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Graphing functions is a crucial skill in mathematics, as it allows us to visually represent and understand the behavior of equations. When graphing a function like the given \( f(x) = \sqrt{2x - x^2} \), we start by identifying the important features such as the **domain**, **range**, and any **symmetries**. These characteristics influence how the graph looks and behaves.

One practical way to graph functions is by using a viewing rectangle, where we specify the limits for the x-axis and y-axis. For instance, in this exercise, the viewing rectangle is \([-5,5]\) by \([-4,4]\). These boundaries help in focusing on the part of the graph that we are interested in analyzing. Plotting different versions of functions, such as \( f(2x) \) or \( f\left(\frac{1}{2}x\right) \), shows how transformations affect the graph's shape and size.

• **Horizontal Compression/Expansion**: Multiplying x by a factor affects the graph's width. If the factor is greater than 1, the function compresses. If less than 1, it expands.
• **Understanding Transformations**: Helps predict function behavior without plotting every time.

By learning to graph and recognize these patterns, we gain insights into the nature of the function and how different transformations impact it.

The domain and range of a function give us crucial information about where the function is defined and what values it can take. For the function \( f(x) = \sqrt{2x - x^2} \), the domain is determined by the requirement that the expression inside the square root must be non-negative. This ensures that all calculations are valid since the square root of a negative number is not real.

To find the domain, solve the inequality \( 2x - x^2 \geq 0 \). Factor this as \( x(2-x) \geq 0 \), which leads us to the interval \( 0 \leq x \leq 2 \). This tells us that the only acceptable x-values are between 0 and 2, inclusive.

The range of the function comes from the values that \( \sqrt{2x - x^2} \) can output. As this is essentially the arc of a semi-circle, it spans from \( y = 0 \) to its peak at \( y = 1 \). Therefore, the range of \( f(x) \) is \( 0 \leq y \leq 1 \). Understanding the domain and range not only helps in graphing but also in solving real-world problems by defining the scope of a function’s applicability.

Quadratic functions are a fundamental type of polynomial that takes the standard form \( ax^2 + bx + c \). The given function, \( f(x) = \sqrt{2x - x^2} \), originally derives from a quadratic expression, \( 2x - x^2 \). Recognizing it in this form allows us to explore its properties and transformations more deeply.

In its quadratic form, \( 2x - x^2 \) can be rewritten to emphasize the parabola's shape: \( -(x^2 - 2x) \). The negative sign indicates that it opens downwards. Completing the square helps to identify the vertex and maximum value, crucial for graphing and transformations:

• Vertex Formula: The vertex occurs at \( x = \frac{-b}{2a} \). Here, \( x = 1 \) gives the maximum point.
• Axis of Symmetry: Provides symmetry line important for graph perfection.

Each transformation such as \( f(2x) \) or \( f\left(\frac{1}{2}x\right) \) can be understood by analyzing its effect on this foundational quadratic. Being comfortable with quadratic functions paves the way for tackling more complex polynomials and mathematical models.