Finding Zeros of Quadratic Functions

Last cycle, we focused on graphing quadratic functions to identify the vertex, axis of symmetry, and the overall shape of the graph. This cycle, we're shifting our focus to where the graph intersects the x-axis. These points are crucial and have several names:

  • x-intercepts: Where the graph crosses the x-axis.
  • Zeros: The x-values that make the function equal to zero.
  • Roots: Another term for the solutions of the quadratic equation.

These three terms all refer to the same points on the graph! Understanding how to find these points is a fundamental skill in algebra.

Methods for Finding Zeros

There are multiple ways to find the zeros of a quadratic function. We will focus on two primary methods:

  1. Graphing: By plotting the quadratic function, you can visually identify the points where the parabola intersects the x-axis. These points represent the zeros of the function.
  2. Factoring: If the quadratic expression can be factored, setting each factor equal to zero and solving will yield the zeros of the function. For example, if we have the quadratic function $f(x) = x^2 + 2x - 3$, we can factor it as $f(x) = (x-1)(x+3)$. Setting each factor to zero, we have $x-1=0$ which gives $x=1$, and $x+3=0$ which gives $x=-3$. Therefore, the zeros are 1 and -3.

Class Activities and Assignments

  • Graphing Calculator Activity: In class, we will be doing an activity using the graphing calculator to explore the relationship between the graph and the zeros of a quadratic function. We will be using the worksheet "Explore Graphs and Factors." For example, we will be graphing quadratics such as $y=(x-2)(x-6)$ and finding the x-intercepts and the axis of symmetry.
  • Group Work: For the double period, we will work in small groups on the "Reading Strategy" worksheet to compare and contrast the methods of finding zeros.
  • Homework: Complete practice problems #1-8 on the practice worksheet to solidify your understanding.

Participation and Discussion

This cycle, discussion questions will contribute to your participation grade. They will account for 1/4 of your participation grade, which is 20% of your final grade (5 points). Make sure to post early as only the first unique response will receive credit.

To receive credit, be sure to respond thoughtfully and uniquely.

Discussion Question of the Day: Since tomorrow is Veterans Day, provide an example of a military application of a parabola. You may link to a picture but must explain what the picture is of and how it relates to the properties of a parabola.

Good luck, and I look forward to seeing your insightful contributions!