Today’s topic is quadratic inequalities and comparing three different models (linear, exponential, and quadratic).

Like other inequalities we have learned, we want to be able to graph a quadratic inequality. Three steps will get you there:

1. Graph the quadratic equation. If the inequality has an equal sign in it, use solid line. Otherwise use dashed line.
2. Pick a point that is not on the graph and test it with the original inequality.
3. Shade the solution area with the test result from Step 2. If the test result is true, the point is in the solution region, shade the region that contains the point. Otherwise, shade the other region.

So far we have learned three data models that can be used to express the relationships for a set of points: linear, exponential, and quadratic.

For linear model, the delta of two consecutive values should be constant. That is, for y = mx + b, y(x+1) – y(x) = m.

For exponential model y = C(1+/-r)^t, we know the ratio of two consecutive values should be contact. That is, y(t+1)/y(t) = 1+/-r.

If neither linear nor exponential relationship can be established, you can try quadratic model.

For homework, visit the following link:

http://blog.newtonchineseschool.org/wangweidong/files/2011/12/Math7_Fall_HW_L12.pdf

We don’t have school for the following two weeks and I will see you on 1/8/2012. Have a wonderful holidays.

Today we talked about the generic formulae for solving a quadratic equation of ax^2 + bx + c = 0

x= (-b +/- sqrt(b^2-4ac))/(2a)

We did exercise of using the formulae to solve some quadratic equations.

We further talked about the real life example of throwing an object, with the following mode:

h = -16t^2 +vt + s

where s is the initial height in feet, v is the initial speed (ft/s), t is the time, and h is th height at t time. When throwing an object upward, v has a positive value; when throwing the object downward, v has a negative value.

The terms inside the radical in the formulae is called discriminant. We talked about the application of the discriminant, and use it to determine how mnay solutions to get.

We further related the formulae to the graphing, and the relationship between the solutions and the x-intercepts.

For homework, visit the following link:

http://blog.newtonchineseschool.org/wangweidong/files/2011/12/Math7_Fall_HW_L11.pdf

Today we covered section 9.3 and 9.4, about graphing a quadratic function and how to solve a quadratic equation with graphing.

We talked about the shape of a quadratic function, a parabola, how to use the value of a to determine whether the u-shaped parabola opens up (when a > 0) or opens down (when a < 0). We have the formulae to find the x value of the vertex: x = – b/2a. We talked about how to find the y value at the vertex. We talked about the axis of symmetry, which is the vertical line passing through the vertex.

As the shape of a parabola is a U-shape, when it opens up, there is a minimum value of the y at the vertex. Similarly, when it opens down, there is a maximum value of y at the vertex.

Students are still uncomfortable working with letters like a, b, c (representing constands) in place of specific numbers, as in y = ax^2 + bx + c. When I ask them with a specific number, they will give me an answer. But if I ask with something with a/b/c, they look at me, puzzled. This takes time, as they are still getting into algebra.

To graph a quadratic function y = ax^2 + bx +c, follow the following steps:

1. Find the vertex with the formulae, find its y value;

2. Make a table with the vertex in the middle, do one side (as the other side is symmetric), +1, +2, +3, find the corresponding y values.

3. Connect those points with a smooth curve.

The solution to a quadratic equations, also called roots, are x-intercepts on the graph. So to solve a quadratic equation with graphing:

1. rewrite the equation into the form of ax^2 + bx +c = 0

2. Graph the quadratic function y = ax^2 + bx + c

3. Find the x-intercepts. These are your solutions.

4. Check the answer algebratically.

Below is the link to the homework for today:

http://blog.newtonchineseschool.org/wangweidong/files/2011/12/Math7_Fall_HW_L10.pdf

Feel free to contact me for any question.