Problem 64 - College Algebra with Modeling and Visualization

Question

Use transformations to sketch a graph of \(f\). \(f(x)=2+\sqrt{-(x-3)}\)

Step-by-Step Solution

Answer

The graph is a leftward curve above \(y=2\) starting from \((3, 2)\).

Step 1: Identify the Base Function

The base function for this exercise is the square root function, which is \(g(x) = \sqrt{x}\). Before we apply any transformations, \(g(x)\) is defined only for \(x \geq 0\) and is a curve starting at the origin and increasing to the right.

Step 2: Apply Horizontal Transformation

The expression under the square root, \(-x + 3\) or \(-(x-3)\), results in a reflection across the y-axis and a horizontal shift. The negative sign reflects the graph of \(g(x)\) across the line \(x = 3\). This makes the function decrease leftward starting from \(x = 3\), technically becoming \(h(x) = \sqrt{-(x-3)}\).

Step 3: Determine the Domain

To ensure the expression under the square root is non-negative, set \(-x + 3 \geq 0\), meaning \(x \leq 3\). The domain of \(f(x)\) is therefore \(x \leq 3\).

Step 4: Apply Vertical Transformation

The function \(f(x) = 2 + \sqrt{-(x-3)}\) adds 2 to each \(y\)-coordinate of the graph of \(h(x)\). This vertical shift moves the whole graph up by 2 units.

Step 5: Sketch the Graph

To sketch the graph, plot starting from the point \((3, 2)\) and moving leftward, since the function decreases with decreasing \(x\). The curve starts at the point \((3, 2)\), decreases and extends leftwards, remaining above the line \(y = 2\).

Key Concepts

Square Root Function

The square root function is one of the basic functions in mathematics. It is typically represented as \( g(x) = \sqrt{x} \). This function is defined only for non-negative values of \( x \) because you cannot take the square root of a negative number without entering the realm of complex numbers.

• The graph of a square root function is a curve that starts at the origin \((0,0)\).
• It continually rises as it moves to the right.

The original function given to us is based on this square root function, but with transformations applied to it, modifying its shape and position on the graph.

Horizontal Shift

Horizontal shifts involve moving the graph of a function left or right along the x-axis. For the given function, \( f(x) = 2 + \sqrt{-(x-3)} \), there is a transformation inside the square root, specifically \(-(x-3)\).

• The minus sign in front of \( x \) results in a reflection across the vertical line \( x = 3 \).
• Effectively, this causes a horizontal shift of the graph to the left from \( x = 3 \).

This means that instead of starting from the origin, like the basic square root function, this graph begins impacting from \( x = 3 \) and spreads to the left.

Vertical Transformation

Vertical transformations change the position of a graph vertically on the coordinate plane. With \( f(x) = 2 + \sqrt{-(x-3)} \), the '2' outside the square root indicates a vertical shift.

• This adds 2 to every \( y \)-value of the transformed square root function.
• It's like taking the graph from the horizontal shift and lifting it upwards by 2 units.

Thus, the starting point of the graph after both horizontal and vertical transformations will be at \((3, 2)\). This elevates the entire graph, maintaining its shape but altering its vertical positioning.

Domain of a Function

The domain of a function is the set of all possible input values \( x \) that it can accept without resulting in undefined or non-real outputs. For \( f(x) = 2 + \sqrt{-(x-3)} \), to ensure the square root is computing a real number, the expression inside must be non-negative.

• Setting \(-x + 3 \geq 0\) leads to solving \( x \leq 3 \).

This calculation shows that our function's domain is \( x \leq 3 \), meaning the function only takes values from negative infinity up to 3, including 3 itself. Understanding the domain is crucial when graphing the function because it tells us the span of \( x \)-values to consider.