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Magazine Chapter 9: Problem 11
Graph the inequality. $$y>x^{2}+1$$
Short Answer
Expert verified
Shade the region above the parabola \(y = x^2 + 1\) with a dashed boundary line, indicating \(y > x^2 + 1\).
Step by step solution
01
Understand the inequality
The inequality given is \( y > x^2 + 1 \). This represents a region above the parabola defined by the equation \( y = x^2 + 1 \). Our task is to graph this inequality, which means identifying and plotting the region where \( y \) is greater than \( x^2 + 1 \).
02
Plot the boundary
First, plot the boundary curve using the equation \( y = x^2 + 1 \). Since the inequality is \( y > x^2 + 1 \), the line itself will be dashed in the graph, indicating that points on the line are not included in the solution.
03
Draw the parabola
The equation \( y = x^2 + 1 \) represents a parabola that opens upwards with a vertex at the point \((0, 1)\). Sketch this parabola on the coordinate plane using a dashed line. At \( x = 0 \), \( y = 1 \). At \( x = \pm 1 \), \( y = 2 \), and so on.
04
Determine the region to shade
Since we need to show where \( y \) is greater than \( x^2 + 1 \), shade the region above the parabola. This area represents all the \((x, y)\) pairs that satisfy the inequality \( y > x^2 + 1 \).
05
Verify with a test point
Pick a test point not on the parabola to ensure it satisfies the inequality. For example, test the point \((0, 2)\). Substitute this into the inequality: \( 2 > 0^2 + 1 \), resulting in \(2 > 1\), which is true. Thus, confirming the correct region is shaded.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Parabolas
A parabola is a U-shaped curve that you often encounter in mathematics. It's defined by the equation of the form \(y = ax^2 + bx + c\). In the exercise, we are focusing on the parabola given by the equation \(y = x^2 + 1\). This specific parabola is simple because it has a few notable properties:
- It opens upwards, which means the arms of the curve extend upwards from the vertex.
- It is symmetric about the vertical line \(x=0\), also known as the axis of symmetry.
- Because \(a=1\) is positive, the parabola does not flip upside down.
Coordinate Plane
The coordinate plane is a two-dimensional surface where each point is defined by a pair of numerical coordinates. These coordinates are usually denoted as \(x, y\). The horizontal line is called the x-axis, while the vertical one is the y-axis. Both axes intersect at the origin \(0, 0\).
- To graph inequalities or functions, you plot points on this plane using their x and y values.
- For example, in the formula \(y = x^2 + 1\), you substitute different x-values to find corresponding y-values and plot them as points.
Inequality Graphing
Inequality graphing involves plotting a range of solutions that satisfy a given inequality. Unlike equations, where we graph specific solutions, inequalities often have multiple solutions. Here's how it works:
- You start by plotting the boundary from the inequality. In our case, the boundary is given by \(y = x^2 + 1\).
- Instead of a solid line, use a dashed line to represent this boundary if the inequality symbol is "greater than" (\(>\)) or "less than" (\(<\)), indicating points on the line are not included in the solution.
- Shade the region of the graph that satisfies the inequality. For \(y > x^2 + 1\), shade above the parabola since we're considering points \(y\) greater than the boundary.
Vertex of a Parabola
The vertex of a parabola is a significant point that represents the parable's maximum or minimum value.For the equation \(y = ax^2 + bx + c\), the vertex can be found using the formula: \(x = -\frac{b}{2a}\). However, for simple equations like \(y = x^2 + 1\), the vertex can often be easily identified:
- Since \(b = 0\) in \(y = x^2 + 1\), the vertex occurs at \(x = 0\).
- Substitute \(x = 0\) into the equation to find \(y = 1\). Thus, the vertex is \( (0, 1) \).
