11.2 Parabolas

Parabolas are fascinating curves with unique properties. They come in vertical and horizontal forms, each with distinct equations and characteristics. Understanding how to graph and interpret parabolas is crucial for mastering quadratic functions and their real-world applications.

From projectile motion to satellite dishes, parabolas are everywhere in our daily lives. Their shape and properties make them ideal for various engineering and scientific applications. Grasping these concepts will help you see math's practical relevance beyond the classroom.

Graphing and Interpreting Parabolas

Graphing vertical and horizontal parabolas

• Vertical parabolas open upward (concave up) or downward (concave down) with the axis of symmetry being a vertical line
  • Equation in standard form: $y = a(x - h)^2 + k$ (quadratic function)
    • $a$ determines the direction of opening and the shape
      • $a > 0$ opens upward (U-shaped)
      • $a < 0$ opens downward (inverted U)
      • $|a| > 1$ results in a narrower shape (compressed vertically)
      • $|a| < 1$ results in a wider shape (stretched vertically)
    • $(h, k)$ represents the vertex (turning point) of the parabola
• Horizontal parabolas open to the left or right with the axis of symmetry being a horizontal line
  • Equation in standard form: $x = a(y - k)^2 + h$
    • $a$ determines the direction of opening and the shape
      • $a > 0$ opens to the right (sideways U)
      • $a < 0$ opens to the left (sideways inverted U)
      • $|a| > 1$ results in a narrower shape (compressed horizontally)
      • $|a| < 1$ results in a wider shape (stretched horizontally)
    • $(h, k)$ represents the vertex (turning point) of the parabola

Key features of parabolas

• Vertex is the point where the parabola changes direction and represents the minimum or maximum point
  • For vertical parabolas: $(h, k)$ in $y = a(x - h)^2 + k$
  • For horizontal parabolas: $(h, k)$ in $x = a(y - k)^2 + h$
• Axis of symmetry is a line that divides the parabola into two equal halves and passes through the vertex
  • For vertical parabolas: $x = h$
  • For horizontal parabolas: $y = k$
• Direction of opening is determined by the value of $a$ in the standard form equation
  • For vertical parabolas:
    • $a > 0$ opens upward (positive quadratic term)
    • $a < 0$ opens downward (negative quadratic term)
  • For horizontal parabolas:
    • $a > 0$ opens to the right (positive quadratic term)
    • $a < 0$ opens to the left (negative quadratic term)
• Roots are the x-intercepts of the parabola, representing where the function equals zero

Applications of Parabolic Functions

Applications of parabolic functions

• Projectile motion describes the path of an object thrown, launched, or shot forming a vertical parabola with downward opening (parabolic trajectory)
  • Equation: $y = -\frac{1}{2}gt^2 + v_0t + h_0$
    • $g$ represents acceleration due to gravity (9.8 m/s²)
    • $v_0$ represents initial velocity
    • $h_0$ represents initial height
• Satellite dishes and headlights utilize the parabolic shape to focus signals or light, forming a vertical parabola with upward opening
• Suspension bridges have main cables supporting the bridge deck in the shape of a vertical parabola with downward opening
• Profit optimization involves a parabolic relationship between price and quantity sold, with the vertex representing the maximum profit point

Additional Concepts

• Discriminant: A value that determines the nature of a quadratic function's roots
• Conic sections: A family of curves that includes parabolas, circles, ellipses, and hyperbolas, formed by intersecting a plane with a double cone