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Magazine Chapter 7: Problem 22
Graph each inequality. $$y<2-3 x^{2}$$
Short Answer
Expert verified
Shade the region below the dashed parabola \(y = 2-3x^2\).
Step by step solution
01
Understand the Inequality
We need to graph the inequality \(y < 2 - 3x^2\). The graph of this inequality consists of a region below the curve \(y = 2 - 3x^2\). Since the inequality is 'less than', it indicates the region below the parabola.
02
Graph the Equation of the Parabola
First, focus on the equation \(y = 2 - 3x^2\). This is a downward-opening parabola with its vertex at the point (0, 2). Plot the vertex and determine the shape by finding additional points. For instance, when \(x = 1\), \(y = -1\), and when \(x = -1\), \(y = -1\).
03
Draw the Boundary as a Dashed Line
Since the original inequality is \(y < 2 - 3x^2\) and not \(y \leq 2 - 3x^2\), the boundary (which is the parabola) is not included. Draw the parabola as a dashed curve to indicate that points on the parabola are not included in the solution set.
04
Shade the Region Below the Parabola
For the inequality \(y < 2 - 3x^2\), shade the region below the dashed parabola. This represents all the points \( (x,y) \) that satisfy the inequality, indicating where \(y\) is less than \(2 - 3x^2\). Sections below the curve are valid solutions.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic Functions
Quadratic functions are mathematical expressions of the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \). These functions describe the relationship between an independent variable \( x \) and a dependent variable \( y \) resulting in a curve called a parabola.
- The standard form of a quadratic function is \( y = ax^2 + bx + c \).
- The coefficient \( a \) determines whether the parabola opens upward (\( a > 0 \)) or downward (\( a < 0 \)).
- Quadratic functions are used in various fields, including physics, engineering, and economics, to model phenomena where relationships follow a parabolic trajectory.
Parabolas
Parabolas are unique curves that can either open upwards or downwards, forming the graph of a quadratic function such as \( y = ax^2 + bx + c \).
- The vertex of the parabola is a critical point that represents the peak or the lowest point of the curve. For a parabola in standard form, the vertex can be found using the formula \( (-\frac{b}{2a}, f(-\frac{b}{2a})) \).
- The axis of symmetry is a vertical line that runs through the vertex, where the parabola is mirrored. Its equation is \( x = -\frac{b}{2a} \).
- The direction of the parabola is determined by the sign of \( a \). If \( a > 0 \), it opens upwards, and if \( a < 0 \), it opens downwards.
Inequalities
Inequalities express a relationship between two expressions that may not be equal. In mathematical terms, they use symbols like \(<\), \(>\), \(\leq\), and \(\geq\) to form statements about the relative sizes or values of two expressions.
- Graphing inequalities involves defining which side of a particular boundary, often a line or curve, includes the solution set.
- In our exercise, the inequality \( y < 2 - 3x^2 \) instructs us to find the values of \( y \) that are less than the ones calculated by the function \( y = 2 - 3x^2 \).
- The boundary is represented by a dashed line when the inequality is strict (\(<\), \(>\)) and a solid line when it is inclusive (\(\leq\), \(\geq\)).
Graphical Representation
Graphical representation brings to life the solutions to equations and inequalities by depicting them on a coordinate plane. It simplifies complex algebraic concepts into a visual form that is often easier to interpret.
- Each point on the graph corresponds to a pair of \( x \) and \( y \) values that satisfy the equation or inequality.
- The parabola \( y = 2 - 3x^2 \) serves as a boundary. Because the inequality is \( y < 2 - 3x^2 \), a dashed line indicates that this boundary is not included in the solution.
- Shading the region below the parabola helps to visually delineate all the \( (x, y) \) pairs that satisfy the inequality.
