Question

graph the equation shown below by transforming the given graph of the parent function.
$y = 4\sqrt{x}$

Explanation:

Step1: Identify the parent function

The parent function for \( y = 4\sqrt{x} \) is \( y=\sqrt{x} \). The graph of \( y = \sqrt{x} \) has key points like \((0,0)\), \((1,1)\), \((4,2)\), \((9,3)\) (as seen in the given graph: the green dot at \((0,0)\), blue dot at \((1,1)\), another at \((4,2)\), and \((9,3)\)).

Step2: Analyze the transformation

The function \( y = 4\sqrt{x} \) is a vertical stretch of the parent function \( y=\sqrt{x} \) by a factor of 4. For a vertical stretch by factor \( a \) (where \( a>1 \)), the transformation rule is \( (x, y) \to (x, a \cdot y) \) for the parent function's points.

Step3: Apply the transformation to key points

• For \( (0,0) \): \( y = 4 \cdot 0 = 0 \), so the point remains \((0,0)\).
• For \( (1,1) \): \( y = 4 \cdot 1 = 4 \), so the new point is \((1,4)\).
• For \( (4,2) \): \( y = 4 \cdot 2 = 8 \), so the new point is \((4,8)\).
• For \( (9,3) \): \( y = 4 \cdot 3 = 12 \), but since the y - axis only goes up to 10 in the given grid, we can still use the transformation concept.

To graph \( y = 4\sqrt{x} \), we take the key points of the parent function \( y=\sqrt{x} \) and multiply their y - coordinates by 4. Then we plot these new points \((0,0)\), \((1,4)\), \((4,8)\), etc., and draw the curve through them, which will be a vertically stretched version of the parent square - root function's graph.

Answer:

To graph \( y = 4\sqrt{x} \), vertically stretch the graph of \( y=\sqrt{x} \) (given) by a factor of 4. Key points transformation: \((0,0)\) stays, \((1,1)\to(1,4)\), \((4,2)\to(4,8)\), \((9,3)\to(9,12)\) (plot these and draw the curve).