Graph the functions in Exercises 37–56.


y = (x + 2)³/² + 1

Verified step by step guidance

1. Identify the basic form of the function: The given function is \( y = (x + 2)^{\frac{3}{2}} + 1 \). This is a transformation of the basic function \( y = x^{\frac{3}{2}} \).
2. Determine the transformations: The expression \( (x + 2) \) indicates a horizontal shift to the left by 2 units, and the \( +1 \) outside the power indicates a vertical shift upwards by 1 unit.
3. Consider the domain: Since the function involves \( (x + 2)^{\frac{3}{2}} \), the expression inside the power must be non-negative. Therefore, \( x + 2 \geq 0 \), which implies \( x \geq -2 \).
4. Sketch the basic function: Start by sketching the graph of \( y = x^{\frac{3}{2}} \), which is defined for \( x \geq 0 \) and resembles a square root function but grows faster.
5. Apply the transformations: Shift the graph of \( y = x^{\frac{3}{2}} \) 2 units to the left and 1 unit up to obtain the graph of \( y = (x + 2)^{\frac{3}{2}} + 1 \). Ensure the graph starts at the point \( (-2, 1) \) and follows the shape of the transformed function.

Key Concepts

Function Transformation

Function transformation involves shifting, stretching, or compressing the graph of a function. In the given function, y = (x + 2)^(3/2) + 1, the term (x + 2) indicates a horizontal shift to the left by 2 units, while the +1 indicates a vertical shift upwards by 1 unit. Understanding these transformations is crucial for accurately graphing the function.

Domain and Range

The domain of a function refers to the set of all possible input values (x-values), while the range refers to the set of all possible output values (y-values). For the function y = (x + 2)^(3/2) + 1, the domain is x ≥ -2, since the expression under the square root must be non-negative. The range starts from 1 and extends to infinity, as the minimum value occurs when x = -2.

Graphing Techniques

Graphing techniques involve plotting points, identifying key features such as intercepts, and understanding the overall shape of the graph. For the function y = (x + 2)^(3/2) + 1, it is important to calculate specific points, such as the vertex and intercepts, and to recognize that the graph will have a characteristic shape due to the cubic root and the transformations applied.