Problem 6 - Algebra 2

Question

Graph each function. \(y=\sqrt{x+1}\)

Step-by-Step Solution

Answer

The graph of \(y=\sqrt{x+1}\) starts at (-1,0) and increases as \(x\) increases. It's in the shape of a half-parabola opened along the x-axis.

Step 1: Determine the Domain of the Function

The domain of the function is all values of \(x\) for which the function is defined. For the function \(y=\sqrt{x+1}\), it is defined for all \(x\) such that \(x+1 \geq 0\). Thus, \(x\geq -1\). So, the domain of the function is all real numbers \(x\) such that \(x\geq -1\).

Step 2: Create a Table of Values

Choose some values for \(x\) from the domain and compute the corresponding \(y\) values. This will provide points which can be plotted on the graph. For example, you can choose \(x= -1,0, 1\). Computing the \(y\) values gives \(y=0,1,\sqrt{2}\) respectively.

Step 3: Plot the Graph

Plot the points from the table of values on a coordinate axis. Draw a smooth curve through the points. Ensure to draw the curve going off to infinite \(x\) in the direction of increasing \(x\) as the square root function is only defined for \(x \geq -1\). The resulting graph starts at the point (-1,0) and increases as x increases.

Key Concepts

Domain of a Function

Understanding the domain of a function is crucial in graphing any mathematical function. The domain refers to the set of all possible input values (usually represented as "x") that can be used in the function such that the outcome is real and defined. For square root functions like \[ y = \sqrt{x+1} \] the function is only defined when the expression under the root is zero or positive. In this case, the expression is \( x + 1 \), and it cannot be negative because the square root of a negative number is not defined in the set of real numbers. Hence, we need \( x+1 \geq 0 \), which simplifies to \( x \geq -1 \). Therefore, the domain for this function is all real numbers greater than or equal to \( -1 \). Being mindful of this domain ensures that our graph correctly represents the function's behavior and limitations.

Table of Values

To effectively graph any function, creating a table of values is an excellent strategy. It helps us find specific points that lie on the graph of a function. When working with the function \( y=\sqrt{x+1} \), you begin by choosing some values for \( x \) that fall within the domain, \( x \geq -1 \). By substituting x values from this domain into the function, we get corresponding \( y \) values. For example:

• If \( x = -1 \), then \( y = \sqrt{-1+1} = 0 \).
• If \( x = 0 \), then \( y = \sqrt{0+1} = 1 \).
• If \( x = 1 \), then \( y = \sqrt{1+1} = \sqrt{2} \approx 1.41 \).

Based on these calculations, the points \((-1, 0), (0, 1), \text{and} (1, \sqrt{2}) \) can then be used to graph the function accurately.

Coordinate Graphing

Once a table of values is established, the next step is plotting these points on a coordinate graph. Coordinate graphing takes place in the Cartesian plane, where each point is defined by an \( (x, y) \) pair. The function \( y=\sqrt{x+1} \) forms a curve rather than a line, and it is essential to connect the plotted points smoothly. Begin at the starting point \((-1, 0)\), which corresponds to the smallest x-value in our domain.

Since square root functions generally produce increasing Y values as X gets larger, plot each subsequent point at \((0, 1)\) and \((1, \sqrt{2})\) and draw a smooth curve passing through them. Beyond the plotted points, ensure the curve extends in the positive x direction. This tail-off aligns with the increasing nature of the square root function as \( x \) tends to infinity while staying unbounded above the calculated range.