Quadratic functions

Understanding Quadratic Functions

  • A quadratic function is a function that can be described by an equation of the form f(x) = ax^2 + bx + c, where a, b, and c are constants.
  • The graph of a quadratic function is a curve called a parabola. Parabolas may open upwards or downwards and vary in width and direction.
  • The highest or lowest point of a parabola, known as the vertex, can be found at the point (-b/2a , f(-b/2a)).
  • If the coefficient a in the function f(x) = ax^2 + bx + c is positive, the parabola opens upward, and if a is negative, the parabola opens downward.
  • The y-intercept of a quadratic function is the point where the parabola crosses the y-axis. This is always at f(x) = c.

Solving Quadratic Functions

  • Quadratic functions can be solved by factoring, completing the square, or using the quadratic formula.
  • Factoring a quadratic function involves breaking it down into two binomial expressions set to zero and then finding the x-values that make the expressions equal to zero.
  • The quadratic formula, x = [ -b ± sqrt(b^2 - 4ac)] / 2a, can be used to find the roots of the quadratic equation when it can’t be easily factored.

Properties of Quadratic Functions

  • The axis of symmetry of a parabola is the vertical line x = -b/2a. This is the line that can divide the parabola into two mirror images.
  • The discriminant in the quadratic formula, denoted by b^2 - 4ac, can determine the number and type of solutions for a quadratic equation.
    • If the discriminant is greater than 0, there are two distinct real solutions.
    • If the discriminant is equal to 0, there is exactly one real solution.
    • If the discriminant is less than 0, there are two complex solutions.
  • The roots or zeros of a quadratic function are the x-values at which the function equals zero (i.e., where the function intersects the x-axis). These can be found by solving the quadratic equation f(x) = 0.
  • The maximum or minimum value of a quadratic function occurs at its vertex. This value is the y-coordinate of the vertex.

Transformations of Quadratic Functions

  • A quadratic function can be transformed by translation, reflection, and stretching or compressing.
  • Translation moves the graph of the function left, right, up, or down in the coordinate plane. This is achieved by adding or subtracting values to x or f(x) in the equation.
  • Reflection ‘flips’ the graph over the x-axis or y-axis. Reflection over the x-axis is achieved by multiplying the whole function by -1.
  • Stretching or compressing changes the ‘width’ of the parabola. This is carried out by multiplying x or f(x) by a constant.
  • The order in which transformations are applied can affect the final outcome.