Problem 65 - Algebra 2

Question

Graph each function. $$ y=\sqrt{x+2} $$

Step-by-Step Solution

Answer

The graph of the function \(y=\sqrt{x+2}\) starts at point (-2,0) and rise to the right similar to half a parabola.

Step 1: Understand the Function Properties

Analyze the function \(y=\sqrt{x+2}\). A simple square root function \(y=\sqrt{x}\) typically starts at the origin (0,0) and rises to the right. This graph only exists when x is larger than or equal to 0. Additionally, the function \(y=\sqrt{x+2}\) has a horizontal shift 2 units to the left from the simple square root function. So, it will exist where \(x+2>=0\) which means for \(x>=-2\). Graph will start on the point (-2,0) and rise to the right.

Step 2: Find Key Points for the Graph

Choose several good points to plot on your function. As your starting point, you always know that a square root function passes through the origin of its basic component. For our function, it will cut the x-axis at x=-2, thus one point to start is (-2,0). After that choose some other points for x-values such as -1, 0, 1, 2, 3 to sketch a rough shape of your graph. Substitute each x-value into the function to calculate the corresponding y-value. For example, for x=0, the y-value will be \(\sqrt{0+2} = \sqrt{2}\).

Step 3: Sketch the Graph

Plot the points calculated in Step 2 on the coordinate system. Connect the points with a smooth curve to produce the graph for the function \(y=\sqrt{x+2}\). The graph should start from the point (-2,0) and rise to the right, producing a shape similar to half of a parabola lying on its side.

Key Concepts

Function Transformation

The concept of function transformation is essential when dealing with various types of functions, including square root functions. A transformation involves changing the position or shape of a graph without altering its essential character. This transformation can be translations (shifts), reflections, dilations (stretching or compressing), or rotations.
For the square root function \(y = \sqrt{x+2}\), we're looking specifically at a horizontal shift. The "+2" inside the square root denotes a shift. Function transformations like this one help us to adjust the function from the basic form \(y = \sqrt{x}\), allowing us to explore a range of graph positions and shapes.

Domain of a Function

Understanding the domain of a function is a crucial step in graphing it accurately. The domain refers to all the possible x-values that can be used in a function to yield real and valid y-values.
In square root functions, we must ensure the expression under the root is non-negative since the square root of a negative number is undefined in the set of real numbers. Starting from the basic square root function \(y = \sqrt{x}\) with a domain of \(x \geq 0\), the transformed function \(y = \sqrt{x + 2}\) shifts this domain accordingly.

Finding the Domain:

• Set the expression under the square root \(x + 2\) \(\geq 0\).
• Solve for x to find the domain: \(x \geq -2\).

Thus, the domain of the function is all real numbers x such that \(x \geq -2\). This indicates where the function is defined and dictates where you start plotting your graph.

Key Points in Graphing

When graphing functions, identifying key points helps to construct an accurate representation of the graph. Begin with the start point, which for the function \(y = \sqrt{x+2}\) occurs at \(x = -2\). Here, the function value is zero, so our first key point is (-2,0).
To further shape the graph:

• Choose additional x-values, such as -1, 0, 1, 2, and 3.
• Calculate the corresponding y-values:
  • For \(x = -1\), \(y = \sqrt{-1+2} = \sqrt{1} = 1\).
  • For \(x = 0\), \(y = \sqrt{0+2} = \sqrt{2}\).
  • Continue to compute for subsequent x-values.

These points provide a visual framework. Plot these (x, y) pairs on a coordinate plane, and connect them smoothly, reflecting the curve's natural path, resembling half a sideways parabola. These key points guide you to shape the overall curve of the graph.

Horizontal Shift

A horizontal shift in graphing occurs when a function is moved left or right along the x-axis. This type of transformation changes the x-coordinates of the function's graph but leaves the y-values unchanged. The extent of the shift is determined by the number adjusted within the function.
In our function \(y = \sqrt{x+2}\), the "+2" inside the square root symbolizes a shift. Exactly 2 units to the left from the origin, in comparison to the original function \(y = \sqrt{x}\). Each function value is adjusted accordingly, starting effectively at \(x = -2\) instead of \(x = 0\).
This concept explains why the graph initiates at the point (-2,0) rather than at the standard origin, reflecting this shift. Recognizing horizontal shifts is crucial to understanding how functions transform and develop their shapes on a graph.