welcome to the webinar “live” review for test 1 (mac 1105) we will start promptly at 8:00 pm...
TRANSCRIPT
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
NoteThe 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
12
12
xxyy
runrise
xy
m
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.l4 is negative
b.l1 , l2 are positive c.l3 is undefined
d.l3 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 – y1 = m(x – x1)
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
89
89
)7(163
xy
m
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 + 3Now, find a line with slope m = -2 and passing through (-2, 3).
y – y1 = m(x – x1)
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: cy = -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 -5 -3 6 7 12
q -1 0 7 -2 0
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)
62 x
positive zero
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) = 350 + 65h Note: This problem involves variables C and h. Using other variables not stated onthe problem would be marked incorrect. Example, C(x) = 350 + 65x.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) = 350 + 65h 350 + 65h = 740 Solving for h we have
350 – 350 + 65h = 740 – 350
65h = 390 65h = 390 65 65 h = 6 hours d. C(h) = 350 + 65h C(4.5) = 350 + 65(4.5) = 350 + 292.5 = 642.5 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 = 26Question 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 = 12Back-substitute y = 12 into one of the original equations to find x. x + (12) = 26x = 14 multiple choice questions
If f(x) = -x3 + 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) = x2 + 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) = 3x2 + 6.
Brenda’s answer: 3x2 + 3h2 + 6
Miranda’s answer: 3x2 + 6xh + 3h2 + 6
T.J.’s answer: 6x2h2 + 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) = 3x2 + 6.
b. Miranda’s answer: 3x2 + 6xh + 3h2 + 6
f(x + h) = 3(x + h)2 + 6
= 3(x2 + 2xh + h2) + 6
= 3x2 + 6xh + 3h2 + 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.
