A quadratic function is a polynomial function of degree two, where the highest exponent of the variable is two. These functions are characterized by a U-shaped graph called a parabola and are widely used in various mathematical and scientific applications.
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Quadratic functions can be written in the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are real numbers, and $a \neq 0$.
The graph of a quadratic function is a parabola, which can open upward or downward depending on the sign of the coefficient $a$.
The vertex of a parabola represents the minimum or maximum value of the quadratic function, and its coordinates can be found using the formula $x = -b/(2a)$.
The axis of symmetry of a parabola is the vertical line that passes through the vertex, and its equation is $x = -b/(2a)$.
Quadratic functions can be transformed by shifting, stretching, or reflecting the graph using various transformations, such as translations, reflections, and dilations.
Review Questions
Explain how a quadratic function is represented and how its graph, the parabola, is characterized.
A quadratic function is represented in the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are real numbers, and $a \neq 0$. The graph of a quadratic function is a parabola, which is a U-shaped curve. The parabola can open upward or downward, depending on the sign of the coefficient $a$. The vertex of the parabola represents the minimum or maximum value of the function, and its coordinates can be found using the formula $x = -b/(2a)$. The axis of symmetry of the parabola is the vertical line that passes through the vertex, and its equation is $x = -b/(2a)$.
Describe how quadratic functions can be transformed using various transformations, and explain the effects of these transformations on the graph of the function.
Quadratic functions can be transformed by applying various transformations, such as translations, reflections, and dilations. Translating the graph of a quadratic function horizontally or vertically will shift the parabola to a new position without changing its shape. Reflecting the graph of a quadratic function about the $x$-axis or $y$-axis will change the orientation of the parabola, causing it to open in the opposite direction. Dilating the graph of a quadratic function will stretch or compress the parabola, changing its width without affecting its overall shape. These transformations allow for the creation of a wide variety of quadratic function graphs, which can be useful in modeling real-world situations and solving problems.
Analyze how the properties of a quadratic function, such as the vertex, axis of symmetry, and concavity, can be used to graph the function and understand its behavior.
The properties of a quadratic function, including the vertex, axis of symmetry, and concavity, can be used to graph the function and understand its behavior. The vertex represents the minimum or maximum value of the function, and its coordinates can be found using the formula $x = -b/(2a)$. The axis of symmetry, which is the vertical line passing through the vertex, can also be determined using the formula $x = -b/(2a)$. The concavity of the parabola, which indicates whether the function is opening upward or downward, is determined by the sign of the coefficient $a$. If $a > 0$, the parabola opens upward, and if $a < 0$, the parabola opens downward. By understanding these properties, one can effectively graph quadratic functions and analyze their behavior, which is crucial in various mathematical and scientific applications.
The point on a parabola where the function changes from increasing to decreasing or vice versa, representing the minimum or maximum value of the function.