Problem 47 - A Graphical Approach to Precalculus with Limits

Question

Use transformations of graphs to sketch a graph of \(y=f(x)\) by hand. $$f(x)=\sqrt{-(x+1)}$$

Step-by-Step Solution

Answer

Reflect \(y=\sqrt{x}\) across the y-axis and shift left by 1.

Step 1: Identify the Base Function

The given function is \(f(x)=\sqrt{-(x+1)}\). This is a transformation of the base function \(y=\sqrt{x}\), which is a square root function.

Step 2: Understand the Inside Transformation

Notice the expression inside the square root: \(-(x+1)\). This can be rewritten as \(-x - 1\). The negative sign indicates a reflection across the y-axis, while the \(+1\) shifts the graph to the left by one unit.

Step 3: Determine the Domain

The square root function requires the input to be non-negative. Thus, solve the inequality: \(-(x+1) \geq 0\), leading to \(x+1 \leq 0\), which simplifies to \(x \leq -1\). The domain is \((-\infty, -1]\).

Step 4: Sketch the Reflected Graph

Start by sketching the base function \(y=\sqrt{x}\), then reflect this graph across the y-axis. This reflection changes \(y=\sqrt{x}\) to \(y = \sqrt{-x}\).

Step 5: Apply the Horizontal Shift

After the reflection, apply the horizontal shift of 1 unit to the left. This moves the graph of \(y = \sqrt{-x}\) to \(y = \sqrt{-(x+1)}\).

Step 6: Final Sketch

Combine these transformations to sketch the final graph. Begin at the point where \(x=-1\) on the x-axis since this is the starting point due to the domain constraint.

Key Concepts

Square Root Function

The square root function is one of the basic building blocks of many algebraic expressions and transformations. It is generally expressed as \( y = \sqrt{x} \). This function starts at the point \( (0, 0) \) and curves upward to the right. It is important to note that the square root function is always non-negative, meaning it only takes values above or on the x-axis.

• Base Shape: The function has a gently increasing slope as x gets larger.
• Domain: The domain of a basic square root function is \( [0, \infty) \), as square roots of negative numbers are not real in this context.
• Graph: Its graph has a characteristic half-parabola shape that only exists in the first quadrant.

Understanding the basic shape and behavior of the square root function is essential when considering transformations such as reflections or shifts.

Reflection Across the Y-axis

A reflection is a transformation that flips a graph over a specific line, much like flipping a picture. In the case of a reflection across the y-axis, every point \( (x, y) \) on the graph is transformed to \( (-x, y) \). In our function \( f(x) = \sqrt{-(x+1)} \), the negative sign inside the square root indicates this reflection.

• New Point Positioning: The positive and negative sides of the x-axis are effectively swapped.
• Effect on Square Root: The graph's direction changes from going left-to-right upwards to right-to-left upwards.

This reflection is crucial as it prepares the graph for further transformation, ensuring its shape conforms to the new equation structure.

Horizontal Shift

Horizontal shifts affect where the new position of a graph starts along the x-axis, particularly moving it left or right. When we talk about shifting horizontally, we generally refer to the transformations inside the function's argument. For the function \( f(x) = \sqrt{-(x+1)} \), we experience a horizontal shift of 1 unit to the left due to the \((x+1)\) part inside the square root. This transformation implies:

• Shift Direction: Since it is \(+1\), the shift goes left.
• Graph Movement: Moves every x-coordinate left by one unit.

These shifts are particularly subtly effective, altering the entire graph's initial appearance while keeping its shape intact, ensuring the function behaves adequately according to its equation.

Domain of a Function

The domain of a function specifies the set of all possible input values (x-values) for which the function is defined. For root functions, this often involves considering when the expression inside the root is non-negative. We need this condition because the square root of a negative number isn't defined within the real number system.For \( f(x) = \sqrt{-(x+1)} \), solving \[ -(x+1) \geq 0 \] gives:

• Simplification: \[ x+1 \leq 0 \] leads to \[ x \leq -1 \]
• Domain Expression: The acceptable domain is \(( -\infty, -1 ]\).

This restriction highlights where the graph starts and ensures calculations stay within real numbers, vital for accurately sketching and interpreting the function graph.