Problem 60 - 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)? \(\begin{array}{llll}{\text { (a) } y=f(x)} & {\text { (b) } y=f(- x)} & {\text { (c) } y=-f\left(-x\right)}\\\ {\text { (d) } y=f(- 2x)} & {\text { (e) } y=f\left(-\frac{1}{2} x\right)}\end{array}\)

Step-by-Step Solution

Answer

The function (a) is a semi-circle, (b) is a reflection over y-axis from (a), (c) flips (b) over x-axis, and (d), (e) yield no visible graphs.

Step 1: Understand the function

The function given is \( f(x) = \sqrt{2x - x^2} \). This is a semi-circle. To determine the domain, set the expression inside the square root greater than or equal to zero: \( 2x - x^2 \geq 0 \). Solving this will help us identify the valid range of \( x \).

Step 2: Solve the inequality

Rewrite \( 2x - x^2 \geq 0 \) as \( x(2-x) \geq 0 \). Solving this, we get critical points at \( x = 0 \) and \( x = 2 \). The inequality holds when \( x \) is in the range \( 0 \leq x \leq 2 \). The graph of \( f(x) \) is then a semi-circle centered at \( x = 1 \) with radius 1.

Step 3: Graph the function (a)

Graph \( y = f(x) = \sqrt{2x - x^2} \) for \( x \) between 0 and 2. This is a semi-circle of radius 1 with center at \( (1, 0) \), only in the positive y-direction.

Step 4: Graph the function (b)

Graph \( y = f(-x) = \sqrt{2(-x) - (-x)^2} \). This equals \( \sqrt{2x - x^2} \), but since \( f(-x) \) reflects the function across the y-axis, it will not be defined outside \( -2 \leq x \leq 0 \). The graph appears as a reflection of (a) on the x-axis from \( x = -2 \) to \( x = 0 \).

Step 5: Graph the function (c)

Graph \( y = -f(-x) \), which is a vertical reflection of \( f(-x) \). Consider the graph of \( f(-x) \) from Step 4 and reflect it across the x-axis, resulting in a semi-circle below the x-axis within \( -2 \leq x \leq 0 \).

Step 6: Graph the function (d)

Graph \( y = f(-2x) = \sqrt{2(-2x) - (-2x)^2} = \sqrt{-4x - 4x^2} \). The domain changes and the function is not real for any \( x \), so it yields no visible graph in the viewing window.

Step 7: Graph the function (e)

Graph \( y = f\left(-\frac{1}{2}x\right) = \sqrt{2\left(-\frac{1}{2}x\right) - \left(-\frac{1}{2}x\right)^2} = \sqrt{-x - \frac{1}{4}x^2} \). The expression under the square root is non-negative in the window displayed, but results in no useful graph because the domain does not correspond to real values within the square root in the given interval.

Key Concepts

Domain of a Function

The domain of a function represents all the possible input values (often x-values) that allow the function to produce real and valid outputs. For the function \( f(x) = \sqrt{2x - x^2} \), determining the domain is crucial as it defines where the graph is located on the coordinate plane.
To find the domain, we need to ensure the expression inside the square root is non-negative. This leads to solving the inequality \( 2x - x^2 \geq 0 \), which can be rewritten as \( x(2-x) \geq 0 \).

• The critical points where the expression changes sign are \( x = 0 \) and \( x = 2 \).
• The inequality holds true for values within the interval \( 0 \leq x \leq 2 \).

Thus, the domain of \( f(x) \) is all x-values between 0 and 2. This essentially means the function forms a semi-circle with these domain boundaries on the horizontal axis.

Transformations of Functions

Transformations involve shifting or scaling a function's graph. They affect how a function's graph appears in relation to its original form, \( f(x) \). One common type of transformation is scaling, which changes the size of the graph either horizontally or vertically.

• For instance, the function \( y = f(-2x) \) implies a horizontal transformation. It compresses the function horizontally by a factor of 2, and reflects it across the y-axis.
• Another example is \( y = f\left(-\frac{1}{2}x\right) \), which horizontally stretches the graph by a factor of 2 and reflects it across the y-axis.

These transformations can sometimes result in the function producing no real values, as seen with these examples. The expressions under the square root must remain non-negative in the transformation, or the graph may not be visible.
Understanding these transformations is key to predicting how the graph of a function will move or change in appearance.

Reflections of Functions

Reflections are specific types of transformations where the graph is flipped across an axis. This can occur across the x-axis, y-axis, or even other lines, changing the function's orientation.

• A reflection across the y-axis is observed in \( y = f(-x) \). It mirrors the graph horizontally, changing the sign of the x-values.
• A reflection across the x-axis is seen with \( y = -f(-x) \). This mirrors the graph vertically, flipping all y-values.

Reflections maintain the general shape of the function but alter its position. For instance, reflecting a semi-circle across the x-axis places it below the axis, while a reflection across the y-axis changes its span horizontally.
Recognizing reflection patterns helps to understand how graph movements relate to the original function's form.