Problem 48 - 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. $$f(x)=2+\sqrt{-(x-3)}$$

Step-by-Step Solution

Answer

The graph of \( f(x) = 2 + \sqrt{-(x-3)} \) is a square root curve reflected across the y-axis, shifted 3 units right, and 2 units up, starting at (3,2).

Step 1: Identify the Base Function

The base function is the square root function, typically written as \( y = \sqrt{x} \). This graph has its vertex at the origin (0,0) and moves to the right in a positive curve.

Step 2: Consider the Inside Transformation

The function is given as \( y = \sqrt{-(x-3)} \). The term \(-(x-3)\) indicates a horizontal transformation. The negative sign means a reflection about the y-axis, and the \(x-3\) indicates a horizontal shift to the right by 3 units.

Step 3: Apply Horizontal Reflection and Shift

Reflect the base function \( y=\sqrt{x} \) across the y-axis. This creates \( y=\sqrt{-x} \), which is defined for \( x \leq 0 \) only. Next, shift this graph horizontally 3 units to the right, changing the vertex from \((0,0)\) to \((3,0)\). This gives the graph of \( y = \sqrt{-(x-3)} \) which is defined for \( x \leq 3 \).

Step 4: Consider the Vertical Shift

Finally, the function \( y = \sqrt{-(x-3)} + 2 \) indicates a vertical shift upwards by 2 units. Adjust every point on the graph up by 2 to account for this.

Step 5: Sketch the Transformed Graph

Start with the vertex at (3,2). The graph extends horizontally towards \( x = -\infty \) while just touching the y-axis at the point where it was shifted vertically, making it a transformed square root curve that is reflected horizontally.

Key Concepts

Square Root Function

The square root function is a fundamental mathematical function represented by \( y = \sqrt{x} \). Its primary characteristic is that it only outputs non-negative values because the square root of any non-negative number is also non-negative. This function forms the base for many graph transformations.

Visually, the square root function begins at the origin \( (0,0) \) and extends to the right, forming a gentle upward curve. This is because as \( x \) increases, \( \sqrt{x} \) also increases, but at a decreasing rate.

• The graph only exists for \( x \geq 0 \).
• The shape is a half parabola opening upwards.

Understanding this base form helps in visualizing how the graph changes with different transformations.

Horizontal Reflection

A horizontal reflection flips the graph of a function over the y-axis. In mathematical terms, for a function \( y = f(x) \), this transformation would be represented as \( y = f(-x) \).

In the case of the square root function, this reflection results in the curve moving leftward, maintaining its half-parabolic shape.

• For \( y = \sqrt{-x} \), the domain changes to \( x \leq 0 \), because the square root is only defined for non-negative values.
• This reflection causes the curve to start at the origin and extend left.

Thus, a horizontal reflection affects the direction of the curve and the domain of the function.

Vertical Shift

A vertical shift involves moving a graph up or down along the y-axis. This transformation can be expressed as \( y = f(x) + c \), where \( c \) is the number of units the graph shifts. If \( c > 0 \), the graph moves upward; if \( c < 0 \), it moves downward.

In our function, the addition of 2 indicates a vertical shift upwards by 2 units.

• Every point moves up by 2 units.
• The entire graph is raised without changing its shape.

This shift does not alter the domain or range besides increasing each y-value by 2.

Horizontal Shift

Horizontal shifts involve moving the graph either left or right along the x-axis. This is achieved by adding or subtracting a constant from the variable \( x \) in the function, represented as \( y = f(x-h) \). Here, \( h \) represents the horizontal shift.

In the given function \( y = \sqrt{-(x-3)} \), the \( x-3 \) indicates a shift 3 units to the right:

• The graph moves right, positioning the vertex from \( (0,0) \) to \( (3,0) \).
• Initially defined for \( x \leq 0 \), the domain adjusts to \( x \leq 3 \).

Horizontal shifting helps reposition the graph while maintaining the same orientation.