Welcome Back to Class!

Yesterday, we began dissecting the Standard Form of a quadratic equation, which looks like this:

$$f(x) = ax^2 + bx + c$$

We learned how the coefficients determine the graph's personality: whether it opens up or down (based on the sign of $a$), where it crosses the y-axis, and where its center lies. Today, we are taking that theory and putting it onto the grid. We are moving from analyzing the equation to graphing the parabola.

Key Concepts: The Graphing Checklist

As we discussed in the lecture (see the attached Class Notes), you need to find specific "landmarks" to sketch an accurate graph. Here is the step-by-step process to solve the problems on your worksheet:

  1. Identify the Direction: Look at $a$. If $a > 0$, the graph opens up (has a minimum). If $a < 0$, it opens down (has a maximum).
  2. Find the Axis of Symmetry (AoS): Use the golden formula from our notes: $$x = \frac{-b}{2a}$$
  3. Find the Vertex: Take the $x$-value from your AoS calculation and plug it back into the original function to find $y$. This gives you your coordinate pair $(x, y)$.
  4. Find the Y-Intercept: This is the easiest point! It is simply $(0, c)$.
  5. Reflect and Plot: Use the Axis of Symmetry like a mirror to plot the reflection of your y-intercept.

Homework Assignment

To practice these skills, please complete the following from the textbook packet (Pg. 72 provided in the attachments).

  • Problems: #15, 16, 17, 18, 19, 20 (The graphing exercises)
  • Word Problems: #30 (Weather/Temperature) and #31 (Golf Ball Height)

Requirements for Graphing (#15-20):
Please use the attached "Template to do your work on". For each function, you must:

  • Graph exactly 5 points (Vertex, Y-int, its reflection, and two other points/zeros).
  • Clearly label the Axis of Symmetry (draw it as a dashed line).
  • Label the Vertex.
  • Label the Y-Intercept.
  • Label the Zeros (if the graph crosses the x-axis).

Discussion Question of the Day

Everyone always thinks there is an easier way to graph these functions! I encourage you to look for patterns. Challenge: If you find a valid method to graph these functions that is faster than the one I taught in class—and I can verify it works consistently—you will receive 20 bonus points on the upcoming test.

Note: Only the first student to present a unique, valid method will receive the credit, so get creative!

Resources: Check the attached files for the Class Notes (including the worked-out examples for finding maximums/minimums), the blank Graphing Template, and the Homework Pages.