Problem 45 - A Graphical Approach to College Algebra

Question

Use transformations of graphs to sketch a graph of \(y=f(x)\) by hand. Do not use a calculator. $$f(x)=-\sqrt{1-x}$$

Step-by-Step Solution

Answer

The graph of \( f(x) = -\sqrt{1-x} \) is reflected over the x-axis and shifted left, starting at \( (1,0) \).

Step 1: Identify the Basic Function

The basic function to start with is the square root function. The graph of the function \( y = \sqrt{x} \) is a curve that starts from the origin (0,0) and moves up to the right.

Step 2: Apply Horizontal Shift

Based on the function \( f(x) = -\sqrt{1-x} \), identify the horizontal shift. Replace \( x \) with \( 1-x \). This indicates a shift left by 1 unit. However, due to subtraction inside the square root, it indicates the range of the original \( \sqrt{x} \) shifts the entire function to reflect over the y-axis. The graph of \( \sqrt{x} \) will shift to start from x=1 and move to the left, with its behavior flipped horizontally.

Step 3: Apply Vertical Reflection

The negative sign in \( -\sqrt{1-x} \) indicates a reflection over the x-axis. Thus, the whole graph that is already shifted and transformed now reflects downwards.

Step 4: Sketch the Graph

Combine all transformations: start with \( y = \sqrt{1-x} \), which is a horizontally flipped and left-shifted square root starting at \( (1, 0) \), then reflect it over the x-axis. The final graph will start at the point \( (1,0) \), decreasing into the second quadrant.

Key Concepts

Square Root Function

The square root function is a fundamental concept in mathematics, forming the basis for many graph transformations. Its basic form, expressed as \( y = \sqrt{x} \), is a curve originating from the point \( (0,0) \) and extending upward to the right. This function has some essential attributes:

• It exists only for non-negative values of \( x \), representing its domain.
• The range is also non-negative, as the square root of any non-zero number is positive.
• The graph exhibits an increasing nature, moving upwards slowly as \( x \) grows larger.

Understanding the characteristics of the square root function helps when applying transformations such as shifts and reflections, allowing you to predict the resultant shape and orientation of the graph.

Horizontal Shift

A horizontal shift is a form of graph transformation that moves the graph left or right along the x-axis. For the function \( f(x) = -\sqrt{1-x} \), identifying the horizontal shift involves looking at the expression inside the square root: \( 1-x \). This transformation impacts the position of the entire graph:

• Unlike \( \sqrt{x} \), the expression \( 1-x \) flips the graph horizontally, reflecting it over the y-axis.
• This is because \( x \) is effectively changed to \( -x+1 \) inside the function.
• The graph then begins at the point \( x=1 \) and extends to the left, rather than to the right.

The transformation creates a new starting point for the graph, giving it a fresh orientation which must be accounted for in further steps like reflections.

Vertical Reflection

Vertical reflection is another transformation where the graph is flipped over a horizontal line, commonly the x-axis. For the given function \( f(x) = -\sqrt{1-x} \), the negative sign in front of the square root introduces this change:

• The entire graph, after being horizontally shifted and reflected across the y-axis, is now reflected downward.
• This means that instead of increasing or maintaining its position, the curve decreases as it moves along the x-axis.
• The graph changes direction because the values of \( \sqrt{1-x} \) are multiplied by \(-1\).

Combining these transformations—a horizontal flip and a vertical reflection—results in a graph that starts at point \( (1,0) \) in the second quadrant, creating a downward arc into the negative y-region. This process showcases how various transformations can dramatically alter the appearance and direction of a graph.