Problem 8 - Algebra 2

Question

Graph each inequality. \(y \geq \sqrt{x+2}-1\)

Step-by-Step Solution

Answer

Graph the function \(y = \sqrt{x+2} - 1\) as a solid curve starting from \(x = -2\) and shade the region above it.

Step 1: Understand the Inequality

The inequality given is \(y \geq \sqrt{x+2} - 1\). This means you need to graph \(y = \sqrt{x+2} - 1\), and then shade the region where \(y\) is greater than or equal to this expression.

Step 2: Identify the Domain

The function \(f(x) = \sqrt{x+2} - 1\) has a domain defined by the condition that the expression inside the square root must be non-negative. Therefore, \(x + 2 \geq 0\) which simplifies to \(x \geq -2\). This means the graph and the shaded region will start from \(x = -2\) and continue to the right.

Step 3: Graph the Boundary

Graph the boundary line of the inequality, which is \(y = \sqrt{x+2} - 1\). Since \(y\) can equal this expression, you draw a solid line. Start by making a table of values. For example: \((-2, -1), (2, 1), (7, 2)\). These points help in plotting the graph accurately.

Step 4: Plot Key Points and Sketch the Curve

Using the points calculated in Step 3, plot them on a coordinate plane. \((-2, -1)\), \((2, 1)\), and \((7, 2)\) are some points to plot. Connect these points smoothly, considering the shape of a square root function, which typically rises slowly and then more steeply.

Step 5: Shade the Appropriate Region

Now, since the inequality is \(y \geq \sqrt{x+2} - 1\), shade the region above the curve. This represents all the points where \(y\) is greater than or equal to \(\sqrt{x+2} - 1\).

Step 6: Verify the Graph

Choose a test point not on the boundary to verify correctness. A point like \((0, 2)\) can be used. Substitute into the inequality: \(2 \geq \sqrt{0+2} - 1\), which simplifies to \(2 \geq \sqrt{2} - 1\), holding true, validates the shaded region being correct.

Key Concepts

Understanding Inequality

Inequalities are expressions that use symbols like \(>\), \(<\), \(\geq\), and \(\leq\) to compare two values. They show where one side is larger or smaller than the other. This is important for representing conditions in real-world scenarios, such as budgets or temperature ranges.
\[\text{For example:}\ y \geq \sqrt{x+2} - 1\] describes all the points where \(y\) is larger than or exactly \(\sqrt{x+2} - 1\).
To solve and graph an inequality, it's crucial to:

• Identify the boundary, where the values are equal. Here it's given by the equation \(y = \sqrt{x+2} - 1\).
• Determine the regions above or below the line according to the inequality.
• Use these concepts correctly to shade the region that satisfies the inequality. Be sure to use a solid line to include the boundary within this region.

Understanding these elements helps in visualizing mathematical situations effectively.

Exploring Quadratic Functions

Quadratic functions are equations where the highest power of the variable is 2. This results in a parabolic curve when graphed, either opening upwards or downwards. A fundamental feature of quadratic functions is their general form \( ax^2 + bx + c \). Although in our original exercise the function is a square root, it behaves similarly in terms of alterations to the standard form.
The function \( f(x) = \sqrt{x+2} - 1 \) stems from the transformation of a quadratic-like curve:

• Shifting: Adding or subtracting from \( x \) or the entire function moves the curve's position in the coordinate plane.
• Rescaling: Multiplying or dividing changes the steepness.
• Morphing: Like with other transformations, the square root adjusts the curve from a quadratic.

By interpreting these functions accurately on the coordinate plane, you can assess solutions and capture representations of varied mathematical concepts.

Navigating the Coordinate Plane

The coordinate plane is a two-dimensional plane where we plot points using a pair of numbers: \(x\) and \(y\) coordinates. The horizontal line is the x-axis and the vertical line is the y-axis. This framework offers a visual way to solve equations and inequalities.
To graph the inequality \(y \geq \sqrt{x+2} - 1\), knowing how to navigate the coordinate plane ensures precision:

• Identify starting points and boundaries, here beginning at \(x = -2\) because \(x+2 \geq 0\).
• Smoothly connect points like \((-2, -1)\), \((2, 1)\), and \((7, 2)\) that are plotted according to any function derived values.
• Shade the region above this curve to encompass all points where \(y\) values fit the inequality \(y \geq f(x)\).

Proficiency in using the coordinate plane allows better visualization of mathematical relationships and easier problem-solving in other aspects of algebra and geometry.