Question

1. consider the graph of the function $f(x) = -sqrt{x - 4} - 6$.

choose the correct answer from each drop - down menu to complete the statement.
all of the points that belong to the graph of $f$ are in quadrant iv. the - choose the correct answer - value of graph of $f$ on its range is - choose the correct answer - .

(there is a clear all button and a drop - down menu with options maximum and minimum shown.)

Explanation:

Step1: Analyze the square root function

The parent function is \( y = \sqrt{x} \), which has a domain \( x \geq 0 \) and range \( y \geq 0 \). For \( y=\sqrt{x - 4}\), the domain becomes \( x-4\geq0\) i.e., \( x\geq4 \), and the range remains \( y\geq0 \).

Step2: Analyze the transformation \( y = -\sqrt{x - 4}\)

Multiplying by -1 reflects the graph over the x - axis. So the range of \( y = -\sqrt{x - 4}\) becomes \( y\leq0 \) (since we reflect \( y\geq0 \) over the x - axis).

Step3: Analyze the transformation \( y=-\sqrt{x - 4}-6\)

Subtracting 6 from \( y = -\sqrt{x - 4}\) shifts the graph down by 6 units. So the range of \( y = -\sqrt{x - 4}-6\) is \( y\leq - 6\).
The function has a maximum value (since the range is all real numbers less than or equal to - 6, the largest value the function can take is - 6). There is no minimum value as \( y\) can go to \( -\infty\) (but wait, actually, let's re - check. Wait, the square root function \( \sqrt{x-4}\geq0\), so \( -\sqrt{x - 4}\leq0\), then \( -\sqrt{x - 4}-6\leq - 6\). So the maximum value of the function is - 6, and as \( x\) increases, \( \sqrt{x - 4}\) increases, so \( -\sqrt{x - 4}\) decreases, and \( -\sqrt{x - 4}-6\) decreases. So the function has a maximum value (the highest point on the graph) and no minimum (since it can go to negative infinity as \( x\) gets larger? Wait, no. Wait, when \( x = 4\), \( f(4)=-\sqrt{4 - 4}-6=-6\). As \( x\) increases, \( \sqrt{x - 4}\) increases, so \( -\sqrt{x - 4}\) decreases, so \( f(x)=-\sqrt{x - 4}-6\) decreases. So the function starts at \( (4,-6) \) and decreases as \( x\) increases. So the maximum value of the function is - 6 (at \( x = 4\)) and it goes to \( -\infty\) as \( x
ightarrow\infty\). So the function has a maximum value.

Answer:

maximum