Welcome to the Webinar “Live” Review for Test 1 (MAC 1105) We will start promptly at 8:00 pm Everyone will be placed on mute I am happy that you are able to join us

Note The following slides present sample problems similar to the test content, including the worked-out solutions. These are a few problems and do not include all the competencies tested on Test 1. To be prepared for the test, you must review the material from all the assigned Chapters 1 and 2 sections and follow the Study Guide posted on Blackboard!

Find the x-and-y-intercepts of the graph of -3x + ¾ y = 18 To find the x-intercept, set y = 0 -3x + ¾ (0) = 18 -3x = 18 x = -6 or (-6, 0) If asked for ordered pair form, write your answer in the form (x, y). If not asked for ordered pair form, then just write the values as specified on the question. To find the y-intercept, set x = 0 -3(0) + ¾ y = 18 ¾ y = 18 ( 4/3 )( ¾) y = 18( 4/3 ) y = 24 or (0, 24)

Slopes of Lines

Choose the graph that matches y = ⅕ x – 5 a. b. c. d.

Graphing y = ⅕ x – 5 using the slope-intercept method First, identify the y-intercept and the slope: (Remember y = mx + b, where m is slope and b is y-intercept) y-intercept is -5 or (0, -5) slope is ⅕ Now, plot the y-intercept and from there, apply “rise over run”

Choose the graph that matches y = ⅕ x – 5 a. b. c. d. y-intercept is (0, -5). So, eliminate a and d. slope is 1/5. So, rise 1, run 5. Answer is b

What can we say about the slopes of the following lines? a.l 4 is negative b.l 1, l 2 are positive c.l 3 is undefined d.l 3 is zero Answer: d

Write the slope-intercept form of the equation for the line passing through the points (-7, 6) and (1, -3). First, find slope: Use the point-slope formula (with any of the given points), and simplify to slope-intercept form: y – y 1 = m(x – x 1 ) y – (-3) = -(9/8)(x – 1) y + 3 = -(9/8)(x – 1) y = -(9/8)x + (9/8) – 3 y = -(9/8)x + (9/8) – (24/8) y = -(9/8)x – 15/8

Parallel lines have negative reciprocal slopes. a. True b. False Your time to work! Poll Question

Parallel lines have negative reciprocal slopes. a. True b. False Answer: False Parallel lines have equal slopes. Perpendicular lines have negative reciprocal slopes.

True or False: The equation of the line that passes through the point (1, 4) and is perpendicular to y = -2x + 6 is y = - ½ x + 9 / 2. Slopes are -2 and - ½, which are not negative reciprocals of each other! Answer: False

Find the point-slope form of the equation of the line through the point (-2, 3) parallel to the line 6x + 3y – 9 = 0. ? First, change to slope-intercept form by isolating the y: 3y = -6x + 9 y = -2x + 3 Now, find a line with slope m = -2 and passing through (-2, 3). y – y 1 = m(x – x 1 ) y – 3 = -2[x – (-2)] y – 3 = -2(x + 2) You have to stop here! (asked for point-slope form)

The line whose equation is y = -6 has slope m = a. -6 b. -1/6 c. 0 d. 6 e. Undefined Your time to work! Poll Question

The line whose equation is y = -6 has slope m = a. -6 b. -1/6 c. 0 d. 6 e. Undefined Vertical Line: x =  Horizontal Line: y =  Answer: c y = -6 is a horizontal line, therefore, m = 0

True or False: The following table represents a function. Function: For each input, there exists only one unique output Answer: True Different inputs can share the same output, like (-3, 0) and (12, 0) p q07-20

True or False: The following graph represents a function. Passes Vertical Line Test? Yes. Answer: True

For the following graph, a.Find C(150) b. Find x if C(x) = 10 a.C(150) means given the input of 150, find the output. Therefore, C(150) = 20 b.If C(x) = 10, we are looking for x values where the output is10. Therefore, x = 50; 300

For the following function, a.Find domain and range b.Find increasing and decreasing intervals a.Domain (Input): 0  x  300 or [0, 300] Range (Output): 5  y  25 or [5, 25] b.Increasing: (0, 200) Decreasing: (200, 300)

Find the domain and range of the function f(x) = Domain: Possible values of the input. Range: Possible values of the output. Domain: or 2x – 6  0 2x  6 x  3 or [3, infinity) Range: y  0 or [0, infinity)

If p is the function ‘f’ of q, then: a. p = f(q) b. the point (q, p) is on the graph of f c. f(0) = p is the vertical intercept of f for some p value d. f(q) = 0 is the horizontal intercept for some q value(s) e. the domain of f are all the possible values of q f.the range of f are all output values p g.all of the above h. none of the above Your time to work! Poll Question Answer using the Questions Box

If p is the function ‘f’ of q, then: a. p = f(q) b. the point (q, p) is on the graph of f c. f(0) = p is the vertical intercept of f for some p value d. f(q) = 0 is the horizontal intercept for some q value(s) e. the domain of f are all the possible values of q f.the range of f are all output values p g.all of the above h. none of the above Answer: g

Dance Rooms Here charges $65 per hour and a $350 deposit. a. Find a linear function that models this situation. Use C for cost and h for hours. b. Find and interpret the slope and the vertical intercept. c. If the total cost was $740, how many hours was the dance room used? d. Find and interpret C(4.5) a. Cost = deposit fee plus rate per hour for each hour of use C(h) = h Note: This problem involves variables C and h. Using other variables not stated on the problem would be marked incorrect. Example, C(x) = x. Make sure to use the variables as specified on each of the test questions! b. Slope = 65. The cost increases at a rate of $65 per hour. The vertical intercept is 350 and it denotes the initial fee.

c. If the total cost was $740, how many hours was the dance room used? d. Find and interpret C(4.5) c. We know C(h) = h h = 740 Solving for h we have 350 – h = 740 – h = h = 6 hours d.C(h) = h C(4.5) = (4.5) = = If the dance room is used for 4.5 hours, the cost is $642.50

A test has 26 questions worth 100 points. The test consists of multiple choice questions worth 2 points each and free response questions worth 6 points each. How many multiple choice questions are on the test? First, establish a system of equations. Let x = multiple choice questions and y = free response questions Number of questions: x + y = 26 Question value: 2x + 6y = 100 Multiply the first equation by -2, to help us eliminate the x-terms: (-2)(x + y) = (26)(-2) So, we have -2x – 2y = -52 2x + 6y = 100 4y = 48 y = 12 Back-substitute y = 12 into one of the original equations to find x. x + (12) = 26 x = 14 multiple choice questions

If f(x) = -x 3 + 3x – 5, find f(-2) f(-2) = -(-2) 3 + 3(-2) – 5 = -(-8) – 6 – 5 = 8 – 6 – 5 = -3 So, f(-2) = -3; that is, when the input is -2, the output is -3.

If f(x) = x 2 + 2x + h, find f(x + h) f(x + h) = (x + h) 2 + 2(x + h) + h Note: (x + h) 2 = (x + h) (x + h) = x² + 2xh + h² Therefore, f(x + h) = (x + h) 2 + 2(x + h) + h = x² + 2xh + h² + 2x + 2h + h = x² + 2xh + h² + 2x + 3h

Professor Vivaldi asked the students to evaluate f(x + h) given the function f(x) = 3x Brenda’s answer: 3x 2 + 3h Miranda’s answer: 3x 2 + 6xh + 3h T.J.’s answer: 6x 2 h 2 + 6xh + 6 Who is correct? a. Brenda b. Miranda c. T.J. d. Neither Your time to work! Answer using the Questions Box

Professor Vivaldi asked the students to evaluate f(x + h) given the function f(x) = 3x b. Miranda’s answer: 3x 2 + 6xh + 3h f(x + h) = 3(x + h) = 3(x 2 + 2xh + h 2 ) + 6 = 3x 2 + 6xh + 3h 2 + 6

A group of friends can spend no more than $50 for popcorn and sodas at a concert. A small bag of popcorn costs $6 and a medium soda is $5. Let x represent popcorn and y represent a soda. Which of the following describes this scenario? a.5x + 6y  50 b.6x + 5y  50 c.5x + 6y  50 d.6x + 5y  50 x: popcorn, $6 each  6x y: soda, $5 each  5y We can eliminate a and c. “No more than $50” implies 50 dollars or less Answer is b.

NOTE: It is important that you follow the instructions on the test. For example, if asked for interval notation, write your answer using interval notation. If you are not asked for interval notation, then just write the values as specified on the question. If asked for ordered pair form, write your answer in the form (x, y). If not asked for ordered pair form, then just write the values as specified on the question. So, read and follow instructions!

This "live" review session has covered content addressed on test 1, but you need to review chapters 1-2 and complete all the problems assigned on the study guide, so that you are well- prepared for the test.

Do well!