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Chapter 17: Problem 37

Use a calculator to display the solution of the given inequality or system of inequalities. $$y \geq 1-x^{2}$$

Short Answer

Expert verified
The solution is the region above and on the parabola \( y = 1 - x^2 \).

Step by step solution

01

Understand the Inequality

We have the inequality \( y \geq 1 - x^{2} \). This represents a region on the coordinate plane where the \( y \) values are greater than or equal to the values on the parabola \( y = 1 - x^2 \).
02

Graph the Equation

To solve the inequality \( y \geq 1 - x^2 \), first plot the graph of \( y = 1 - x^2 \). This is a downward-facing parabola with a vertex at \((0, 1)\). The parabola intersects the x-axis at \((1, 0)\) and \((-1, 0)\).
03

Test a Point

Choose a test point not on the boundary of \( y = 1 - x^2 \), such as \((0, 2)\). Substitute into the inequality to verify if \( 2 \geq 1 - 0^2 \) holds true. This simplifies to \(2 \geq 1\), which is correct.
04

Shade the Valid Region

Since the test point yields a true statement, the solution region lies above and on the parabola defined by \( y = 1 - x^2 \). This includes the vertex, the parabola itself, and the area above the curve.
05

Use a Calculator

To verify, use a graphing calculator to plot \( y = 1 - x^2 \) and validate that the region above including the curve confirms the inequality \( y \geq 1 - x^2 \). Ensure you use the appropriate function to test values in this region if needed.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing
Graphing is a powerful method in mathematics that helps visualize equations and inequalities on a coordinate plane. By plotting graphs, you can easily see where certain conditions, such as inequalities, are met. Let's break it down:
  • Start by identifying the equation or inequality you need to graph. In our case, it's the inequality \( y \geq 1 - x^2 \).
  • The next step is to plot the parabola described by the equation \( y = 1 - x^2 \) since the inequality is concerned with values relative to this curve.
  • Once the graph of the curve is drawn, determine which side of the parabola satisfies the inequality by testing points.
This process is crucial because it helps you find regions on the graph where conditions hold true, like in our inequality where the solution set is everything on and above the parabola. Always remember to use a calculator or graphing software to verify your plot, checking that the shaded region aligns with the mathematical solution.
Parabola
A parabola is a U-shaped curve that can open upwards or downwards depending on the equation. It is a common feature in quadratic equations of the form \( y = ax^2 + bx + c \). In this exercise, we deal with a specific type of parabola:
  • The function \( y = 1 - x^2 \) is a downward-facing parabola. This means it curves downwards as it moves away from the vertex.
  • The vertex of this parabola is at the point (0, 1). This is the peak of the curve when it faces downwards.
  • The parabola intersects the x-axis at (1, 0) and (-1, 0), where the curve crosses the horizontal axis.
Understanding parabolas is essential, as they frequently appear in various mathematical contexts. To solve inequalities involving parabolas, you must accurately graph them and understand how different points relate to the vertex and axis of symmetry.
Coordinate Plane
The coordinate plane is a two-dimensional surface formed by the intersection of a horizontal line called the x-axis and a vertical line called the y-axis. It's where we graph equations and visualize their solutions. Here's how you can use it effectively:
  • Each point on the plane is defined by a pair of numbers (x, y), known as coordinates. For example, the origin is at (0, 0).
  • When graphing, choose a range for the axes that accommodate the essential features of your function, such as the vertex and endpoints of a parabola.
  • The coordinate plane allows you to visually find solutions to inequalities, such as identifying areas where one side of an equation is greater than or less than another.
Working with the coordinate plane provides a clear visual context for solutions and is a fundamental skill in both algebra and geometry. It helps in interpreting the position and path of various functions, supporting deeper mathematical insight.

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Most popular questions from this chapter

Use a calculator to display the solution of the given inequality or system of inequalities. $$\begin{aligned} &y>x^{2}+2 x-8\\\ &y<\frac{1}{x}-2 \end{aligned}$$

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