A parabola is a curved, U-shaped line or surface that is the graph of a quadratic function. It is one of the fundamental conic sections, along with the circle, ellipse, and hyperbola. Parabolas have many important applications in mathematics, physics, and engineering.
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Parabolas are the graphs of quadratic functions, which are used to solve quadratic equations in the topics 9.1, 9.2, and 9.3.
The properties of parabolas, such as the vertex and axis of symmetry, are used to graph quadratic functions in topics 9.6 and 9.7.
Quadratic inequalities, which involve parabolas, are solved in topic 9.8.
Parabolas are one of the fundamental conic sections studied in topic 11.2.
Systems of nonlinear equations involving parabolas are solved in topic 11.5.
Review Questions
Explain how the concept of a parabola is connected to the topics of solving quadratic equations using the square root property, completing the square, and the quadratic formula.
The parabola is the graph of a quadratic function, which is the foundation for solving quadratic equations. In topics 9.1, 9.2, and 9.3, students learn different methods to solve quadratic equations, such as the square root property, completing the square, and the quadratic formula. These methods all rely on the properties of parabolas, as the solutions to the quadratic equations correspond to the x-intercepts of the parabolic graph.
Describe how the properties of a parabola, including the vertex and axis of symmetry, are used to graph quadratic functions in topics 9.6 and 9.7.
The vertex and axis of symmetry of a parabola are crucial in graphing quadratic functions. In topic 9.6, students learn to use the vertex and other properties, such as the y-intercept and any x-intercepts, to sketch the graph of a quadratic function. In topic 9.7, they explore how transformations, such as translations, reflections, and dilations, affect the shape and position of the parabolic graph.
Analyze how the properties of parabolas are applied to solve quadratic inequalities in topic 9.8 and systems of nonlinear equations involving parabolas in topic 11.5.
The understanding of parabolas is essential for solving quadratic inequalities in topic 9.8. The graph of a quadratic inequality is a parabolic region, and the solutions are determined by the points where the parabola intersects the x-axis. In topic 11.5, students encounter systems of nonlinear equations that may involve parabolas. Analyzing the properties of the parabolic graphs, such as their vertices and axes of symmetry, is crucial for finding the points of intersection between the parabola and other curves, which represent the solutions to the system of equations.
A quadratic function is a polynomial function of degree two, written in the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are real numbers and $a \neq 0$.