The Math reading in Ch 3.2 for this week is short but mathematically dense. Take deep breaths, give yourself time, and don’t panic. Carefully mark each line in the reading that you don’t understand or can’t follow so we can have a productive lecture/seminar on Monday. This holds especially for the step-by-step algebraic manipulations – mark each step you don’t understand.

You may find it helpful to spend some time reading algebra tutorials online if your algebra is rusty.

Some specifics reading tips:

- First, here’s some terminology to help get started on the reading. A
*quadratic*always has a squared variable term (like x^{2}). For example, on p163 in Example 1 the area formula A(L) has the squared variable term L^{2}. When you graph a quadratic you get a U-shaped (or upside down U-shaped) curve called a*parabola*that is either opens up or opens down (see the graph on p164). The bottom (or top) of the parabola is called a*vertex*and it is a minimum (or maximum) point of the function. The vertex in Example 2 p164 is (-2,-3). - Study the vertex form on p165 before you try to read the example on p164. For the time being, don’t worry about the discussion of transformations on p164, rather, just try to match up the vertex of the graph of Example 2 with the vertex form defined on p165. The vertex form of a quadratic lets you just plug in the vertex. For example, the general vertex form is f(x) = a*(x – (h))
^{2}– (k) and the parabola of Example 2 p164 is g(x) = ½ * (x – (-2))^{2}– (-3). Watch out, you have to be careful about plugging in negative numbers in the vertex form — that’s why I put the h and k in parentheses (h) and (k). - The discussion in the bottom half of p165 is a good challenge. One of our goals for the week is to learn the algebra to turn the vertex form of a quadratic into standard form and standard form of a quadratic into vertex form. The bottom half of p165 is about how to transform a standard form into a vertex form. If you have trouble following it, then skip to the top of p166 to see the conclusion. Then go back and take it slow and read closely to follow each word and formula, again marking each step that you don’t understand. Again, keep the general purpose of that part of the reading in mind:
*to derive a formula for transforming a standard form of any quadratic into the vertex form*! The*h*and the*k*are the vertex coordinates and the formulas for*h*and*k*are given at the top of p. 166 using the standard form parameters*a*,*b*, and*c*(or*A*,*B*,*C*as you’ll see in Math Lab 3 or in LoggerPro). - Do the try-it-now 2 on p166 to challenge and test your understanding.
- Skip Example 5 pp167-68
- Study the three graphs on p168. Horizontal intercepts are also called roots. We find roots by
*solving*quadratics (reverse evaluation). When we solved linear equations we got single answers. The graphs on p168 show that solving quadratics can give 0, 1, or 2 answers. - Example 7 uses the definitions at the top of p166. The vertex form is an easy way to find roots, and Example 7 shows how.
- Do the try-it-now 3 on p169 to challenge and test your understanding.
- Closely following the algebra steps on pp169-70 to see if you can understand each step. The last sentence on p169 claims the formula at the bottom of the page is “Based on previous work…”. Where does that formula come from? Can you see how to start with the vertex form, plug in values for h and k, and get the f(x) formula at the bottom of the page? The algebra on p169-170 derives the general quadratic formula for roots of a quadratic. Follow each step carefully to make sure you understand the algebra. Come to class with questions or readiness to help your classmates.
- Use Desmos to do the try-it-now 4 on p171. See if you can figure out how you know from looking at a quadratic whether it has a maximum or a minimum (cupped up or cupped down).