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12.3 The Parabola

3 min read•june 24, 2024

Parabolas are U-shaped curves with fascinating properties. They're symmetrical, have a as their , and can open upward or downward. Understanding their equation is key to graphing and analyzing their behavior.

Parabolas have real-world applications in physics, engineering, and architecture. They're defined by key features like the and , and their shape is determined by the coefficient 'a' in their equation. Mastering parabolas opens doors to solving practical problems.

Parabola Fundamentals

Graphing parabolas

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  • U-shaped curves symmetrical about a line called the
    • passes through the vertex (, either or )
  • Open upward (vertex is minimum) or downward (vertex is maximum)
  • with vertex at origin: y=ax2y = ax^2
    • aa determines shape and orientation
      • a>0a > 0: opens upward
      • a<0a < 0: opens downward
  • General form with vertex at (h,k)(h, k): y=a(x−h)2+ky = a(x - h)^2 + k
    • (h,k)(h, k) represents vertex coordinates
    • Axis of symmetry is vertical line x=hx = h

Equations in standard form

  • : y=a(x−h)2+ky = a(x - h)^2 + k
    • aa determines shape and orientation
    • (h,k)(h, k) represents vertex coordinates
  • Writing equation in standard form:
    1. Identify vertex (h,k)(h, k)
    2. Determine aa based on shape and orientation
    3. Substitute hh, kk, and aa into standard form equation
  • If given in general form y=ax2+bx+cy = ax^2 + bx + c, complete the square to rewrite in standard form

Parabola Properties and Applications

Key features of parabolas

  • Vertex: turning point (minimum or maximum)
  • : point inside that defines shape
    • Lies on axis of symmetry
    • Distance from vertex to focus: 14a\frac{1}{4a} (aa is coefficient of x2x^2 in standard form)
  • : line perpendicular to axis of symmetry that defines shape
    • Distance from vertex to directrix also 14a\frac{1}{4a}
    • is set of all points equidistant from focus and directrix
  • : width at focus, equal to 1a\frac{1}{a} (aa is coefficient of x2x^2 in standard form)
  • : line segment passing through the focus, perpendicular to the axis of symmetry, with endpoints on the parabola

Applications of parabolas

  • Real-world applications:
    • Path of projectile under gravity ()
    • Shape of satellite dishes and car headlights
    • Design of suspension bridges and arches
  • Solving real-world problems:
    1. Identify given information and question to be answered
    2. Determine appropriate parabolic equation based on context
    3. Substitute given values into equation
    4. Solve equation to find desired quantity
  • Example: Ball thrown upward at 20 m/s from 1.5 m height, height after tt seconds: h(t)=−4.9t2+20t+1.5h(t) = -4.9t^2 + 20t + 1.5
    1. Vertex represents maximum height
    2. Equation in general form, complete square to find vertex
    3. Vertex at (209.8,101.54.9)\left(\frac{20}{9.8}, \frac{101.5}{4.9}\right), maximum height approximately 20.7 m

Parabolas as Conic Sections

  • Parabolas are a type of , formed by intersecting a plane with a cone
  • of a parabola is always equal to 1
  • : Light rays parallel to the axis of symmetry reflect off the parabola and pass through the focus

Key Terms to Review (44)

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.
Conic section: A conic section is a curve obtained by intersecting a cone with a plane. The primary types of conic sections are circles, ellipses, parabolas, and hyperbolas.
Conic Section: A conic section is a two-dimensional shape that is formed by the intersection of a plane and a three-dimensional cone. These shapes include circles, ellipses, parabolas, and hyperbolas, and they have numerous applications in mathematics, science, and engineering.
Constant Term: The constant term is a numerical value in a polynomial or equation that does not depend on any variable. It is the term that remains unchanged regardless of the values assigned to the variables in the expression.
Directrix: A directrix is a fixed line used in the geometric definition of a conic section. For a parabola, it is equidistant from any point on the curve to the focus and the directrix.
Directrix: The directrix is a fixed line used in the definition of a conic section, such as an ellipse, hyperbola, or parabola. It serves as a reference point in the geometric construction and mathematical description of these curves.
Eccentricity: Eccentricity measures the deviation of a conic section from being circular. It is denoted by $e$ and determines the shape of the conic.
Eccentricity: Eccentricity is a measure of the shape or deviation of a conic section from a perfect circle. It is a dimensionless quantity that describes the elongation or flattening of a conic section, such as an ellipse, hyperbola, or parabola, and is a fundamental property that characterizes these geometric shapes.
Focal Length: Focal length is a measure of the distance between the optical center of a lens or curved mirror and the point at which light rays converge or diverge. It is a fundamental property that determines the magnification and field of view of an optical system, and plays a crucial role in the behavior of lenses, mirrors, and other optical devices.
Focal Width: Focal width, in the context of parabolas, refers to the distance between the vertex of a parabolic curve and its focus. The focus is the point at which all the rays parallel to the axis of symmetry converge after reflection from the parabolic surface. The focal width is a crucial parameter that determines the shape and properties of a parabola.
Focus: A focus (plural: foci) is a point used to define and describe conic sections such as ellipses, parabolas, and hyperbolas. In these shapes, distances to the focus have special geometric properties.
Focus: The focal point or point of concentration, the center of interest or activity. In the context of conic sections, the focus refers to a specific point that defines the shape and properties of these geometric figures.
General Form: The general form of an equation is a standardized way of expressing the equation that reveals its underlying structure and characteristics. This term is particularly relevant in the context of various mathematical functions and conic sections, as it allows for a concise and informative representation of these entities.
Horizontal Shift: A horizontal shift is a transformation of a function that moves the graph of the function left or right along the x-axis, without changing the shape or orientation of the graph. This concept is applicable to various types of functions, including transformations of functions, absolute value functions, exponential functions, trigonometric functions, and the parabola.
Latus rectum: The latus rectum of a parabola is the line segment perpendicular to the axis of symmetry that passes through the focus and whose endpoints lie on the parabola. It is used to describe certain geometric properties of parabolas.
Latus Rectum: The latus rectum is a line segment that passes through the focus of a conic section and is perpendicular to the major axis. It is an important geometric property that helps define the shape and characteristics of ellipses, hyperbolas, and parabolas.
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: In mathematics, a maximum refers to the highest point or value of a function, particularly in the context of quadratic functions represented by parabolas. This peak value is significant as it helps identify the behavior of the function, indicating where it reaches its greatest output. In a parabola that opens downward, the maximum point is also known as the vertex, representing the highest point on the graph.
Minimum: The minimum is the smallest or lowest value in a set of numbers or a function. It represents the point at which a quantity or variable reaches its lowest possible point or level.
Opening Downward: The term 'opening downward' refers to the orientation of a parabolic function, where the vertex of the parabola is positioned at the top, and the parabola opens downward, forming a concave-down shape. This is in contrast to a parabola that opens upward, which has a vertex at the bottom and a convex-up shape.
Opening Upward: The term 'opening upward' refers to the orientation or shape of a parabolic curve, where the curve opens or points in an upward direction on a coordinate plane. This characteristic of a parabola is a crucial aspect of understanding the behavior and properties of this type of quadratic function.
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.
Parabolic Reflector: A parabolic reflector is a concave, curved surface that is shaped like a parabola. It is commonly used in various applications, such as telescopes, headlights, and satellite dishes, to reflect and focus electromagnetic waves or light rays in a specific direction.
Parabolic Trajectory: A parabolic trajectory refers to the curved path taken by an object that is launched or projected into the air, such as a projectile or a ball. This trajectory is governed by the principles of gravity and follows a parabolic shape, which is a specific type of quadratic function.
Quadratic equation: A quadratic equation is a second-degree polynomial equation in a single variable, typically written as $ax^2 + bx + c = 0$, where $a \neq 0$. The solutions to the quadratic equation are known as the roots of the equation.
Quadratic Equation: A quadratic equation is a polynomial equation of the second degree, where the highest exponent of the variable is 2. These equations take the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are real numbers and $a$ is not equal to 0. Quadratic equations are fundamental in mathematics and have applications in various fields, including physics, engineering, and economics.
Reflective Property: The reflective property is a characteristic of certain geometric shapes, particularly parabolas, where the shape reflects or mirrors itself about a specific axis or line. This property has important implications in the study of parabolic curves and their applications.
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.
Turning point: A turning point is a point on the graph of a polynomial function where the graph changes direction from increasing to decreasing or vice versa. It occurs at local maxima or minima.
Turning Point: A turning point is a critical moment or event that marks a significant change or shift in direction, often serving as a pivotal point that can alter the course of something. This term is particularly relevant in the context of analyzing the behavior and characteristics of various mathematical functions, including polynomials, power functions, and parabolas.
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.
Vertex Form: The vertex form of a quadratic equation is a way to express the equation in a form that highlights the vertex of the parabolic graph. It is a useful representation that provides information about the maximum or minimum point of the parabola, as well as its orientation and symmetry.
Vertex form of a quadratic function: The vertex form of a quadratic function is given by $y = a(x-h)^2 + k$, where $(h, k)$ is the vertex of the parabola. It is useful for identifying the maximum or minimum point and the axis of symmetry.
Vertical shift: A vertical shift is a transformation that moves a graph up or down in the coordinate plane by adding or subtracting a constant to the function's output. It does not affect the shape of the graph, only its position.
Vertical Shift: Vertical shift refers to the movement of a graph or function up or down the y-axis, without affecting the shape or orientation of the graph. This transformation changes the y-intercept of the function, but leaves the x-intercepts and the overall shape unchanged.
X-Intercepts: The x-intercepts of a graph are the points where the graph crosses the x-axis, or where the function's y-value is equal to zero. They represent the values of x for which the function has a solution of y = 0, and they provide important information about the behavior and characteristics of the function.
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.
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12.4 Rotation of Axes