Welcome to class! As we continue our exploration of quadratics, we are launching a special three-part video series. The video above goes a bit deeper than what we covered in class today, but don't worry—I will be posting a different video each day (today, Friday, and Monday) to explain these concepts from different angles. By the end of this series, you will be a pro at analyzing these graphs!

Today's Focus: The Standard Form

For today's lesson, we are keeping it simple and looking for three specific things in the Standard Form of a quadratic function:

$$f(x) = ax^2 + bx + c$$
  • Identify Coefficients: What are the values of $a$, $b$, and $c$?
  • Direction: Does the parabola open up or down?
  • Y-Intercept: Where does the graph cross the y-axis?

Key Concepts from Class Notes

Based on our slides, here are the crucial properties you need to remember:

1. The Coefficient $a$ (Direction)

The value of $a$ tells us the "mood" of the parabola:

  • If $a$ is positive ($a > 0$), the parabola opens upward.
    Helpful Hint: It looks like a smile or a "U" (Happy). This graph has a minimum value.
  • If $a$ is negative ($a < 0$), the parabola opens downward.
    Helpful Hint: It looks like a frown (Sad). This graph has a maximum value.

2. The Constant $c$ (Y-Intercept)

The value of $c$ is your y-intercept. It represents the point $(0, c)$ where the parabola crosses the vertical axis.

Class Materials

Homework Assignment

Please complete the following exercises from Page 72 in your textbook:

Problems: #5-7, 15-23

For each problem, ensure you answer the following:

  1. Find the values of $a$, $b$, and $c$.
  2. Tell whether the graph will open upwards or downwards.
  3. Does the graph have a minimum or maximum?
  4. What is the y-intercept of the function?

Discussion Question of the Day

The y-intercept isn't just a point on a graph; it has meaning in real-world modeling. For example, in a physics problem, it might represent the starting height of an object.

Your Task: Give me a real-life example of a parabola (like a projectile path or a bridge) and explain what the y-intercept would represent in that specific context.