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Magazine Chapter 7: Problem 8
Graph each inequality. \(y \geq \sqrt{x+2}-1\)
Short Answer
Expert verified
Graph the function \(y = \sqrt{x+2} - 1\) as a solid curve starting from \(x = -2\) and shade the region above it.
Step by step solution
01
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.
02
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.
03
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.
04
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.
05
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 \(\sqrt{x+2} - 1\).
06
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.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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:
\[\text{For example:}\ y \geq \sqrt{x+2} - 1\] 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.
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:
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.
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:
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 \(y \geq f(x)\).
