Question

the function $f(x) = \sqrt{x}$ is graphed below in black. choose the graph that shows the function $g(x) = \sqrt{x + 3} - 7$ in blue.

Explanation:

Step1: Analyze Horizontal Shift

The function \( f(x) = \sqrt{x} \) has a horizontal shift in \( g(x) = \sqrt{x + 3} - 7 \). The \( x + 3 \) inside the square root means a shift left by 3 units. So the vertex of \( f(x) \) (at \( (0,0) \)) will move to \( (-3, 0) \) for the \( \sqrt{x + 3} \) part.

Step2: Analyze Vertical Shift

Then, the \( -7 \) outside the square root is a vertical shift down by 7 units. So from \( (-3, 0) \), we move down 7 units to \( (-3, -7) \). This is the vertex of \( g(x) \). The shape of the square root function remains the same, just shifted left 3 and down 7. So the graph of \( g(x) \) should have its starting point (vertex) at \( (-3, -7) \) and follow the same curve as \( f(x) \) but shifted accordingly.

Answer:

To graph \( g(x)=\sqrt{x + 3}-7 \), shift the graph of \( f(x)=\sqrt{x} \) left 3 units (so the vertex moves from \( (0,0) \) to \( (-3,0) \)) and then down 7 units (so the vertex moves to \( (-3, -7) \)). The curve will have the same shape as \( \sqrt{x} \) but with the vertex at \( (-3, -7) \), increasing slowly from there as \( x \) increases.