Identifying Parabolas in Quadratic Functions

Parabola Identification

• Parabolas are graphs of quadratic functions. A quadratic function is a polynomial function of degree 2.
• The general form of a quadratic function is y = ax² + bx + c, where a, b, and c are constants.
• The value of 'a' determines the direction of opening of the parabola. If 'a' is positive, the parabola opens upward; if 'a' is negative, the parabola opens downward.

Identifying Parabolas Graphically

• Opening Direction: The first thing to note in a parabola graph is the direction of opening.

  • If the parabola opens upwards, the 'a' value in the quadratic equation is positive.
  • If the parabola opens downwards, the 'a' value in the quadratic equation is negative.

• Vertex: The vertex is the highest or lowest point on the parabola. The x-coordinate of the vertex can be found using the formula x = -b / 2a.

• Axis of Symmetry: A vertical line that divides the parabola into two symmetrical halves, passing through the vertex. Its equation is x = -b / 2a.

• y-intercept: The point where the parabola intersects the y-axis. To find it, substitute x = 0 in the quadratic equation.

• x-intercepts (roots/zeros): The points where the parabola intersects the x-axis. They are found by setting y = 0 and solving the quadratic equation. The solutions to the quadratic equation are the x-intercepts.

Examples

• Consider a parabola that opens upwards and has a vertex at (2, 1). This parabola could be represented by a quadratic equation of the form y = a(x - 2)² + 1, where 'a' is a positive constant.
• A downward-opening parabola with x-intercepts at (1, 0) and (5, 0) and a vertex at (3, ?) can be represented by a quadratic equation like y = -b(x-3)² + (?) where ‘b’ is a positive constant.

Key features to look for in a graph of a parabola.

• Shape: A smooth curve that is symmetrical about a vertical axis. The parabola will continue upward or downward without any sharp turns, angles, or discontinuities.
• Symmetry: The shape of one half of the parabola is mirrored on the other side of the axis of symmetry.
• Direction: Look for whether the parabola opens upward or downward. This determines the sign of the coefficient 'a'.

Quadratic Equation Selection

• Given a Graph: Students should determine the direction of the parabola's opening (up or down). This tells the sign of the coefficient 'a'.
• Vertex and Point: Given the coordinates of the vertex and another point on the graph, substitute these coordinates into the equation y = a(x - h)² + k, where (h, k) is the vertex, to solve for 'a'.
• X-intercepts: If the x-intercepts are known, use the factored form of a quadratic equation (y = a(x - p)(x - q)). Then substitute the point, or solve for 'a' by substituting a known point into the equation.
• Given the a, b, c values: If the equation is already written in standard form (y = ax² + bx + c) and only the equation is needed, students can choose the closest quadratic expression accordingly based on their observations.

Important Generalizations

• Stretching/Compressing: The absolute value of 'a' determines how wide or narrow the parabola is. A larger absolute value of 'a' makes the parabola narrower (steeper), while a smaller absolute value makes the parabola wider.
• Parabolas are not Lines or Curves: Parabolas are a specific type of curve; they are characterized by their symmetry. Lines and other curves have different properties.
• Vertex Forms/Standard Form: Use the appropriate equation form depending on the given information. The formula for finding the x-coordinate of the vertex will help you solve for the values of the equation. The selection among alternatives will often focus on the direction of the parabola opening, and other key characteristics noted during observation.

This quiz focuses on identifying and understanding parabolas in the context of quadratic functions. You'll learn about the characteristics of parabolas, including their direction of opening, vertices, and axes of symmetry. Test your knowledge on how to graph and interpret these essential features of quadratic functions.