Question

graph the equation shown below by transforming the given graph of the parent function.
$y = \sqrt3{x + 2} + 4$

Explanation:

Step1: Identify Parent Function

The parent function for a cube root function is \( y = \sqrt[3]{x} \). The graph of \( y=\sqrt[3]{x} \) passes through the origin \((0,0)\), and has the shape shown (the given graph).

Step2: Analyze Horizontal Shift

For the function \( y=\sqrt[3]{x + 2}+4 \), the term \( x + 2 \) inside the cube root indicates a horizontal shift. The rule for horizontal shifts is: if we have \( y=\sqrt[3]{x - h} \), it shifts \( h \) units horizontally. Here, \( x+2=x - (-2) \), so we shift the parent graph 2 units to the left.

Step3: Analyze Vertical Shift

The \( +4 \) outside the cube root indicates a vertical shift. The rule for vertical shifts is: if we have \( y=\sqrt[3]{x}+k \), it shifts \( k \) units vertically. Here, \( k = 4 \), so we shift the graph (after the horizontal shift) 4 units up.

Step4: Apply Shifts to Key Points

• Parent function \( y = \sqrt[3]{x} \) has key points like \((0,0)\), \((1,1)\), \((-1,-1)\), \((8,2)\), \((-8,-2)\).
• After horizontal shift (2 units left): For \((0,0)\) becomes \((0 - 2,0)=(-2,0)\); \((1,1)\) becomes \((1 - 2,1)=(-1,1)\); \((-1,-1)\) becomes \((-1 - 2,-1)=(-3,-1)\); \((8,2)\) becomes \((8 - 2,2)=(6,2)\); \((-8,-2)\) becomes \((-8 - 2,-2)=(-10,-2)\).
• After vertical shift (4 units up): \((-2,0)\) becomes \((-2,0 + 4)=(-2,4)\); \((-1,1)\) becomes \((-1,1 + 4)=(-1,5)\); \((-3,-1)\) becomes \((-3,-1 + 4)=(-3,3)\); \((6,2)\) becomes \((6,2 + 4)=(6,6)\); \((-10,-2)\) becomes \((-10,-2 + 4)=(-10,2)\).

Plot these new points \((-2,4)\), \((-1,5)\), \((-3,3)\), \((6,6)\), \((-10,2)\) and draw the curve through them, which is the graph of \( y=\sqrt[3]{x + 2}+4 \).

Answer:

To graph \( y=\sqrt[3]{x + 2}+4 \), shift the parent cube - root function \( y = \sqrt[3]{x} \) 2 units to the left and 4 units up. Key points of the transformed function can be found by applying the horizontal (2 units left) and vertical (4 units up) shifts to key points of the parent function, and then drawing the curve through these transformed points.