Problem 74 - Algebra and Trigonometry

Question

If \(f(x)=\sqrt{2 x-x^{2}}\) graph the following functions in the viewing rectangle \([-5,5]\) by \([-4,4] .\) How is each graph related to the graph in part (a)? $$ \begin{array}{ll}{\text { (a) } y=f(x)} & {\text { (b) } y=f(-x)} \\ {\text { (c) } y=-f(-x)} & {\text { (d) } y=f(-2 x)} \\ {\text { (e) } y=f\left(-\frac{1}{2} x\right)}\end{array} $$

Step-by-Step Solution

Answer

Reflect, flip, and scale graphs based on given functions; relate changes to the original graph.

Step 1: Understand the Parent Function

The given function is \( f(x) = \sqrt{2x - x^2} \). This is the square root of a quadratic expression. The domain (values of \( x \)) must make the expression inside the square root non-negative, i.e., \( 2x - x^2 \geq 0 \). Determine this inequality's solution to find the domain.

Step 2: Simplify the Domain

Factor the quadratic expression: \( 2x - x^2 = x(2 - x) \). Set each factor \( \geq 0 \):\( x \geq 0 \) and \( 2 - x \geq 0 \), giving \( 0 \leq x \leq 2 \). This is the domain of the function \( f(x) \). Graph \( f(x) \) for \( x \) within this interval.

Step 3: Graph y=f(-x)

The function \( y = f(-x) \) is the reflection of \( f(x) \) across the y-axis. Replace \( x \) with \( -x \) in the domain: \( -2 \leq -x \leq 0 \) translated to \( 0 \leq x \leq 2 \). Therefore, the graph is the same as part (a), symmetric over the y-axis.

Step 4: Graph y=-f(-x)

The function \( y = -f(-x) \) is the reflection of \( f(-x) \) over the x-axis. This involves negating the output values of \( f(-x) \). Therefore, it is still defined over \( 0 \leq x \leq 2 \) with the graph flipped vertically from that in part (b).

Step 5: Graph y=f(-2x)

The function \( y = f(-2x) \) involves a horizontal scaling and reflection. Solve for \( x \) in \( 2(-2x) - (-2x)^2 \geq 0 \) to find the domain. Solving gives \( -1 \leq x \leq 0 \), so plot the function over this interval using the same values of \( y \) as part (a).

Step 6: Graph y=f(-\frac{1}{2}x)

The function \( y = f\left(-\frac{1}{2}x\right)\) involves a horizontal stretch and reflection. Solve \( 2\left(-\frac{1}{2}x\right) - \left(-\frac{1}{2}x\right)^2 \geq 0 \) to get the domain \( -4 \leq x \leq 0 \). Plot over this interval similarly to part (a).

Key Concepts

Function Domain

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

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

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.

Quadratic Equation

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.