Problem 69 - Algebra 2

Question

a. Graph \(y=\sqrt{-x}, y=\sqrt{1-x},\) and \(y=\sqrt{2-x}\) b. How does the graph of \(y=\sqrt{h-x}\) differ from the graph of \(y=\sqrt{x-h} ?\)

Step-by-Step Solution

Answer

The graph of \(y=\sqrt{-x}\) flips the parent function \(y=\sqrt{x}\) around the y-axis. \(y=\sqrt{1-x}\) starts at the point (1,0) and continues to the left, while \(y=\sqrt{2-x}\) starts at the point (2,0) and moves to the left. For \(y=\sqrt{h-x}\), the graph will shift h units to right but then moves to the left while \(y=\sqrt{x-h}\) shifts h units to right and continues to the right on the horizontal axis.

Step 1: Graphing \(y=\sqrt{-x}\)

The graph of \(y=\sqrt{x}\) starts at the origin and continues to increase as x increases. However, if we negate the x, the horizontal input, the graph will flip horizontally. So the graph will start at the origin and continue to the left as the x decreases.

Step 2: Graphing \(y=\sqrt{1-x}\)

The graph \(y=\sqrt{-x}\) flipped horizontally originates at the origin. To graph \(y=\sqrt{1-x}\), it is translated one unit to the right from the origin because of the 1 in 1-x. So, the line will start from (1,0) and continue to the left.

Step 3: Graphing \(y=\sqrt{2-x}\)

Following the same logic as applied in step 2, the graph of \(y=\sqrt{2-x}\) will start at (2,0) and continue to the left, because the value of 'h' in 'h-x' is 2 in this case.

Step 4: Comparing \(y=\sqrt{h-x}\) and \(y=\sqrt{x-h}\)

The function \(y=\sqrt{h-x}\) has the graph flipped horizontally and then translated h units to the right, while \(y=\sqrt{x-h}\) just has a horizontal translation of h units to the right from the origin. In other words, if h is positive, \(y=\sqrt{h-x}\) starts from (h,0) and continues to the left, whereas \(y=\sqrt{x-h}\) starts at (h,0) and continues to the right.

Key Concepts

Function Transformations

Function transformations are important tools in algebra that change the position or shape of a graph. They include translations, reflections, stretches, and compressions. When graphing, a transformation affects the whole set of points that constitute the graph of a function. Transformations can be grouped into two main categories: vertical and horizontal. Each transformation plays a significant role in altering how a function appears when plotted on a coordinate grid. For example, if you multiply the function by a number, it stretches or compresses the graph vertically. Similarly, horizontal transformations affect the x-coordinates. These include shifts and reflections across the y-axis, fundamentally changing the graph's orientation along the x-axis.

Horizontal Shifts

Horizontal shifts are a type of function transformation specifically involving the x-coordinates. They move the graph left or right on the coordinate plane. In the context of square root functions, horizontal shifts occur when the expression under the square root sign has an added or subtracted number. For instance, if we start with the basic function \(y = \sqrt{x}\), changing it to \(y = \sqrt{x - h}\) results in a horizontal shift of h units to the right. If instead we have \(y = \sqrt{-x}\), it implies the function is reflected over the y-axis. Consequently, the graph now progresses towards negative x-values instead of positive, flipping horizontally.

• If \(h\) is positive, the shift is to the right.
• If \(h\) is negative, the shift is to the left.

Graphing Square Root Functions

Graphing square root functions can be simple once you understand transformations and shifts. The parent function, \(y = \sqrt{x}\), begins at the origin and extends to the right in the first quadrant. By adding transformations, we can adjust how this graph appears, starting with reflections and shifts.
To graph functions like \(y = \sqrt{-x}\) or \(y = \sqrt{2-x}\):

• A negative under the square root, as in \(y = \sqrt{-x}\), flips the graph horizontally.
• The expression \(1-x\) shifts the graph to the right by 1 unit, starting at (1,0).
• The expression \(2-x\) shifts it 2 units right, initiating at (2,0).

These graphs behave similarly, reflecting and shifting based on the constants that modify \(x\). Remember, by understanding how each transformation affects the basic shape, you can predict and sketch the behavior of any square root function efficiently. This way, both comparison and graphing become intuitive tasks.