Question

which of the following is the graph of this square root function? f(x) = \sqrt{x + 2} - 4

Explanation:

Step1: Find the domain of the function

For the square root function \( f(x)=\sqrt{x + 2}-4 \), the expression inside the square root must be non - negative. So we set \( x + 2\geq0 \), which gives \( x\geq - 2 \). This means the graph of the function starts at \( x=-2 \).

Step2: Find the y - intercept (when \( x = 0 \))

Substitute \( x = 0 \) into the function: \( f(0)=\sqrt{0 + 2}-4=\sqrt{2}-4\approx1.414 - 4=-2.586 \).

Step3: Find the starting point (when \( x=-2 \))

Substitute \( x=-2 \) into the function: \( f(-2)=\sqrt{-2 + 2}-4=0 - 4=-4 \). So the starting point of the graph is \( (-2,-4) \).

Now let's analyze the three graphs:

• The first graph: Check the starting point. If we look at the x - coordinate of the starting point, it should be \( x = - 2 \). Let's check the y - coordinate. When \( x=-2 \), \( y=-4 \). The first graph seems to have a starting point with \( x\) value greater than \( - 2 \) (from the grid), so it's not correct.
• The second graph: The y - coordinate of the starting point is around \( 4 \), but we know that when \( x=-2 \), \( y=-4 \), so this graph is not correct.
• The third graph: The starting point should be at \( x=-2,y = - 4 \). Let's check the domain (starts at \( x=-2 \)) and the general shape. When \( x = 0 \), \( y=\sqrt{2}-4\approx - 2.586\), which matches the trend of the third graph.

Answer:

The third graph (the one with x - axis from - 5 to 5 and y - axis from - 6 to 3, with the curve starting at \( x=-2,y=-4 \) and increasing)