A quadratic function is a polynomial function of degree two, meaning it contains a variable raised to the power of two. These functions are characterized by a parabolic shape 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 leading coefficient $a$.
The vertex of a parabola represents the minimum or maximum value of the function, and its coordinates can be found using the formula $x = -b/2a$.
Quadratic functions have a domain of all real numbers and a range that depends on the sign of the leading coefficient $a$.
Quadratic functions are widely used in various fields, such as physics, engineering, and economics, to model phenomena involving acceleration, projectile motion, and optimization problems.
Review Questions
Explain how the general form of a quadratic function, $f(x) = ax^2 + bx + c$, relates to the properties of the function, such as its graph and critical points.
The coefficients $a$, $b$, and $c$ in the general form of a quadratic function, $f(x) = ax^2 + bx + c$, directly impact the properties of the function. The sign of the leading coefficient $a$ determines the orientation of the parabolic graph, with $a > 0$ resulting in an upward-opening parabola and $a < 0$ resulting in a downward-opening parabola. The coefficient $b$ affects the horizontal shift of the parabola, and the constant term $c$ determines the vertical shift. The vertex of the parabola, which represents the critical point of the function, can be found using the formula $x = -b/2a$, highlighting the relationship between the coefficients and the function's key features.
Describe how the domain and range of a quadratic function are determined and how they relate to the function's graph.
The domain of a quadratic function is the set of all real numbers, as the function is defined for any real input value. The range of a quadratic function, however, depends on the sign of the leading coefficient $a$. If $a > 0$, the function has a minimum value, and the range is $[f(x_\text{min}), \infty)$. If $a < 0$, the function has a maximum value, and the range is $(-\infty, f(x_\text{max})]$. The vertex of the parabolic graph represents the point where the function changes from increasing to decreasing or vice versa, corresponding to the minimum or maximum value of the function and the endpoints of the range.
Analyze how quadratic functions can be used to model real-world situations, and explain the significance of the function's parameters in these applications.
Quadratic functions are widely used to model various real-world phenomena, such as projectile motion, optimization problems, and economic models. In these applications, the coefficients $a$, $b$, and $c$ in the function $f(x) = ax^2 + bx + c$ hold important meaning. The leading coefficient $a$ represents the rate of change or acceleration, which is crucial in modeling motion and optimization problems. The coefficient $b$ reflects the initial conditions or starting values, while the constant term $c$ represents the offset or baseline value. By understanding the relationships between the function's parameters and the real-world context, we can use quadratic functions to make accurate predictions, optimize processes, and gain insights into complex systems.
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.
A function that is the sum of a finite number of terms, each of which consists of a variable raised to a non-negative integer power and multiplied by a coefficient.