Standard Form of the Quadratic Equation

Using the information we've been learning about over the last week, we started looking at the quadratic equation in the standard form. We realized that if a = 0, we end up with a linear equation, and it follows all of the rules we've learned about previously. We learned how to write quadratic equations in standard form, and then looked at the quadratic equation in non-standard form. Students quickly identified 2 key parts of this equation: the discriminant and the equation for the line of symmetry. I highly encourage students to memorize the quadratic equation in both of the forms we looked at today.

Tonight's Homework: Home Book, pages 172-3 (42, 45-48, 50-55) and page 179 (46-47)

What's A Discriminant?

We discussed what the discriminant is, and how it helps you determine how many solutions a quadratic equation will have. If the discriminant is positive, the equation will have two (2) solutions. If the discriminant is 0, then the equation will have one (1) solution. If the discriminant is negative, the equation will have no solutions. Solutions relate to the number of x-intercepts the parabola has.

Tonight's Homework: Home Book, read pages 180-181. Do page 182, (1-15, 18-24).

Vertical Motion Model and Graphing Quadratic Inequalities

We expanded our understanding of graphing quadratics to include the graphing of quadratic inequalities. These follow the same basic rules of quadratics, and then have the following two extra steps:

1. The line will be dotted if you see an inequality ( < or > ) sign.
2. Shading will be either inside or outside of the function. Use a test point to determine where the shading goes.

We also talked a bit about the vertical motion model, and how it fits into our study of quadratics.

Tonight's Homework: Home Book, pages 170-171 (2-34, even)

Eureka! We Found It!

As part of our school's Secretary Appreciation Day, students wrote a note of thanks to one of our school secretaries. These were delivered to them this afternoon.

Students shared their summaries about the relationship of a and b to the location of the line of symmetry and the vertex for a particular parabola. Most found that the line of symmetry was the opposite of half of the b value. This is true, but only when a = 1. We did a little bit more work on this and determined the formula for the line of symmetry to be x = - b/ 2a.

Tonight's Homework: Read pages 711-716 in the CME book, and then do problems 11 and 15 on page 719.

Looking at Patterns in Quadratic Equations

Students were encouraged to look at their data in another way today. Specifically, I asked students to consider grouping data where a and b were both positive, and then look for a relationship between the line of symmetry and the values. Similarly, group data where a and b are both negative and see if their is a relationship between the line of symmetry and the a and b values.

Tonight's Homework: Continue to work on the quadratic equation project you've been working on. Focus on the relationships between the a and b variables and the location of the line of symmetry and the vertex of the parabola.

Where Will the Vertex Be?

The beginning of a list of vocabulary related to the study of quadratics was shared with students this morning. Words like parabola, vertex, line of symmetry, maximum and minimum were defined based on our investigation last week. Students then used class time to work on their investigation about the relationship of "a", "b" and the effect they have on the placement of the vertex and line of symmetry. Data tables, graph and analysis for this is due by tomorrow.

Tonight's Homework: Get your grade slip signed. Finish the investigation described above.

Mouse Scares Boys (and a few girls) in Class

Yes, it's true, we had a mouse (well, actually 2) in class today, which distracted many students, especially boys for about 10 minutes. After getting settled, students successfully discovered that the coefficient of the x^2 term (a) determines whether a parabola opens up or down, as well as how wide or skinny it will be. We talked about some vocabulary (vertex, line of symmetry, slope, and y-intercept) and related these terms to the parabola. Students also were successful in determining that "c" moves the parabola up or down on the coordinate grid. Determining what "b" does was a little more mysterious, and that's where we will focus our investigation today.

Tonight's Homework: Using the general formula of y = x^2 + bx, try to determine the relationship between a and b that governs the location of the parabola on the coordinate grid. Focus on the vertex and the line of symmetry, as these are big clues.

Concept Tests 11 and 12 Have Finally Been Completed!

Students took the last part of the concept tests covering concepts 11 and 12. Those who finished early were given time to continue working on the quadratic investigation that we started in class on Tuesday. The three part assignment (table of values, graphs and analysis for each equation type) is due on Friday, and will not be accepted late.

Tonight's Homework: Finish the quadratic investigation so you can turn it in on Friday. If you are on the track team, here's hoping you perform well in all of your events. Go Mustangs!

Investigating Quadratics, Part Two

Because I have been asked to attend a district meeting on Thursday, the final part of the tests covering concepts 11 and 12 will be delayed until Thursday (thus making the plans for the guest teacher much simpler for this large class). Students worked on the second and third parts of the quadratic investigation today. Here's a recap of the assignment:

Part 1: y = ax^2
Part 2: y = x^2 + c
Part 3: y = x^2 + bx

For each part, make a table of data (5 points minimum) for 6 examples of the basic equation, graph the six functions on one large graph, and write an analysis describing how the variable (a, b, or c) changes the graph.

Tonight's Homework: Continue working on the quadratic equation investigation. Plan ahead, especially if you will be at the track meet on Thursday. The last part of concept test 11 will be given in class tomorrow. See unit 11 in the home book if you want to review.

Transitioning to the Second Degree Equation

Students received feedback from yesterday's quiz. Most did very well! Before taking yet another part of the concept test, we talked about what a second degree equation would look like in its simplest form. Students offered many variations on the basic equation (y = x^2) and were encouraged to use many of these examples in completing their homework tonight.

Tonight's Homework: Complete part one of the quadratic equation investigation we started in class today. The three parts of the assignment are: six different data tables, one graph with six different colored plots graphed, and an analysis of what you found.

Quiz over Exponential Growth and Decay Completed

Students took a quiz over exponential growth and decay today. This is a part of the concept 11 test, and the rest of the test will be taken in class later in the week.

Tonight's Homework: Home Book p. 58, p. 66 and p. 85, selected problems. The purpose of doing these problems is to help students review for an upcoming part of the concept test over ezponents.

Concept 12 Test (part 1) Taken in Class Today

We concluded our discussion about exponential growth and decay by noticing that when the base of the power is greater than 1, the graph shows an increase, while if the base of the power is a decimal number (between 0 and 1), the graph shows a decay or decrease. We reviewed how the compounding period (annually, semi-annually, quarterly, or daily) impacts the amount of interest earned. Students then took a quick 10 minutes quiz over graphing square root functions. This is part of Concept Test #12. More of the test will be taken next week.

Tonight's Homework: Home Book, pg. 90-92 (1-14 and 62-67) and 100-101 (60-77). Be sure to prepare for the 10 minute quiz you'll take on Monday over exponential growth and decay. Bring your calculator!

More on Depreciation and Compounding Interest

As we continued our investigation into exponential growth and decay, we looked at how compounding annually compares to compounding on a semi-annual or quarterly basis. One must remember to cut the interest down (by half or a fourth, in these cases) and then increase the value of n before calculating the interest. This sounds confusing, but it really isn't. In most cases, you'll gain more interest by compounding over shorter periods of time. For instance, if given the chance to earn interest monthly rather than quarterly, take it! Of course, the opposite is true if you are paying interest on a purchase. In this case, take the longest compounding period offered.

Tonight's Homework: Complete problems 1-12 on the Compound Interest worksheets. Be sure to show the key sequence (process) you used to complete each problem. Remember to prepare for the quiz tomorrow over square root functions by reviewing the green and ivory square root worksheets in your notebook.

Compounding Interest and Depreciation

After running through several long multi-step problems to determine the balance of a bank account earning interest over a period of several years, we learned how to use an exponential growth formula to help us solve the problems in a fraction of the time. Students learned how to use a specific button on their calculators (either ^ or y^x) to help them with both exponential growth and decay problems.

Tonight's Homework: Complete both sides of the yellow Exponential Growth and Decay WS. Be sure to include the key sequence or process you used when punching numbers into your calculator. Round your answers at the very end, and be sure to truncate, rather than round to the nearest.

Modeling Exponential Growth and Decay

Students participated in a set of activities to model radioactive decay and exponential growth. First, students flipped pennies and class data was collected. Those who flipped a head remained in the game, while those flipping a tail where out for the rest of the game. We did this three times to get a complete set of class data. Next we looked at bacterial growth related to the culturing of bacteria causing strep throat.

Tonight's Homework: Make a graph of the penny flipping data. Find a rule that includes an exponent to describe the data. Then make a graph of the bacterial growth data. Again, try to find a rule that includes an exponent to describe the data.