Math 125 Little Book Homework Chapters 7, 10, 11, and 12 Do NOT copy the book follow the guidelines given for each section. NO CREDIT will be given if you copy the book! You earn 2 points if you turn in your homework on the due date; 1 point if you turn it in the next meeting. Turn in on Tuesday, 10/23 7 ANALYTIC GEOMETRY 7.1 The Rectangular Coordinate System - Write the distance formula given on page 168. Redo Examples 1 and 2. - Write the midpoint formula given on page 169. Redo Example 3. - Redo Example 4 on page 170. This is an example of an analytic geometry problem. 7.2 Lines - Write the standard form of a linear equation in 2 variables given on page 171 or 172. - Explain how to find the x- and y-intercepts (see 2 nd paragraph on page 172). - Redo Examples 1 and 2 on pages 171-172. - Write the definition of the slope of a line (boxed at the bottom of page 172). - Redo Example 3 (Just sketch the graph and count no need to draw the creatures). - Write the formula for finding he slope of a line given two points (boxed at the bottom of pahe 173). - Redo Example 4. - Write the slope-intercept form of the equation of a line. This gives a third way of finding the slope. - Redo Example 5 (Copy the equation and convince yourself that the slope is 2 read the rest). - Redo Example 6. This shows that we need to isolate y first before we can read the slope. - Redo Examples 7 and 8. Pay attention to the properties and equations of horizontal and vertical lines. - Write the relationships of the slopes of parallel lines and perpendicular lines given on pages 176 and 177. - Redo Examples 9 through 12. - Redo Examples 13 and 14 to see how to graph a line using the slope and a point. 7.3 Writing Equations of Lines - Redo Examples 1 through 6. These lead to equations of horizontal and vertical lines. - Write the equation needed to set up the equation of an oblique line. This is given on the middle of y y = m x x. Feel free to use page 182. All other math books write this in the point-slope form ( ) 1 1 this when redoing the examples. - Redo Examples 7 through 10. - Another way to write an equation of a line is given on page 185. This method starts from y = mx + b. - Redo Examples 7 through 10 if you want to practice this method. Turn in on Thursday, 10/25 7.4 Parabolas: Equations Given in Standard Form - Graph the basic vertical and horizontal parabolas given on page 189. Note that the vertex (0,0) is the most important point to get and then we just need one point on each side. - Redo Example 1. This shows how a negative sign reflects the basic vertical parabola along the x-axis. 2 - Redo Examples 2 and 3. These show how adding or subtracting a constant to x shifts the basic vertical parabola vertically.
- Redo Examples 4 and 5. These show how adding or subtracting a constant to x shifts the basic vertical parabola horizontally. - Redo Examples 6 and 7 to see how the basic vertical parabola can have both horizontal and vertical shifts. - Redo Example 8 to see three transformations of the basic vertical parabola. - Redo Example 9 to see how the basic vertical parabola can be compressed. - Redo Example 10 to see how the basic vertical parabola can be stretched. - Redo Example 11 to see how the four transformations of the basic vertical parabola. 7.5 Parabolas: Equations Given in General Form - This is the unreadable version of the equation of a parabola. This means all the information except for how the parabola opens is lost. We need to find the vertex. - There are two ways to get the vertex: complete the square or use a formula. - Redo Examples 1 through 4 using your preferred method. I suggest using the formula given on page 205. 7.7 Circles - Draw a circle. Label the center and the radius. - Write the center-radius form of the equation of a circle (boxed, middle of page 214). - Redo Examples 1 through 5 to see how to graph a circle whose center-radius form of the equation is given. - Redo Examples 6 through 9 to see how to write the center-radius form of the equation given certain properties of a circle - Redo Example 10 to see how to find the center and radius from the general equation of a circle. Turn in on Tuesday, 10/30 7.8 Ellipses and Hyperbolas - Redo Examples 1 through 4 to see how to draw an ellipse. Note that we just need to find the x- and y- intercepts (vertices) and connect them. - Redo Examples 6 and 7 to see how to graph a hyperbola. Just do these: (1) Find the vertices and the imaginary intercepts. (2) Draw the rectangle. (3) Draw the diagonals. (4) Draw the hyperbola. 10 FUNCTIONS 10.1 Definitions - Write the definition of a function. - Write the four ways of representing relations and functions. - Redo Examples 1 through 4 to see how to determine whether a given relation is a function or not when a mapping of two sets is given. - Redo Examples 5 and 6 to see how when a relation or function is defined as a set of ordered pairs. - Redo Examples 7 through 11 to see how when a relation or function is given by an equation or inequality. - Write the summary given on top of page 310. f x (read this as f of x ). - Redo Examples 14 through 18 to understand the function notation ( ) 10.2 The Domain and the Range - Redo the two examples on page 312 to see how to find the domain and the range for a mapping and a set of ordered pairs.
- Redo the examples on page 313 to see how to find the domain and the range when the graph is given. The first graph consists of one piece; the second graph has three pieces (Sorry, the page is too busy!). Just copy the graphs and write the answers for the domain and the range given in the Algebear clouds. Note that we go from left to right for the domain and bottom to top for the range. 10.3 Graphs of Basic Functions - Write the general and basic equations and sketch the graphs of the linear, quadratic, square root, and absolute value functions. Turn in on Thursday, 11/1 10.4 Graphing Transformations - Redo Examples 1 through 11. The idea is the same as what we saw in Chapter 7: the basic graphs can be shifted horizontally, vertically, flipped along the x- or y-axis, compressed or stretched. 10.5 The Zeros of a Function - Redo Examples 1 through 6; no need to sketch the graphs. Note that to get the zero(s) of a function, f x by 0 and solve for x. Graphically, the zero(s) is(are) the x-intercepts of the graph. simply replace ( ) 10.6 Operations on Functions (No need to copy the graphs just perform the operation; also ignore finding the domain) - Redo Examples 1 through 4. Note that addition, subtraction, multiplication, and division of functions are natural operations. - Write the definition of composition of functions given on the top of page 334. - Redo Examples 5 and 6. Hopefully you will see that to perform the composition of two functions, all we do is replace all the x s of the outer function by the formula for the inner function. - Redo Examples 7 and 8 to see how to find the value of the composition of two functions at a given value of x. - Redo Examples 9 through 12 to get more practice on composing functions. 10.7 The Inverse of a Function - Copy the pair of mappings given on pages 336 and 337 to understand the notion of inverting a function. Note that the resulting relation is not necessarily a function (second pair given on page 337). - Write the definition of a one-to-one function given on the bottom of page 337. Note that this will guarantee that the inverse will be a function. - Redo Examples 1 through 3 on pages 338 340 to see how to check if a function is one-to-one. - There are three main parts in discussing the inverse of a function: (1) Find the formula for the inverse. (2) Verify that the functions invert each other. (3) Graph both functions together to see that one is the mirror image of the other along y= x. - Redo Examples 1 through 7 on pages 342-345. I did the graphs last but you can put them where they belong in each example.
Turn in on Tuesday, 11/6 11 EXPONENTIAL AND LOGARITHMIC FUNCTIONS 11.1 The Exponential Function - Redo Examples 1 through 3. These show what the graph looks like when the base is larger than 1. - Redo Examples 4 and 5. These show what the graph looks like when the base is a positive number less than 1. 11.2 The Logarithmic Function - The logarithmic function is the inverse of the exponential function. Write the boxed equivalent equations given in the middle of page 364. - Redo Examples 1 through 6 using the boxed equations given on page 364 - Redo Examples 1 and 4 (page 365) using the notion of applying (composing) the inverse function the notion of undoing. You may choose whichever method you prefer. - Redo Examples 7 and 8 to see how to graph the log function. I used the notion of the inverse. I think you will find it easier to just write the equivalent exponential equation. So, for Example 7, write f x = log x as x = 2 y and graph this second equation by assigning values to y (-1, 0, 1 should be ( ) 2 enough) and solving the corresponding x s and graphing the points (x, y). 11.3 Properties of Logarithms - Write the 5 properties of logarithms listed on page 367-368. - Redo Examples 1 through 5. - I explained how to evaluate logarithms by solving an equation. This method is too long! Instead, write the property that I gave at the bottom of page 369 (Property (6)). - Redo Examples 6 through 8 using Property (6). - Write properties 7 9 given on page 371. - Write the Negatron explanation of properties (7) (9) - Redo Examples 9 and 10 on pages 373 374 to see how to evaluate logarithms using (7) (9). - Redo Examples 11 through 14 to see how to expand log expressions using (7) (9). - Redo Examples 15 through 17 to see how to compress log expressions using (7) (9). Turn in on Thursday, 11/8 11.4 Solving Exponential and Logarithmic Equations Exponential Equations - Redo Examples 1 through 6. These show that after making the bases the same, all we do is equate the exponents. - Redo Examples 7 and 8. These show that when it is impossible to make the bases the same, then we take the log (base 10) of both sides. Logarithmic Equations - Redo Example 9 to see that when both sides have a single log expression (same base) on both sides, we simply ignore the logs and equate the arguments. - Redo Example 10. This example has log on one side and a number on the other. I showed how to snail or undo to solve. - Redo Examples 11 through 13. These examples show that we first need to compress the log expressions into one using (7) (9) before we can snail or undo.
11.5 Applications - Write the compound interest formula and what the letters stand for. (Algebear cloud p 385) - Redo Example 1 to see how to calculate the amount after t years. - Redo Example 2 to see how to find the number of years it will take for the principal to grow to a given amount. Turn in on Tuesday, 11/13 7.10 Chapter Review - Write all the information given on pages 178 180 except for the ellipse and hyperbola on page 180. For these, just write the equations from pages 220 and 224. 10.9 Chapter Review (Typo again it says 10.8) - Write most of the information from this section except the stuff on Variation given on page 356. 11.6 Chapter Review - Sketch the exponential graphs given on top of page 388. - Sketch the logarithmic graphs given on page 389. - Write the nine properties of logarithms (bottom of page 389). - Redo Examples 1 through 8. - Write the compound interest formula and the definition of the letters given on page 392. Turn in on Tuesday, 11/27 12 SEQUENCES AND SERIES 12.1 Sequences - Write the definition of a sequence (page 393). - Redo the graph given on page 393. Note that the domain is the set of positive integers and the range defines the sequence. - Write the notation used to denote the terms of a sequence (Negatron p 394). 12.2 Series and Summation Notation - Write the definition of a series. - Write the finite sum S5 = a1 + a2 + a3 + a4 + a5. - Write the same finite sum using the summation notation S 5 5 = ak. k= 1 - Redo Examples 1 through 5 to see how to write an expanded sum in summation notation. - Redo Examples 6 through 9 to see how to switch from summation notation to expanded sum. Turn in on Thursday, 11/29 12.3 Arithmetic Sequences and Series - Write the definition of an arithmetic sequence. - Write the formula for finding the nth term (bottom of page 401). - Write the formulas for finding the sum of the first n terms (page 404). - Redo Examples 1 through 7.
12.4 Geometric Sequences and Series - Write the definition of a geometric sequence. - Write the formula for finding the nth term (page 407 before example 3). - Write the formula for finding the sum of the first n terms (first box page 409). - Write the formula for finding the infinite sum (second box page 409). - Redo Examples 1 through 8. Turn in on Tuesday, 12/4 12.5 Chapter Review - Jot down most of the information given in this section.