# Math 140 - Week 2 Notes

## Monday, February 22

Today we talked about how to graph functions. We started with these examples where factoring lets you find the roots (also known as zeros or x-intercepts).

1. $$y = 4-x^2$$.

2. $$f(x) = x^3 - 3x^2 - 18 x$$.

3. $$y = 2x - 6$$.

4. $$g(x) = x^4 - 3x^3 + 2 x^2$$.

We also reviewed the following six basic graphs that you should know:

 x y y = mx+b x y y = c x y y = x² x y y = |x| x y y = 1/x x y y = √x

We finished by using these 6 basic graphs to graph variations like:

1. $$y = 3|x|$$

2. $$y = \dfrac{1}{x-2}$$

3. $$y = x^2 + 5$$

4. $$y = |x-3|$$

5. $$y = -x^2$$

6. $$y = \sqrt{-x}$$

## Wednesday, February 24

Today we focused on linear functions. You need to know these formulas for linear functions:

• Slope-Intercept Form $$y = mx + b$$
• Point-Slope Form $$y-y_0 = m(x - x_0)$$

You also need to understand slope very well:

• Slope $$m = \dfrac{\text{rise}}{\text{run}} = \dfrac{\Delta y}{\Delta x} = \dfrac{ y_2 - y_1}{x_2 - x_1}$$

We did the following exercises in class:

1. Find a formula to convert Celsius to Fahrenheit.

2. Find a formula to convert Fahrenheit to Celsius.

3. Find an equation for the line passing through $$(-1,4)$$ and $$(3,-4)$$.

4. Find the slope and $$y$$ intercept of the equation $$6x − 5y + 15 = 0$$.

5. Find a formula for a line with slope 2 passing through $$(2,5)$$.

6. $$\dfrac{x+3}{4} = 3$$

7. $$\dfrac{3}{x} - 5 = \dfrac{2}{x}$$

8. Pressure underwater (measured in ATMs) is a linear function of the depth $$x$$ in meters given by $$P = \frac{1}{10}x+1$$. What is the slope and what are its units?

9. In 2021, the Virginia state income tax for people making over \$17,000 is $$T = 720 + 0.0575 (I - 17000)$$. Graph this linear function. What is the slope? What does the input variable $$I$$ represent? What about the output variable $$T$$?

## Friday, February 26

Today we talked about powers and radicals. We covered the basic rules for powers and radicals, and some of the common mistakes that people make. Here is a quick summary of what you should know.

We did the following exercises in class:

Simplify the following.

1. $$8^{2/3}$$

2. $$4^{5/2}$$

3. $$16^{1/4} + 16^{-1/4}$$

4. $$\sqrt{x^2+x^2+x^2+x^2}$$

5. $$\displaystyle \frac{m^2 n^3}{a^2 c^{-3} } \cdot \frac{a^{-7} n^{-2}}{m^2 c^4}$$

6. $$\displaystyle \sqrt{\frac{18x}{25x^3}}$$

Solve.

1. $$\dfrac{2^{57}}{2^x} = 8$$

2. $$(x^{-1} + 2^{-1})^{-1} = \tfrac{1}{3}$$