Question

graph the function.
$f(x) = sqrt{x} - 5$
choose the correct graph
a.
graph a

b.
graph b

c.
graph c

d.
graph d

Explanation:

Step1: Analyze the parent function

The parent function is \( y = \sqrt{x} \), which has a domain \( x \geq 0 \) and starts at the origin \((0,0)\), increasing slowly.

Step2: Analyze the transformation

The given function is \( g(x)=\sqrt{x}-2 \). This is a vertical shift of the parent function \( y = \sqrt{x} \) down by 2 units. So the graph of \( g(x) \) should have the same shape as \( y = \sqrt{x} \) but every point \((x,y)\) on \( y = \sqrt{x} \) will be \((x,y - 2)\) on \( g(x) \).

Step3: Check the y - intercept

For the parent function \( y=\sqrt{x} \), when \( x = 0 \), \( y=0 \). For \( g(x)=\sqrt{x}-2 \), when \( x = 0 \), \( g(0)=\sqrt{0}-2=- 2 \). So the y - intercept is \((0,-2)\).

Step4: Analyze the options

• Option A: If the graph does not have a y - intercept at \((0, - 2)\) or the shape is incorrect, it can be eliminated.
• Option B: Check the y - intercept and the shape. If the graph is shifted down by 2 units from \( y=\sqrt{x} \), with domain \( x\geq0 \) and passing through \((0, - 2)\) and increasing, this is a candidate.
• Option C: If the graph has a different domain (e.g., negative x - values) or incorrect y - intercept, it can be eliminated since the square root function is only defined for \( x\geq0 \).
• Option D: Similar to option C, if the domain is incorrect or the y - intercept is wrong, it can be eliminated.

Looking at the graphs, the correct graph should have the same shape as \( y = \sqrt{x} \) (defined for \( x\geq0 \), increasing) but shifted down by 2 units, so the y - intercept is \((0,-2)\).

Answer:

The correct graph is the one that represents a square - root - shaped graph (increasing, defined for \( x\geq0 \)) with a y - intercept at \((0, - 2)\). (Assuming from the options, the one that matches the vertical shift down by 2 of \( y=\sqrt{x} \), typically the graph with the correct y - intercept \((0,-2)\) and the correct domain and shape. If we assume the options are labeled and the correct one is, for example, the one where the graph starts at \((0, - 2)\) and has the square - root shape, say Option B (depending on the actual visual, but based on the transformation, the graph with \( x\geq0 \), y - intercept \((0,-2)\) and increasing is correct))