# Algebra Discussion

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week 2 reflections:

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Week 2 application:

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Note 1 : This is college algebra. Don't make it tough. Use simple English.

Note 2:  For reflection you have to answer in each question with 1-2 sentences (check sample answer from other students below)

Note 3 : for week 2 application, again dont make it lengthy. 100-150 words are more than enough.

I will attach other 2 pdf which we studied this week.

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2 Graphs and Functions Copyright © 2017, 2013, 2009 Pearson Education, Inc. 1 2.2 Circles • Center-Radius Form • General Form • An Application Copyright © 2017, 2013, 2009 Pearson Education, Inc. 2 Circle-Radius Form By definition, a circle is the set of all points in a plane that lie a given distance from a given point. The given distance is the radius of the circle, and the given point is the center. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 3 Center-Radius Form of the Equation of a Circle A circle with center (h, k) and radius r has equation which is the center-radius form of the equation of the circle. A circle with center (0, 0) and radius r has equation Copyright © 2017, 2013, 2009 Pearson Education, Inc. 4 Example 1 FINDING THE CENTER-RADIUS FORM Find the center-radius form of the equation of each circle described. (a) center (– 3, 4), radius 6 Solution Center-radius form Substitute. Watch signs here. Let (h, k) = (–3, 4) and r = 6. Simplify. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 5 Example 1 FINDING THE CENTER-RADIUS FORM Find the center-radius form of the equation of each circle described. (b) center (0, 0), radius 3 Solution The center is the origin and r = 3. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 6 Example 2 GRAPHING CIRCLES Graph each circle discussed in Example 1. (a) Solution Writing the given equation in center-radius form gives (– 3, 4) as the center and 6 as the radius. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 7 Example 2 GRAPHING CIRCLES Graph each circle discussed in Example 1. (b) Solution The graph with center (0, 0) and radius 3 is shown. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 8 General Form of the Equation of a Circle For some real numbers c, d, and e, the equation can have a graph that is a circle or a point, or is nonexistent. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 9 General Form of the Equation of a Circle Consider There are three possibilities for the graph based on the value of c. 1. If c > 0, then r 2 = c, and the graph of the equation is a circle with radius 2. If c = 0, then the graph of the equation is the single point (h, k). 3. If c < 0, then no points satisfy the equation and the graph is nonexistent. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 10 Example 3 FINDING THE CENTER AND RADIUS BY COMPLETING THE SQUARE Show that x2 – 6x + y2 +10y + 18 = 0 has a circle as its graph. Find the center and radius. Solution We complete the square twice, once for x and once for y. and Copyright © 2017, 2013, 2009 Pearson Education, Inc. 11 Example 3 FINDING THE CENTER AND RADIUS BY COMPLETING THE SQUARE Add 9 and 25 on the left to complete the two squares, and to compensate, add 9 and 25 on the right. Add 9 and 25 on both sides. Complete the square. Factor. Because 42 = 16 and 13 > 0, the equation represents a circle with center (3, – 5) and radius 4. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 12 Example 4 FINDING THE CENTER AND RADIUS BY COMPLETING THE SQUARE Show that 2x2 + 2y2 – 6x +10y = 1 has a circle as its graph. Find the center and radius. Solution To complete the square, the coefficients of the x2- and y2-terms must be 1. Divide by 2. Rearrange and regroup terms. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 13 Example 4 FINDING THE CENTER AND RADIUS BY COMPLETING THE SQUARE Complete the square for both x and y. Factor and add. Center-radius form The equation has a circle with center at and radius 3 as its graph. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 14 Example 5 DETERMINING WHETHER A GRAPH IS A POINT OR NONEXISTENT The graph of the equation x2 + 10x + y2 – 4y + 33 = 0 is either a point or is nonexistent. Which is it? Solution Subtract 33. and Copyright © 2017, 2013, 2009 Pearson Education, Inc. 15 Example 5 DETERMINING WHETHER A GRAPH IS A POINT OR NONEXISTENT Complete the square. Factor; add. Since – 4 < 0, there are no ordered pairs (x, y), with both x and y both real numbers, satisfying the equation. The graph of the given equation is nonexistent—it contains no points. (If the constant on the right side were 0, the graph would consist of the single point (– 5, 2).) Copyright © 2017, 2013, 2009 Pearson Education, Inc. 16 Example 6 LOCATING THE EPICENTER OF AN EARTHQUAKE Suppose receiving stations A, B, and C are located on a coordinate plane at the points (1, 4), (–3, –1), and (5, 2). Let the distances from the earthquake epicenter to these stations be 2 units, 5 units, and 4 units, respectively. Where on the coordinate plane is the epicenter located? Solution Graph the three circles. From the graph it appears that the epicenter is located at (1, 2). To check this algebraically, determine the equation for each circle and substitute x = 1 and y = 2. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 17 Example 6 LOCATING THE EPICENTER OF AN EARTHQUAKE Station A: Copyright © 2017, 2013, 2009 Pearson Education, Inc. 18 Example 6 LOCATING THE EPICENTER OF AN EARTHQUAKE Station B: Copyright © 2017, 2013, 2009 Pearson Education, Inc. 19 Example 6 LOCATING THE EPICENTER OF AN EARTHQUAKE Station C: Copyright © 2017, 2013, 2009 Pearson Education, Inc. 20 Example 6 LOCATING THE EPICENTER OF AN EARTHQUAKE The point (1, 2) lies on all three graphs. Thus, we can conclude that the epicenter is at (1, 2). Copyright © 2017, 2013, 2009 Pearson Education, Inc. 21 2 Graphs and Functions Copyright © 2017, 2013, 2009 Pearson Education, Inc. 1 2.1 Rectangular Coordinates and Graphs • Ordered Pairs • The Rectangular Coordinate System • The Distance Formula • The Midpoint formula • Equations in Two Variables Copyright © 2017, 2013, 2009 Pearson Education, Inc. 2 Ordered Pairs An ordered pair consists of two components, written inside parentheses. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 3 Example 1 WRITING ORDERED PAIRS Use the table to write ordered pairs to express the relationship between each category and the amount spent on it. Category food Amount Spent \$8506 housing \$21,374 transportation \$12,153 health care \$4917 apparel and services \$2076 entertainment \$3240 Copyright © 2017, 2013, 2009 Pearson Education, Inc. 4 WRITING ORDERED PAIRS Example 1 (a) housing Category Solution Use the data in the second row: (housing, \$21,374). food Amount Spent \$8506 housing \$21,374 transportation \$12,153 health care \$4917 apparel and services \$2076 entertainment \$3240 Copyright © 2017, 2013, 2009 Pearson Education, Inc. 5 Example 1 WRITING ORDERED PAIRS (b) entertainment Category Solution Use the data in the last row: (entertainment, \$3240). food Amount Spent \$8506 housing \$21,374 transportation \$12,153 health care \$4917 apparel and services \$2076 entertainment \$3240 Copyright © 2017, 2013, 2009 Pearson Education, Inc. 6 The Rectangular Coordinate System y-axis Quadrant I Quadrant II P(a, b) b a Quadrant III x-axis 0 Quadrant IV Copyright © 2017, 2013, 2009 Pearson Education, Inc. 7 The Distance Formula Using the coordinates of ordered pairs, we can find the distance between any two y points in a plane. Horizontal side of the triangle has length P(– 4, 3) Q(8, 3) x R(8, – 2) Definition of distance Copyright © 2017, 2013, 2009 Pearson Education, Inc. 8 The Distance Formula y Vertical side of the triangle has length P(– 4, 3) Q(8, 3) x R(8, – 2) Copyright © 2017, 2013, 2009 Pearson Education, Inc. 9 The Distance Formula y By the Pythagorean theorem, the length of the remaining side of the triangle is P(– 4, 3) Copyright © 2017, 2013, 2009 Pearson Education, Inc. Q(8, 3) x R(8, – 2) 10 The Distance Formula y So the distance between (– 4, 3) and (8, – 2) is 13. P(– 4, 3) Q(8, 3) x R(8, – 2) Copyright © 2017, 2013, 2009 Pearson Education, Inc. 11 The Distance Formula To obtain a general formula for the distance between two points in a coordinate plane, let P(x1, y1) and R(x2, y2) be any two distinct points in a plane. P(x1, y1) Complete a triangle by locating point Q with coordinates (x2, y1). The Pythagorean theorem gives the distance between P and R. Copyright © 2017, 2013, 2009 Pearson Education, Inc. y Q(x2, y1) x R(x2, y2) 12 Distance Formula Suppose that P(x1, y1) and R(x2, y2) are two points in a coordinate plane. The distance between P and R, written d(P, R) is given by the following formula. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 13 Example 2 USING THE DISTANCE FORMULA Find the distance between P(– 8, 4) and Q(3, – 2.) Solution Be careful when subtracting a negative number. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 14 Using the Distance Formula If the sides a, b, and c of a triangle satisfy a2 + b2 = c2, then the triangle is a right triangle with legs having lengths a and b and hypotenuse having length c. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 15 Example 3 APPLYING THE DISTANCE FORMULA Determine whether the points M(– 2, 5), N(12, 3), and Q(10, – 11) the vertices of a right triangle? Solution A triangle with the three given points as vertices is a right triangle if the square of the length of the longest side equals the sum of the squares of the lengths of the other two sides. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 16 Example 3 APPLYING THE DISTANCE FORMULA Copyright © 2017, 2013, 2009 Pearson Education, Inc. 17 Example 3 APPLYING THE DISTANCE FORMULA Copyright © 2017, 2013, 2009 Pearson Education, Inc. 18 Example 3 APPLYING THE DISTANCE FORMULA Copyright © 2017, 2013, 2009 Pearson Education, Inc. 19 Example 3 APPLYING THE DISTANCE FORMULA The longest side, of length 20 units, is chosen as the possible hypotenuse. Since is true, the triangle is a right triangle with hypotenuse joining M and Q. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 20 Collinear Points We can tell if three points are collinear, that is, if they lie on a straight line, using a similar procedure. Three points are collinear if the sum of the distances between two pairs of points is equal to the distance between the remaining pair of points. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 21 Example 4 APPLYING THE DISTANCE FORMULA Determine whether the points P(– 1, 5), Q(2, – 4), and R(4, – 10) are collinear. Solution Copyright © 2017, 2013, 2009 Pearson Education, Inc. 22 Example 4 APPLYING THE DISTANCE FORMULA Determine whether the points P(– 1, 5), Q(2, – 4), and R(4, – 10) are collinear. Solution Copyright © 2017, 2013, 2009 Pearson Education, Inc. 23 Midpoint Formula The coordinates of the midpoint M of the line segment with endpoints P(x1, y1) and Q(x2, y2) are given by the following. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 24 Example 5 USING THE MIDPOINT FORMULA Use the midpoint formula to do each of the following. (a)Find the coordinates of the midpoint M of the line segment with endpoints (8, – 4) and (– 6,1). Solution Substitute in the midpoint formula. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 25 Example 5 USING THE MIDPOINT FORMULA Use the midpoint formula to do each of the following. (b) Find the coordinates of the other endpoint Q of a segment with one endpoint P(– 6, 12) and midpoint M(8, – 2). Solution Let (x, y) represent the coordinates of Q. Use the midpoint formula twice. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 26 Example 5 x-value of P USING THE MIDPOINT FORMULA x-value of M x-value of P y-value of M Substitute carefully. The coordinates of endpoint Q are (22, – 16). Copyright © 2017, 2013, 2009 Pearson Education, Inc. 27 Example 6 APPLYING THE MIDPOINT FORMULA The following figure depicts how a graph might indicate the increase in the revenue generated by fast-food restaurants in the United States from \$69.8 billion in 1990 to \$195.1 billion in 2014. Use the midpoint formula and the two given points to estimate the revenue from fastfood restaurants in 2002, and compare it to the actual figure of \$138.3 billion. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 28 Example 6 APPLYING THE MIDPOINT FORMULA Copyright © 2017, 2013, 2009 Pearson Education, Inc. 29 Example 6 APPLYING THE MIDPOINT FORMULA Solution The year 2002 lies halfway between 1990 and 2014, so we must find the coordinates of the midpoint of the segment that has endpoints (1990, 69.8) and (2014, 195.1). Thus, our estimate is \$132.5 billion, which is less than the actual figure of \$138.3 billion. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 30 Example 7 FINDING ORDERED-PAIR SOLUTIONS OF EQUATIONS For each equation, find at least three ordered pairs that are solutions (a) Solution Choose any real number for x or y and substitute in the equation to get the corresponding value of the other variable. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 31 Example 7 FINDING ORDERED-PAIR SOLUTIONS OF EQUATIONS (a) Solution Let x = – 2. Let y = 3. Multiply. Add 1. Divide by 4. This gives the ordered pairs (– 2, – 9) and (1, 3). Verify that the ordered pair (0, –1) is also a solution. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 32 Example 7 (b) Solution FINDING ORDERED-PAIR SOLUTIONS OF EQUATIONS Given equation Let x = 1. Square each side. Add 1. One ordered pair is (1, 2). Verify that the ordered pairs (0, 1) and (2, 5) are also solutions of the equation. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 33 Example 7 FINDING ORDERED-PAIR SOLUTIONS OF EQUATIONS (c) A table provides an organized method for determining ordered pairs. x y Solution –2 0 –1 –3 0 –4 1 –3 2 0 Five ordered pairs are (–2, 0), (–1, –3), (0, –4), (1, –3), and (2, 0). Copyright © 2017, 2013, 2009 Pearson Education, Inc. 34 Graphing an Equation by Point Plotting Step 1 Find the intercepts. Step 2 Find as many additional ordered pairs as needed. Step 3 Plot the ordered pairs from Steps 1 and 2. Step 4 Join the points from Step 3 with a smooth line or curve. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 35 GRAPHING EQUATIONS Example 8 (a) Graph the equation Solution Step 1 Let y = 0 to find the x-intercept, and let x = 0 to find the y-intercept. The intercepts are Copyright © 2017, 2013, 2009 Pearson Education, Inc. 36 Example 8 GRAPHING EQUATIONS (a) Graph the equation Solution Step 2 We find some other ordered pairs (also found in Example 7(a)). Let x = – 2. Let y = 3. Multiply. Add 1. Divide by 4. This gives the ordered pairs (– 2, – 9) and (1, 3). Copyright © 2017, 2013, 2009 Pearson Education, Inc. 37 GRAPHING EQUATIONS Example 8 (a) Graph the equation Solution Step 3 Plot the four ordered pairs from Steps 1 and 2. Step 4 Join the points with a straight line. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 38 Example 8 GRAPHING EQUATIONS (b) Graph the equation Solution Plot the ordered pairs found in Example 7b, and then join the points with a smooth curve. To confirm the direction the curve will take as x increases, we find another solution, (3, 10). Copyright © 2017, 2013, 2009 Pearson Education, Inc. 39 Example 8 GRAPHING EQUATIONS (c) Graph the equation Solution Plot the points and join them with a smooth curve. x y –2 0 –1 –3 0 –4 1 –3 2 0 This curve is called a parabola. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 40 Wk2 Applications Pick one concept presented this week. Discuss one real-life scenario in which this concept is applicable. In other words, state one way in which you can use this concept in every day life. 8 Wk2 Reflections lease share a summary of the week's activity. The following are things you can consider to include in your reflection: 1. Summarize what you learned. 2. Identify problem areas. 3. Discuss strengths. What you like about class? 4. Discuss weaknesses. What you dislike about class? 5. Discuss ways to improve class. What can you do to improve your performance? What can your instructor do to improve your performance? 1. I learned how to solve for range, domain, radius, graphing, how to read graphs, and how to determine if something is a function or not. 2. I had a little bit of trouble in each section, but once I keep doing the problems over again I figured it out. 3. I like that I can keep working the problems over and over again until I g to know them and learn them. 4. What i dislike about the class is not having someone to show me how to work these problems. 5. I can Improve my performance by doing more practice problems and asking someone for help.
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Running head: ALGEBRA WEEK 2 DISCUSSION

Algebra Week 2 Discussion
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ALGEBRA WEEK 2 DISCUSSION

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Algebra Week 2 Discussion
Wk2 Application

The main concept of this week was reading and evaluating graphs. Graphs can
be used in the real life situation to help in quick understanding of information or a...

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