1. **Problem Statement:** Match each equation with its corresponding graph based on the transformations of the square root function. 2. **Recall the base function:** The basic square root function is $f(x) = \sqrt{x}$, which starts at $(0,0)$ and increases gradually to the right. 3. **Understand transformations:** - Horizontal shifts: $f(x) = \sqrt{x - h}$ shifts the graph right by $h$ units. - Horizontal shifts: $f(x) = \sqrt{x + h}$ shifts the graph left by $h$ units. - Vertical shifts: $f(x) = \sqrt{x} + k$ shifts the graph up by $k$ units. - Vertical shifts: $f(x) = \sqrt{x} - k$ shifts the graph down by $k$ units. - Vertical stretch/compression: $f(x) = a\sqrt{x}$ stretches the graph vertically by factor $a$. - Vertical translation combined with stretch: $f(x) = a + \sqrt{x}$ shifts up by $a$. 4. **Analyze each equation and match to graph:** - $f(x) = \sqrt{2x}$: This is a vertical stretch by factor $\sqrt{2}$ and horizontal compression by $\frac{1}{2}$ (since inside is $2x$). Starts at $(0,0)$ and rises faster. - $f(x) = \sqrt{x} - 2$: This shifts the basic graph down by 2 units. So it starts at $(0,-2)$ and rises. - $f(x) = \sqrt{x - 2}$: This shifts the graph right by 2 units. So it starts at $(2,0)$ and rises. - $f(x) = 2 + \sqrt{x}$: This shifts the graph up by 2 units. So it starts at $(0,2)$ and rises. - $f(x) = 2\sqrt{x}$: This vertically stretches the graph by factor 2. Starts at $(0,0)$ and rises faster. - $f(x) = \sqrt{x + 2}$: This shifts the graph left by 2 units. So it starts at $(-2,0)$ and rises. 5. **Match with given graphs:** - Graph a starts near $(-1,-2)$ and rises, crossing $(0,0)$, indicating a downward shift and left shift. The closest is $f(x) = \sqrt{x} - 2$ (down 2) or $\sqrt{x + 2}$ (left 2). Since it starts near $(-1,-2)$, the vertical shift down by 2 is more prominent. So graph a matches $f(x) = \sqrt{x} - 2$. - Graph b starts above $(0,2)$ and rises, indicating a vertical shift up by 2. So graph b matches $f(x) = 2 + \sqrt{x}$. - Graph c starts just after $(2,0)$ and rises, indicating a right shift by 2. So graph c matches $f(x) = \sqrt{x - 2}$. 6. **Summary of matches:** - Graph a: $f(x) = \sqrt{x} - 2$ - Graph b: $f(x) = 2 + \sqrt{x}$ - Graph c: $f(x) = \sqrt{x - 2}$ 7. **Note:** The other functions $f(x) = \sqrt{2x}$, $f(x) = 2\sqrt{x}$, and $f(x) = \sqrt{x + 2}$ are not matched to these three graphs based on the descriptions.