Begin by graphing the cube root function, f(x)=√(x) . Then use transformations of this graph to

step 1 Okay, we've got R of X equals 1 half the cubed root of x minus 2 plus 2. So this tells us we're

Begin by graphing the cube root function, f(x)=√(x) . Then...

Key Concepts

Cube Root Function

The cube root function, represented by f(x) = ?x, is a basic odd function that passes through the origin and exhibits symmetry about the origin. It is defined for all real numbers and has the property of increasing continuously, meaning as x increases, the cube root of x also increases. This function sets the foundation for understanding more complex transformations.

Function Transformations

Function transformations involve changing the basic graph of a function using operations such as translations, scalings, reflections, and rotations. The effect of each transformation can be analyzed separately, and when combined, they yield a new graph that retains the overall shape of the original function, albeit shifted, stretched, compressed, or flipped.

Horizontal Translation

A horizontal translation occurs when the input (x) of a function is altered by an addition or subtraction. In the expression ?(x + 2), the graph of the cube root function is shifted horizontally. Specifically, replacing x with (x+2) shifts the graph 2 units to the left.

Vertical Scaling

Vertical scaling refers to multiplying a function by a constant factor, affecting the steepness or compression of the graph. In the given function, multiplying by 1/2 compresses the cube root graph vertically, making the graph less steep. This means that for a given change in x, the change in the function's output is reduced by half.

Vertical Translation

Vertical translation involves adding or subtracting a constant value to the output of a function, shifting the graph upward or downward. In the function r(x) = 1/2?(x+2) - 2, subtracting 2 shifts the graph 2 units downward.

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