Hey Professor Baker's Math Class! As we wrap up our unit on quadratic functions, it's time to focus on finding the roots of these functions. Your test is tomorrow, so let's make sure you're well-prepared. If you comment on this post, you'll be allowed to use a note card on tomorrow's test – a great incentive to engage with the material!
The test will have three sections: Vocabulary, Process, and Applications, just like before. Let's quickly review the key terms and methods.
Key Vocabulary
- Axis of Symmetry: The vertical line that divides the parabola into two symmetrical halves.
- Discriminant: The part of the quadratic formula under the square root ($b^2 - 4ac$), which determines the number and type of roots.
- Quadratic Formula: The formula used to find the roots of a quadratic equation: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
- Parabola: The U-shaped curve that represents a quadratic function.
- Plus and Minus ($\pm$): Indicates that there are two possible solutions when taking the square root.
- Roots: The solutions to the quadratic equation; the points where the parabola intersects the x-axis. Also called zeros or x-intercepts.
- Vertex: The maximum or minimum point on the parabola.
- X-Intercept: The point(s) where the parabola crosses the x-axis (where y = 0).
- Y-Intercept: The point where the parabola crosses the y-axis (where x = 0).
- Zeros: Another term for the roots of the quadratic function.
Four Methods for Solving Quadratic Functions
1. Graphing to Find the Roots
By graphing the quadratic equation, we can visually identify the roots as the x-intercepts of the parabola.
- Worksheet to practice graphing quadratics. (Answers included)
- Interactive site for graphing quadratic functions
2. Factoring to Find the Roots
Factoring involves breaking down the quadratic expression into a product of two binomials. Setting each binomial equal to zero and solving gives us the roots. For example, from attached factoring worksheet, the factors for the expression $3p^2 - 2p - 5$ are $(3p - 5)(p + 1)$.
- Worksheet for practicing factoring. (Answers included)
- Site explaining various factoring methods
- Game to practice factoring
3. Solving by Using Square Roots
This method is applicable when the quadratic equation can be written in the form $(ax + b)^2 = c$. By taking the square root of both sides, we can solve for x.
- Site explaining the square root method
- Interactive practice quiz for solving by square roots
- Worksheet with answers for practice
4. Solving Using the Quadratic Formula
The quadratic formula is a universal method for finding the roots of any quadratic equation in the form $ax^2 + bx + c = 0$. As a reminder, the formula is: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
- Site explaining how to use the quadratic formula
- Interactive practice quiz for the quadratic formula
- Worksheet with answers to practice
Applications of Roots
Just like we used the vertex to find minimum or maximum values in modeling situations, we use roots to find when a value equals zero. This is often useful for finding when an object hits the ground or when a company's profit reaches zero.
Discussion Question of the Day:
Today was a review day! I want you to post what you think you will get on the test tomorrow. If you guess you will pass (greater than 60) and are within 5 points of your actual score, I will add 5 points to your grade! (Example: guess 83, get 86, final grade is 91). If you can't leave a comment, show me and I'll give you something to comment on. Good luck studying!