Problem 47 - Beginning and Intermediate Algebra

Question

Use the transformation techniques to graph each of the following functions. $$f(x)=-\sqrt{x+5}$$

Step-by-Step Solution

Answer

To graph \(f(x)=-\sqrt{x+5}\) using transformation techniques, we first identify the parent function, which is \(y=\sqrt{x}\). The given function has two transformations: a reflection across the x-axis due to the negative sign, and a horizontal shift 5 units to the left due to the "+5" inside the square root function. Apply the reflection and horizontal shift to points on the parent function and plot the transformed points, connecting them with a smooth curve. The graph should start at point (-5,0) and extend downwards and to the right. Verify the graph represents the function by checking additional points, such as (-4,-1).

Step 1: Identify the parent function

The parent function is the square root function \(y = \sqrt{x}\), which has a graph that looks like a curve starting at the origin (0,0) and extending upwards and to the right.

Step 2: Identify the transformations

The given function is \(f(x) = -\sqrt{x + 5}\). We can see that there are two transformations: 1. A negative sign in front of the square root, which will result in a reflection across the x-axis. 2. A horizontal shift to the left by 5 units, because of the "+5" inside the square root function.

Step 3: Apply the reflection

First, we will apply the reflection across the x-axis. For this, every point on the parent function \(y = \sqrt{x}\) will be mirrored along the x-axis. For example, if there is a point (1,1) on the parent function, after the reflection, it will become the point (1,-1).

Step 4: Apply the horizontal shift

Next, we will apply the horizontal shift to the left by 5 units. All the points on the reflected function will be moved 5 units to the left. If there is a point (1,-1) on the reflected function, after the horizontal shift, it will become the point (-4, -1).

Step 5: Graphing the function

Now, we can graph the function using the transformations. Plot the points that we have found by applying the transformations and connect them with a smooth curve to obtain the graph of the function \(f(x) = -\sqrt{x + 5}\). Remember that the graph should start at the point (-5,0) and extend downwards and to the right, since it is a reflection of the square root function and has a horizontal shift of 5 units to the left.

Step 6: Verify the graph

Verify that the graph represents the function by checking a few additional points. For example, you could plug x = -4 into the function, obtaining \(f(-4) = -\sqrt{-4 + 5} = -\sqrt{1} = -1\), which corresponds to the point (-4,-1) on the graph. Similarly, you can verify other points on the graph to ensure that the graph represents the given function, \(f(x) = -\sqrt{x + 5}\).

Key Concepts

Graphing Functions

Graphing a function involves plotting points that satisfy the equation of the function and connecting them to visualize the curve or line that they form. It gives us a visual representation of mathematical relationships, allowing us to understand shifts, stretches, and reflections.

• Each function has a 'parent function,' a simpler form that you start with before applying transformations.
• Transformations can shift, reflect, stretch, or compress the graph.
• Graphing helps in identifying key features like intercepts, maxima/minima, and asymptotes.

Being able to graph functions accurately is essential in mathematics as it allows you to predict and interpret real-world phenomena based on equations and data.

Square Root Function

The square root function, denoted as \( y = \sqrt{x} \), is one of the basic parent functions in algebra. Its graph starts at the origin (0,0) and curves gently upwards and to the right.
This function is defined only for non-negative values of \( x \), as the square root of a negative number is not a real number.

• The domain of \( y = \sqrt{x} \) is \( x \geq 0 \).
• The range is also non-negative, \( y \geq 0 \).
• It represents a half-parabola in the first quadrant of the coordinate plane.

Understanding the square root function is crucial as it is a foundation for grappling with transformations and more complex algebraic functions.

Horizontal Shifts

A horizontal shift in graphing means moving the graph of a function left or right on the coordinate plane. This type of transformation does not affect the size or shape of the graph, only its position.

• To shift the graph to the left, you add a constant to \( x \) within the function: \( y = f(x + c) \).
• To shift to the right, you subtract a constant from \( x \): \( y = f(x - c) \).

In the function \( f(x) = -\sqrt{x + 5} \), the \(+5\) within the square root indicates a horizontal shift of 5 units to the left.
This adjustment transforms the starting point of our graph from (0,0) to (-5,0), reflecting the function's new rooted beginning.

Reflection Across Axes

Reflection is a transformation that flips a graph over a specific axis. In a reflection, the shape and size of the graph remain unchanged, but its orientation and position are altered.

• Reflecting across the x-axis involves multiplying the function by -1, resulting in \( y = -f(x) \).
• This transformation changes all positive \( y \)-values to negative, and vice versa, flipping the graph upside down.

For the function \( f(x) = -\sqrt{x + 5} \), the negative sign in front of the square root sign means that every point on the graph of \( y = \sqrt{x} \) is reflected across the x-axis.
The reflection flips the curve so that it decreases instead of increases, changing its direction while keeping the same set of \( x \)-values.