Square Root Transformations knowledge point
The standard square root function \(f(x) = \sqrt{x}\) starts at the origin \((0,0)\).
The given graph of \(g(x)\) starts at the point \((-3, 3)\).

Determine the horizontal and vertical shifts

Using the Function Transformations knowledge point
The shift from \((0,0)\) to \((-3, 3)\) represents:

• A horizontal shift left by 3 units: \(x \to x + 3\)
• A vertical shift up by 3 units: \(+ 3\)

This gives the general form:

$$g(x) = a\sqrt{x + 3} + 3$$

Solve for the vertical stretch factor

Using the Function Transformations knowledge point
Identify another clear point on the grid that the graph passes through, which is \((1, 5)\).
Substitute \(x = 1\) and \(g(x) = 5\) into the equation:

$$
LATEXBLOCK0
$$

Write the final equation

Using the Square Root Transformations knowledge point

$$g(x) = \sqrt{x + 3} + 3$$

Answer:

The graph above is a transformation of the function \(\sqrt{x}\)

Write an equation for the function graphed above

\(g(x) =\) <blank>\(\sqrt{x + 3} + 3\)</blank>