Problem 72 - Precalculus Essentials

Question

Begin by graphing the square root function, \(f(x)=\sqrt{x} .\) Then use transformations of this graph to graph the given function. $$h(x)=-\sqrt{x+1}$$

Step-by-Step Solution

Answer

The graph of the function \(h(x)=-\sqrt{x+1}\) can be obtained from the graph of \(f(x)=\sqrt{x}\) by reflecting it in the x-axis and then shifting it to the left by 1 unit.

Step 1: Graph the original function

Start by graphing the original function, \(f(x)=\sqrt{x}\). This is a basic square root function which starts at the origin (0,0) and increases slowly. The part of the graph corresponding to negative values of x does not exist because the square root of a negative number is not a real number.

Step 2: Apply the reflection

Apply a reflection in the x-axis. This is done because of the negative sign in front of the square root in the function \(h(x)\). This changes the graph of \(f(x)=\sqrt{x}\) to the graph of \(f(x)=-\sqrt{x}\). Now, instead of increasing slowly, the graph starts at the origin and decreases slowly.

Step 3: Apply the horizontal shift

Now apply a horizontal shift to the left by 1 unit. This is done because of the \(x+1\) inside the square root in the function \(h(x)\). This shift moves every point on the graph of \(f(x)=-\sqrt{x}\) one unit to the left, resulting in the graph of \(h(x)=-\sqrt{x+1}\). Now, the graph starts at (-1,0) and decreases slowly.

Key Concepts

Graphing Functions

Graphing functions is the process of drawing the curve that represents a function on a coordinate plane. It involves plotting the graph of an equation to see how the function behaves. For example, with a function like\(f(x) = \sqrt{x}\), the graph starts at the origin, \((0,0)\), and extends to the right, as negative x-values don't yield real results for square root functions. This approach lets you visually analyze a function's properties, such as intercepts, slopes, and asymptotic behavior.
The graph shows the relationship between x-values and their corresponding y-values. Through graphing, one can observe how transformations like shifts, reflections, stretches, or compressions affect the graph's shape and position.

• Basic graphing begins with identifying the function's domain and range.
• For transformational graphing, identify shifts or changes needed.
• Draw the graph by plotting key points and joining them to form a smooth curve.

Square Root Function

The square root function, \(f(x) = \sqrt{x}\), is a basic algebraic function with a characteristic graph. This function produces a graph that is a gradual curve, starting at the origin, and moving only through positive y-values. The graph is not defined for negative x-values as it doesn't produce real y-values for those inputs.
The square root function has distinctive features that make it unique. It is continuous for all non-negative x-values and has a domain of \([0, \infty)\) and a range of \([0, \infty)\) as well.

• Starts at \((0,0)\), the lowest point of the function.
• Increases at a decreasing rate as x becomes larger.

Understanding this foundational graph is crucial for grasping more complex transformations.

Horizontal Shift

A horizontal shift is a transformation that moves the graph of a function left or right on the coordinate plane. This shift impacts where the graph begins or ends horizontally. When we speak of a horizontal shift, we're referring to changes made within the function's input, indicated by adding or subtracting a value within the expression. For the function \(h(x) = -\sqrt{x+1}\), the graph shifts one unit left because of the \(x+1\), where the number inside determines the direction and magnitude:

• \(x - c\) shifts the graph c units to the right.
• \(x + c\) shifts the graph c units to the left.

In this exercise, we see the effect of \(x+1\), which moves the entire graph of \(-\sqrt{x}\) one unit left, starting it off from the point \((-1,0)\). This transformation helps adjust the function's starting point to align with desired properties of a new graph.

Reflection Across X-Axis

Reflection across the x-axis is a transformation that flips the graph of a function over this axis. This means that if you have a point \((x, y)\) on the graph of \(f(x)\), after a reflection across the x-axis, the corresponding point on the transformed graph becomes \((x, -y)\). This transformation is represented by placing a negative sign in front of the entire function.
For example, taking the function \(f(x) = \sqrt{x}\) and applying a reflection results in \(-\sqrt{x}\) which mirrors the original function. Therefore, instead of increasing as \(f(x)\) does, the reflection \(-\sqrt{x}\) decreases, flipping all y-values to their negatives. This is crucial in understanding how reflections affect the function and aids in graphing, especially in visualizing systems that require symmetry or comparisons.