5.1 Quadratic Functions
3 min read•june 24, 2024
Quadratic functions and parabolas are essential in algebra, describing U-shaped curves with unique properties. These functions have wide-ranging applications, from modeling projectile motion to optimizing business decisions.
Understanding the features of parabolas, like the and , is key to graphing and analyzing quadratic functions. Mastering techniques for solving quadratic equations opens doors to more advanced mathematical concepts and real-world problem-solving.
Quadratic Functions and Parabolas
Features of parabolas
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- U-shaped curve symmetrical about a vertical line called the
- Defined by a in the form
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- Turning point of the represents the minimum or
- Coordinates can be found using the formula for
- Axis of symmetry
- Vertical line passing through the vertex divides the parabola into two equal halves
- Equation of the axis of symmetry is for
- Direction of opening
- Parabola opens upward () if the is positive ()
- Parabola opens downward () if the is negative ()
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- Points where the parabola intersects the x-axis, also known as x-intercepts or zeros of the function
Graphing quadratic functions
- :
- Leading coefficient determines the direction of opening and width of the parabola
- Coefficient affects the axis of symmetry and x-coordinate of the vertex
- Constant term represents the and shifts the parabola vertically
- Graphing steps
- Identify the direction of opening based on the sign of the leading coefficient
- Find the vertex coordinates using the formula
- Plot the y-intercept point on the y-axis
- Calculate additional points by substituting x-values into the
- Connect the plotted points to form the parabolic curve
Extrema of quadratic functions
- Vertex represents the minimum or () of the quadratic function
- If , the vertex is a (lowest value of the function)
- If , the vertex is a maximum point (highest value of the function)
- Finding the minimum or maximum value
- Calculate the x-coordinate of the vertex using the formula
- Substitute the x-coordinate into the quadratic function to find the corresponding y-coordinate
- Interpreting the extrema
- is the lowest output value of the function ()
- Maximum value is the highest output value of the function ()
Applications of quadratic optimization
- Optimization problems involve finding the minimum or maximum value of a quadratic function in real-world scenarios
- Steps to solve optimization problems
- Identify given information and the quantity to be optimized (maximized or minimized)
- Define variables and express the quantity as a quadratic function in terms of the variables
- Find the minimum or maximum value of the quadratic function using vertex formula
- Interpret the result in the context of the original problem
- Examples of
- Maximizing the area of a rectangle with a fixed perimeter (fencing problem)
- Minimizing the cost of production while maximizing profit (business optimization)
- Determining the optimal dimensions of a container to minimize surface area (packaging design)
Solving Quadratic Equations
- : A method to find by expressing the quadratic equation as a product of linear factors
- : A technique to rewrite the quadratic equation in vertex form, useful for finding the vertex and solving equations
- : A general formula to find roots of any quadratic equation, derived from the method
- : A value that determines the nature of the roots (real and distinct, real and repeated, or complex) in the
Key Terms to Review (38)
Axis of symmetry: The axis of symmetry is a vertical line that divides a parabola into two mirror-image halves. It passes through the vertex of the parabola and has the equation $x = -\frac{b}{2a}$ for a quadratic function in standard form.
Axis of Symmetry: The axis of symmetry is a line that divides a symmetrical figure, such as a parabola or absolute value function, into two equal halves. It represents the midpoint or line of reflection for the function, where the left and right sides are mirror images of each other.
Co-vertex: The co-vertices of an ellipse are the endpoints of the minor axis. They are perpendicular to and lie at the midpoint of the major axis.
Completing the square: Completing the square is a method to solve quadratic equations by converting them into a perfect square trinomial. This facilitates easier solving and helps in deriving the quadratic formula.
Completing the Square: Completing the square is a technique used to solve quadratic equations and transform quadratic functions into a more useful form. It involves rearranging a quadratic expression into a perfect square plus or minus a constant, allowing for easier analysis and manipulation of the equation or function.
Concave Down: Concave down is a term used to describe the shape of a quadratic function when its graph curves downward, forming a bowl-like shape. This type of function has a negative leading coefficient, indicating that the function decreases as the input variable increases.
Concave Up: Concave up is a term used to describe the shape of a curve on a graph, where the curve bends upward, forming a U-shape. This term is particularly relevant in the context of quadratic functions and exponential functions, as it describes the overall shape and behavior of these types of functions.
Discriminant: The discriminant is a value calculated from the coefficients of a quadratic equation $ax^2 + bx + c = 0$. It determines the nature and number of roots of the quadratic equation.
Discriminant: The discriminant is a value that determines the nature of the solutions to a quadratic equation. It provides information about the number and type of solutions, and is a crucial concept in the study of quadratic functions and the rotation of axes.
Extremum: An extremum, in the context of mathematics, refers to a point at which a function reaches either a maximum or a minimum value. It is a critical point where the function's derivative is equal to zero or undefined, and the function's value is greater than or less than the values in the immediate vicinity.
Factoring: Factoring is the process of breaking down an expression into a product of simpler expressions, often polynomials. It simplifies solving equations by expressing them as a product of factors.
Factoring: Factoring is the process of breaking down a polynomial or algebraic expression into a product of smaller, simpler expressions. It involves identifying common factors and using various techniques to express a polynomial as a product of its factors. Factoring is a fundamental algebraic skill that is essential for understanding and manipulating polynomials, rational expressions, quadratic equations, and other types of equations and functions.
General form of a quadratic function: The general form of a quadratic function is expressed as $ax^2 + bx + c = 0$ where $a$, $b$, and $c$ are constants and $a \neq 0$. This representation is crucial for solving quadratic equations and analyzing their properties.
Global maximum: A global maximum is the highest point over the entire domain of a function. For polynomial functions, it is where the function attains its greatest value.
Global Maximum: The global maximum of a function is the highest point or value that the function attains over its entire domain. It represents the absolute maximum value of the function, as opposed to a local maximum which is the highest point within a specific region of the function's graph.
Global minimum: The global minimum of a function is the lowest point over its entire domain. It represents the smallest value that the function can take.
Global Minimum: The global minimum of a function is the point on the function's graph where the function achieves its absolute lowest value. It represents the point at which the function reaches its global or overall minimum, as opposed to a local minimum which is the lowest point in a particular region of the graph.
Leading coefficient: The leading coefficient of a polynomial is the coefficient of the term with the highest degree. It plays a crucial role in determining the end behavior of the polynomial function.
Leading Coefficient: The leading coefficient of a polynomial is the numerical coefficient of the term with the highest degree. It is the first, or leading, coefficient in the standard form of a polynomial expression. The leading coefficient plays a crucial role in understanding and analyzing various topics in college algebra, including polynomials, quadratic equations, functions, and more.
Maximum Point: The maximum point of a quadratic function is the vertex of the parabola, which represents the highest or lowest point of the function's graph. It is the point where the function changes from increasing to decreasing or vice versa, and it is the point at which the function reaches its maximum or minimum value.
Maximum value: The maximum value of a quadratic function refers to the highest point on its graph, which occurs at its vertex if the parabola opens downward. It is relevant when the leading coefficient of the quadratic term is negative.
Minimum Point: The minimum point of a quadratic function is the point on the graph where the function reaches its lowest value. It represents the vertex of the parabolic curve, which is the point where the function changes from decreasing to increasing or vice versa.
Minimum value: The minimum value of a quadratic function is the lowest point on its graph, which occurs at the vertex if the parabola opens upwards. It is found using the vertex form or by completing the square.
Parabola: A parabola is a symmetric curve that represents the graph of a quadratic function. It can open upward or downward depending on the sign of the quadratic coefficient.
Parabola: A parabola is a curved, U-shaped line that is the graph of a quadratic function. It is one of the fundamental conic sections, along with the circle, ellipse, and hyperbola, and has many important applications in mathematics, science, and engineering.
Quadratic formula: The quadratic formula is used to find the roots of a quadratic equation of the form $ax^2 + bx + c = 0$. It is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
Quadratic Formula: The quadratic formula is a mathematical equation used to solve quadratic equations, which are polynomial equations of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are real numbers and $a \neq 0$. This formula provides a systematic way to find the solutions, or roots, of a quadratic equation.
Quadratic function: A quadratic function is a polynomial function of degree 2, typically expressed in the form $f(x) = ax^2 + bx + c$ where $a$, $b$, and $c$ are constants and $a \neq 0$. The graph of a quadratic function is a parabola that opens either upwards or downwards.
Quadratic Function: A quadratic function is a polynomial function of degree two, where the highest exponent of the independent variable is two. Quadratic functions are widely used in various mathematical and scientific applications, including physics, engineering, and economics.
Quadratic Optimization: Quadratic optimization is the process of finding the optimal solution to a problem involving a quadratic objective function, subject to linear or nonlinear constraints. It is a fundamental concept in mathematical programming and has applications in various fields, including engineering, economics, and decision-making.
Roots: Roots of a polynomial are the values of the variable that make the polynomial equal to zero. They are also known as solutions or zeros of the equation.
Roots: In mathematics, the term 'roots' refers to the solutions or values of a polynomial equation that make the equation equal to zero. Roots are an essential concept in various topics related to polynomial functions and equations, including quadratic equations, power functions, and the graphs of polynomial functions.
Standard form: Standard form of a linear equation in one variable is written as $Ax + B = 0$, where $A$ and $B$ are constants and $x$ is the variable. The coefficient $A$ should not be zero.
Standard Form: Standard form is a way of expressing mathematical equations or functions in a specific, organized format. It provides a consistent structure that allows for easier manipulation, comparison, and analysis of these mathematical representations across various topics in algebra and beyond.
Standard form of a quadratic function: The standard form of a quadratic function is given by $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants and $a \neq 0$. This form is useful for identifying the coefficients directly.
Vertex: The vertex is the highest or lowest point on the graph of a quadratic function. It represents the maximum or minimum value of the function.
Vertex: The vertex is a critical point in various mathematical functions and geometric shapes. It represents the point of maximum or minimum value, or the point where a curve changes direction. This term is particularly important in the context of quadratic equations, functions, absolute value functions, and conic sections such as the ellipse and parabola.
Y-intercept: The y-intercept is the point at which a line or curve intersects the y-axis, representing the value of the dependent variable (y) when the independent variable (x) is zero. It is a crucial concept in understanding the behavior and properties of various mathematical functions and their graphical representations.