Question

using the starting point, \\((5, -3)\\), find the graph of this function.

\\y = \sqrt{x - 5} - 3\\

Answer:

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<reasoning>

Identify the starting point of the function

The given function is:
\[
y = \sqrt{x - 5} - 3
\]
The starting point of a standard square root function \(y = a\sqrt{x - h} + k\) is \((h, k)\).
Here, \(h = 5\) and \(k = -3\), so the starting point is \((5, -3)\).

Evaluate another point on the graph

Substitute \(x = 6\) into the function to find a second point:
\[
y = \sqrt{6 - 5} - 3 = \sqrt{1} - 3 = 1 - 3 = -2
\]
The graph must pass through \((5, -3)\) and \((6, -2)\).

Match with the given options

• The first graph starts at \((5, -2)\) and goes through \((6, -1.5)\). This is incorrect.
• The second graph starts at \((-5, -3)\) and goes to the right. This is incorrect.
• The third graph starts at \((5, 3)\) and goes to the right. This is incorrect.
• Looking closely at the first graph, the starting point is actually plotted at \((5, -2)\). Let's re-examine the coordinates of the first graph: the horizontal axis has \(5\) and the vertical axis has \(-2\).
• Let's re-examine the options:
• Left graph: Starts at \((5, -2)\).
• Middle graph: Starts at \((-5, -3)\).
• Right graph: Starts at \((5, 3)\) or \((5, 4)\).
• Wait, let's look at the left graph again. The y-axis has labels \(1, 2, 3, 4\) above the x-axis, and \(-1, -2, -3, -4, -5\) below. The curve starts at \(x = 5\). The y-coordinate of this starting point is at \(-2\)? No, looking at the grid lines: the x-axis is the thick horizontal line. Below it, the grid lines are \(-1, -2, -3, -4, -5\). The starting point of the curve in the first graph is at \(x = 5\), and it lies on the grid line for \(y = -2\).
• Let's check if there is a typo in the question's prompt "Using the starting point, (5, -3)" versus the actual graph plotted. If the starting point is indeed \((5, -3)\), the first graph is the closest match despite a potential 1-unit vertical shift typo in the drawing, or the first graph's starting point is indeed \((5, -2)\) due to a common textbook mismatch. Let's re-verify the grid: the curve starts at \(x=5\). The y-value is 2 units below the x-axis, which is \(-2\).
• Thus, the first option (left graph) is the intended correct choice representing the shape and approximate position of \(y = \sqrt{x-5}-3\).

</reasoning>

<answer>
<mcq-correct>(A) The first graph (left), which starts at \(x = 5\) and curves upwards to the right in the fourth quadrant.</mcq-correct>
<mcq-option>(B) The second graph (middle), which starts in the third quadrant at \(x = -5\).</mcq-option>
<mcq-option>(C) The third graph (right), which starts in the first quadrant at \(y > 0\).</mcq-option>
</answer>

<plot>
{
"elements": [
{
"type": "point",
"params": [[5, -3]],
"properties": {
"name": "(5, -3)",
"size": 4,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "functiongraph",
"params": [{"js": "Math.sqrt(x - 5) - 3", "latex": "\sqrt{x - 5} - 3"}, 5, 10],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 3
}
}
],
"timestamps": [0.5, 1.0]
}
</plot>

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