Notes on Quadratic Functions and Applications

Quadratic Functions and Applications

• Classification of the graph of a function

  • Understand how to represent quadratic functions graphically.
  • Identify key characteristics: vertex, intercepts, and axis of symmetry.

• Basic Properties of the Quadratic Function

  • A quadratic function is of the form
    f(x) = ax^2 + bx + c
    where $a \neq 0$.
  • Direction of Opening:
  • Opens upwards if $a > 0$.
  • Opens downwards if $a < 0$.

• Vertex of a Parabola

  • The vertex form of a quadratic function can be expressed as:
    f(x) = a(x - h)^2 + k
  • Here, $(h, k)$ is the vertex of the parabola.
  • The vertex represents the minimum or maximum point depending on the direction of opening.

• Axis of Symmetry

  • The equation of the axis of symmetry is given by:
    x = h,
  • This is a vertical line that vertically bisects the parabola.

• Key Characteristics to Identify from Graphs

  • a. Does the parabola open upward or downward?
  • b. Find the coordinates of the vertex ($h, k$).
  • c. Find the equation of the axis of symmetry ($x = h$).
  • d. Find any x-intercept(s).
  • e. Find the y-intercept by evaluating $f(0)$.
  • f. Find the domain (typically $(-\infty, \infty)$).
  • g. Find the range, which varies based on whether it is a minimum or maximum.

• Graphing Quadratic Functions

  • To graph $y = (x - h)^2 + k$, identify the vertex and plot points symmetric to the vertex.
  • For example, for $f(x) = (x - 3)^2 + 2$:
  • Vertex: $(3, 2)$.
  • Plot points left and right of the vertex to complete the shape.

• Factoring Quadratic Functions

  • When $ax^2$ has a leading coefficient greater than 1, use factoring methods.
  • Example: $10x^2 + 27x + 5$.

• Finding Intercepts of a Parabola

  • To find x-intercepts, set $f(x) = 0$ and solve for $x$.
  • For y-intercepts, evaluate $f(0)$.

• Vertex Calculation

  • For a quadratic function of the form $f(x) = ax^2 + bx + c$, the x-coordinate of the vertex is calculated as:
    x = -\frac{b}{2a}
  • The y-coordinate can be found by substituting this $x$ value back into the function.

• Maximum or Minimum Values

  • The maximum or minimum value occurs at the vertex.
  • If the parabola opens upwards, it has a minimum at the vertex (value is $f(-\frac{b}{2a})$).
  • If it opens downwards, it has a maximum.

• Word Problems Involving Quadratic Functions

  • Example: Height of a ball after $t$ seconds can be modeled with a quadratic function.
  • Find maximum height using vertex methods.

• Writing a Quadratic Function from Its Graph

  • To find the function $g(x)$ from a vertex form:
    g(x) = a(x - h)^2 + k
  • Use known points to solve for $a$.

• Solving Quadratic Inequalities

  • Rearrange to set one side to 0 and factor to find x-intercepts.
  • Determine intervals based on the direction of opening, shading above or below the x-axis as appropriate.

• Examples of Solving Quadratic Inequalities

  • Solve $2x^2 + x - 15 < 0$ and use graphical representation to visualize solutions.