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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 vertex and axis of symmetry, 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

  • Parabola
    • U-shaped curve symmetrical about a vertical line called the axis of symmetry
    • Defined by a quadratic function in the form $f(x)=ax^2+bx+c$
  • Vertex
    • Turning point of the parabola represents the minimum or maximum point
    • Coordinates can be found using the formula $(\frac{-b}{2a}, f(\frac{-b}{2a}))$ for $f(x)=ax^2+bx+c$
  • Axis of symmetry
    • Vertical line passing through the vertex divides the parabola into two equal halves
    • Equation of the axis of symmetry is $x=\frac{-b}{2a}$ for $f(x)=ax^2+bx+c$
  • Direction of opening
    • Parabola opens upward (concave up) if the leading coefficient $a$ is positive ($a>0$)
    • Parabola opens downward (concave down) if the leading coefficient $a$ is negative ($a<0$)
  • Roots
    • Points where the parabola intersects the x-axis, also known as x-intercepts or zeros of the function

Graphing quadratic functions

  • Standard form of a quadratic function: $f(x)=ax^2+bx+c$
    • Leading coefficient $a$ determines the direction of opening and width of the parabola
    • Coefficient $b$ affects the axis of symmetry and x-coordinate of the vertex
    • Constant term $c$ represents the y-intercept and shifts the parabola vertically
  • Graphing steps
    1. Identify the direction of opening based on the sign of the leading coefficient $a$
    2. Find the vertex coordinates using the formula $(\frac{-b}{2a}, f(\frac{-b}{2a}))$
    3. Plot the y-intercept point $(0, c)$ on the y-axis
    4. Calculate additional points by substituting x-values into the quadratic function
    5. Connect the plotted points to form the parabolic curve

Extrema of quadratic functions

  • Vertex represents the minimum or maximum value (extremum) of the quadratic function
    • If $a>0$, the vertex is a minimum point (lowest value of the function)
    • If $a<0$, the vertex is a maximum point (highest value of the function)
  • Finding the minimum or maximum value
    1. Calculate the x-coordinate of the vertex using the formula $x=\frac{-b}{2a}$
    2. Substitute the x-coordinate into the quadratic function to find the corresponding y-coordinate
  • Interpreting the extrema
    • Minimum value is the lowest output value of the function (global minimum)
    • Maximum value is the highest output value of the function (global maximum)

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
    1. Identify given information and the quantity to be optimized (maximized or minimized)
    2. Define variables and express the quantity as a quadratic function in terms of the variables
    3. Find the minimum or maximum value of the quadratic function using vertex formula
    4. Interpret the result in the context of the original problem
  • Examples of quadratic optimization
    • 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

  • Factoring: A method to find roots by expressing the quadratic equation as a product of linear factors
  • Completing the square: A technique to rewrite the quadratic equation in vertex form, useful for finding the vertex and solving equations
  • Quadratic formula: A general formula to find roots of any quadratic equation, derived from the completing the square method
  • Discriminant: A value that determines the nature of the roots (real and distinct, real and repeated, or complex) in the quadratic formula