Question

graph the function below;
y = \sqrt3{x + 3} + 3
show your work here

Explanation:

Step1: Identify Parent Function

The parent function is \( y = \sqrt[3]{x} \), which has a point at \((0,0)\), \((1,1)\), \((-1,-1)\), \((8,2)\), \((-8,-2)\).

Step2: Analyze Transformations

The given function is \( y = \sqrt[3]{x + 3} + 3 \).

• Horizontal shift: \( x + 3 \) means shift the parent graph 3 units to the left (replace \( x \) with \( x + 3 \)).
• Vertical shift: \( + 3 \) means shift the graph 3 units up.

Step3: Transform Key Points

• For \((0,0)\) in parent:

Horizontal shift: \( 0 - 3 = -3 \) (x-coordinate), Vertical shift: \( 0 + 3 = 3 \) (y-coordinate) → \((-3, 3)\).

• For \((1,1)\) in parent:

Horizontal shift: \( 1 - 3 = -2 \), Vertical shift: \( 1 + 3 = 4 \) → \((-2, 4)\).

• For \((-1,-1)\) in parent:

Horizontal shift: \( -1 - 3 = -4 \), Vertical shift: \( -1 + 3 = 2 \) → \((-4, 2)\).

• For \((8,2)\) in parent:

Horizontal shift: \( 8 - 3 = 5 \), Vertical shift: \( 2 + 3 = 5 \) → \((5, 5)\).

• For \((-8,-2)\) in parent:

Horizontal shift: \( -8 - 3 = -11 \), Vertical shift: \( -2 + 3 = 1 \) → \((-11, 1)\).

Step4: Plot Transformed Points

Plot the points \((-3, 3)\), \((-2, 4)\), \((-4, 2)\), \((5, 5)\), \((-11, 1)\) and draw a smooth curve through them, matching the shape of the cube root function (S-shaped, passing through these points with the horizontal/vertical shifts applied).

Answer:

The graph is the cube root function \( y = \sqrt[3]{x} \) shifted 3 units left and 3 units up, plotted using transformed key points like \((-3, 3)\), \((-2, 4)\), \((-4, 2)\), etc., with a smooth S - shaped curve.