Question

graph the following function by moving the green and blue dots (if necessary).
$y = 3\sqrt3{x}$

Explanation:

Step1: Recall the parent function

The parent function is \( y = \sqrt[3]{x} \), which passes through \((0,0)\), \((1,1)\), and \((-1,-1)\), \((8,2)\), \((-8,-2)\) etc.

Step2: Analyze the transformation

The given function is \( y = 3\sqrt[3]{x} \), which is a vertical stretch of the parent function by a factor of 3. So we multiply the \( y \)-values of the parent function's points by 3.

• For \( x = 0 \), \( y = 3\sqrt[3]{0}=0 \), so the point is \((0,0)\) (the green dot, which is already at the origin).
• For \( x = 1 \), \( y = 3\sqrt[3]{1}=3\times1 = 3 \), so the point on the right side (positive \( x \)) should be \((1,3)\) (the blue dot near \( x = 1 \) should be moved up to \( y = 3 \)).
• For \( x = -1 \), \( y = 3\sqrt[3]{-1}=3\times(-1)= - 3 \), so the point on the left side (negative \( x \)) should be \((-1,-3)\) (the blue dot near \( x=-1\) should be moved down to \( y = - 3 \)).
• For \( x = 8 \), \( y = 3\sqrt[3]{8}=3\times2 = 6 \), so the right - most blue dot (near \( x = 8 \)) should be at \((8,6)\).
• For \( x=-8\), \( y = 3\sqrt[3]{-8}=3\times(-2)=-6\), so the left - most blue dot (near \( x = - 8 \)) should be at \((-8,-6)\).

To graph \( y = 3\sqrt[3]{x} \):

1. Keep the green dot at \((0,0)\) since when \( x = 0 \), \( y = 0 \).
2. For the blue dot with \( x = 1 \), move it to \( y = 3 \) (so the point is \((1,3)\)).
3. For the blue dot with \( x=-1\), move it to \( y=-3\) (so the point is \((-1,-3)\)).
4. For the blue dot with \( x = 8 \), move it to \( y = 6 \) (so the point is \((8,6)\)).
5. For the blue dot with \( x=-8\), move it to \( y=-6\) (so the point is \((-8,-6)\)).

Answer:

To graph \( y = 3\sqrt[3]{x} \), adjust the blue dots as follows:

• The blue dot at \( x = 1 \) should be at \((1,3)\).
• The blue dot at \( x=-1\) should be at \((-1,-3)\).
• The blue dot at \( x = 8 \) should be at \((8,6)\).
• The blue dot at \( x=-8\) should be at \((-8,-6)\).

The green dot remains at \((0,0)\).