Quadratic Functions and Their Properties

Introduction to Quadratic Functions

• Quadratic functions are expressed in the form: f(x) = ax^2 + bx + c

  • Where

  • $a$ determines the direction of the parabola (upward if $a > 0$, downward if $a < 0$).

  • $b$ and $c$ are coefficients that affect the position and orientation of the graph.

Key Components of Quadratic Functions

The Quadratic Equation

• The general format is: y = ax^2 + bx + c

  • This function has a parabolic shape.

• Intercepts:

  • Y-Intercept is found by substituting $x=0$:
    y = c

  • X-Intercepts are found by solving:
    ax^2 + bx + c = 0

  • Can be determined using the quadratic formula: x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

    • Discriminant:

    • $D = b^2 - 4ac$

      • If $D > 0$: Two distinct x-intercepts.

      • If $D = 0$: One repeated x-intercept.

      • If $D < 0$: No x-intercepts.

Vertex of the Parabola

• The vertex is the highest or lowest point on the graph, depending on the direction of the parabola.

  • X-Coordinate of Vertex:
    x_v = \frac{-b}{2a}

  • Y-Coordinate of Vertex:

  • Found by substituting $xv$ back into the function: yv = f(x_v) = a\left(\frac{-b}{2a}\right)^2 + b\left(\frac{-b}{2a}\right) + c

Graphing Quadratic Functions

General Characteristics

• The graph is symmetric about a vertical line called the axis of symmetry, located at
  x = \frac{-b}{2a}

• The direction of the parabola is defined by the sign of $a$:

  • Opens upward if $a > 0$, downward if $a < 0$.

Vertex Example

• For the function:
  y = 3x^2 + 5x + 2

• Calculate x-coordinate of vertex:
  x_v = \frac{-5}{2(3)} = \frac{-5}{6}

• To find $y$:

  • Substitute $x = -\frac{5}{6}$ in the function:
    y_v = 3\left(\frac{-5}{6}\right)^2 + 5\left(\frac{-5}{6}\right) + 2

Finding Intercepts

Calculation of X-Intercepts

• To find x-intercepts, solve for when $y=0$:

  • Use factorization or quadratic formula.

  • Example for f(x) = -x^2 + x :

    • Factor out $x$:
      x(-x + 1) = 0

    • Solutions: $x = 0$ and $x = 1$.

  • If graphing, plot x-intercepts and y-intercepts accordingly.

Example of Y-Intercept

• For function f(x) = 3x^2 + 5x + 2 ,

  • The y-intercept will be:

  • Directly from $c$:
    y = 2

  • Thus, point (0, 2) on the graph.

Real-World Applications of Quadratic Functions

• Quadratic equations model various phenomena, such as projectile motion or revenue maximization problems.

• Revenue Optimization Example:

  • Assuming revenue $R(x) = -5x^2 + 50x$,

  • Vertex provides maximum revenue point.

Summary

• Quadratic functions have distinct properties based on coefficients, which allow us to predict the shape and intersections of their graphs.

• Tools include calculating intercepts, finding the vertex, and evaluating the discriminant to determine possible solutions.

• Understanding these functions is crucial, especially when applied to calculus in later studies, such as analyzing behavior near critical points and optimization problems.