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The domain of a function refers to all the possible input values (or the "x" values) that the function can accept. For the function \( f(x) = \sqrt{2x - x^2} \), we need to ensure that the expression inside the square root, \( 2x - x^2 \), is non-negative. This is because square roots of negative numbers are not defined in the realm of real numbers.

To find the domain, we solve the inequality \( 2x - x^2 \geq 0 \). By factoring, we get \( x(2 - x) \geq 0 \). This tells us that both \( x \) and \( 2 - x \) must be non-negative (greater than or equal to zero).

• If \( x \geq 0 \), then \( 2 - x \geq 0 \) implies \( x \leq 2 \).
• Combining these, we find \( 0 \leq x \leq 2 \).

This interval \([0, 2]\) is the domain of the function. It represents all the x-values for which \( f(x) \) is defined.

Reflection in graph transformations means flipping a graph over a specified axis. Two common types of reflections in graphing functions are across the x-axis and the y-axis.

When we talk about \( y = f(-x) \), this represents a reflection over the y-axis. We essentially negate the x-values of the original function. For the function \( f(x) = \sqrt{2x - x^2} \), reflecting it over the y-axis would mean substituting \( x \) with \( -x \). However, since negative x-values within \([-2, 0]\) change back to positive \([0, 2]\) when squared and combined, the domain effectively remains the same as the original.

Similarly, \( y = -f(-x) \) involves a double reflection: first over the y-axis, resulting in \( f(-x) \), and then over the x-axis. This flips the graph upside down from the configuration achieved in \( f(-x) \). Through these processes, reflections help visualize symmetry in function graphs.

Horizontal scaling involves stretching or compressing a graph along the x-axis. It alters how wide or narrow the graph appears without changing its shape.

The transformation \( y = f(-2x) \) involves both a reflection and horizontal compression. The \(-2\) inside the function affects x directly, compressing the horizontal axis by a factor of two and reflecting it. To understand the domain of this transformation, solve \( 2(-2x) - (-2x)^2 \geq 0 \). Doing so gives us \(-1 \leq x \leq 0\).

For \( y = f\left(-\frac{1}{2}x\right) \), we are performing a horizontal stretch by a factor of two. By solving \( 2\left(-\frac{1}{2}x\right) - \left(-\frac{1}{2}x\right)^2 \geq 0 \), we determine the domain to be \(-4 \leq x \leq 0\). Here, the graph appears wider, reflecting a horizontal stretch.

Horizontal scaling changes how we perceive the spread of the graph but does not alter its general direction.

A quadratic equation is a polynomial equation of degree two, typically in the form \( ax^2 + bx + c = 0 \). In our context, the function \( f(x) = \sqrt{2x - x^2} \) involves a quadratic expression under a square root.

The quadratic expression \( 2x - x^2 \) can be rewritten as \( x(2 - x) \), helping us solve for domain-related values. These factorized forms are crucial to determining where the graph of the function lies on the x-axis.

Quadratics like \( 2x - x^2 \) exhibit a characteristic parabolic shape, either opening upwards or downwards. Here, since it is expressed under a square root, the function only considers the top half of the parabola, where the expression is non-negative.

Understanding quadratics involves:

• Identifying its standard form \( ax^2 + bx + c \).
• Factoring to simplify and find roots or intervals.
• Utilizing graph transformations like reflection and scaling to understand how changes affect the parabola's graph.

These insights about quadratic equations can aid significantly in manipulating and graphing functions.