.) how many ways are there to place testing dummies into the rollercoaster? b.) how many ways are there to place dummies into the roller coaster if the front car must have 3 or fewer testing dummies in it? c.) how many ways are there to place dummies into the roller coaster if the back car must be empty? d.) how many ways are there to place dummies into the roller coaster if the neither the front or back car can be
Answers
(a) There are 2²⁸ = 268,435,456 ways to place testing dummies into the rollercoaster
(b) There are 15 × 2²⁴ = 402,653,184 ways to place dummies into the rollercoaster if the front car must have 3 or fewer testing dummies.
(c) There is only 1 × 2²⁴ = 16,777,216 way to place dummies into the rollercoaster if the back car must be empty.
(d) There are 15 × 2²⁰ = 15,728,640 ways to place dummies into the rollercoaster if neither the front nor the back car can be full.
a) There are 7 cars and each car has 4 seats, there are a total of 7 × 4 = 28 seats. Each seat can either be occupied by a testing dummy or left empty.
For each seat, there are two possibilities (dummy or empty). Since each seat can be treated independently, the total number of ways to place the testing dummies is 2²⁸.
2²⁸ = 268,435,456
b) If the front car must have 3 or fewer testing dummies,
The front car can have 0, 1, 2, or 3 testing dummies.
There are C(4, 0) + C(4, 1) + C(4, 2) + C(4, 3) = 1 + 4 + 6 + 4 = 15 ways to select the number of dummies for the front car.
The remaining 6 cars can have any number of dummies between 0 and 4. Each car has 4 seats, and each seat can be occupied by a dummy or left empty. So, there are 2⁴ possibilities for each of the 6 remaining cars.
Total number of ways = 15 × (2⁴)⁶ = 15 × 2²⁴.
15 × 2²⁴ = 402,653,184
c) If the back car must be empty, we can calculate the number of arrangements by considering the possibilities for the back car and then multiplying it by the remaining possibilities for the other cars.
The back car can have 0 dummies. There is only 1 possibility for the back car.
The remaining 6 cars can have any number of dummies between 0 and 4. Each car has 4 seats, and each seat can be occupied by a dummy or left empty. So, there are 2^4 possibilities for each of the 6 remaining cars.
Total number of ways = 1 × (2⁴)⁶ = 1 × 224.
1 × 2²⁴ = 16,777,216
d) If neither the front nor back car can be full, we can calculate the number of arrangements by considering the possibilities for both the front and back cars and then multiplying it by the remaining possibilities for the other cars.
The front car can have 0, 1, 2, or 3 testing dummies. There are C(4, 0) + C(4, 1) + C(4, 2) + C(4, 3) = 1 + 4 + 6 + 4 = 15 ways to select the number of dummies for the front car.
The back car can have 0 dummies. There is only 1 possibility for the back car.
The remaining 5 cars (excluding the front and back cars) can have any number of dummies between 0 and 4. Each car has 4 seats, and each seat can be occupied by a dummy or left empty. So, there are 2^4 possibilities for each of the 5 remaining cars.
Total number of ways = 15 × 1 × (2⁴)⁵ = 15 × 2²⁰.
15 × 2²⁰ = 15,728,640
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The question is incomplete the complete question is :
an amusement park is testing a roller coaster ride for safety. the roller coaster has 7 distinguishable cars, each of which contains 4 distinguishable seats. each seat can either be occupied by a testing dummy or left empty. the dummies are indistinguishable from one another, and there are enough to fill every car. for all sub-problems below, you are allowed to leave the entire ride empty, fill every seat in the ride, or anything in between. (update 2/17/23: the cars cannot be rearranged. they are fixed in one order.) a.) how many ways are there to place testing dummies into the rollercoaster? b.) how many ways are there to place dummies into the roller coaster if the front car must have 3 or fewer testing dummies in it? c.) how many ways are there to place dummies into the roller coaster if the back car must be empty? d.) how many ways are there to place dummies into the roller coaster if the neither the front or back car can be full?
Related Questions
For questions 7–9, imagine a segment in a plane. Use that information to help you answer the questions. How many midpoints does the segment have? A. one B. two C. infinitely many D. none
Answers
A translation maps P(2, 3) onto Q(-2, 6). Find the coordinates of the image R(5, 10) under the same translation.
Answers
Answer:8,4
Step-by-step explanation:
Just put it it’s right
Validation of the model and answering the question "what are my options" occur in the ___ phase of the IDC.
A. choice
B. design
C. intelligence
D. implantation
Answers
Validation of the model and answering the question "what are my options" occur in the design phase of the IDC (Intelligence, Design, and Choice) framework.
The IDC framework is a decision-making process that consists of three phases: Intelligence, Design, and Choice. Each phase corresponds to a specific set of activities and objectives.
In the intelligence phase, the focus is on gathering information, identifying the problem or decision to be made, and understanding the factors and variables involved. This phase involves data collection, analysis, and exploration to gain insights and knowledge about the problem domain.
In the design phase, the emphasis is on developing and evaluating potential options or solutions to address the problem or decision at hand. This phase involves creating models, prototypes, or simulations to represent the problem and exploring different alternatives.
Validation of the model is an important aspect of this phase to ensure that the proposed solutions align with the problem requirements and objectives.
The question "what are my options" is a fundamental question that arises during the design phase. It implies the exploration and generation of various possible choices or solutions that can be evaluated and compared.
Therefore, the design phase of the IDC framework encompasses the activities of validating the model and answering the question "what are my options." It involves refining and testing potential solutions to make informed decisions in the subsequent choice phase.
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What is the solution to the equation below? Round your answer to two decimal places In x = 0.4
A. -0.40
B. -0.92
C. 2.51
D. 1.49
Answers
Answer:
D. 1.49
Step-by-step explanation:
_________________
Answer:
D. 1.49
Step-by-step explanation:
That is the answer, I have no explanation why.
HELPPPP
Which expression is a perfect square?
A. x² - 4x + 4
B. x2 - 4x - 4
C. 12 - 9x + 9
D. 12 - 9x – 9
Answers
Answer: Choice A
========================================================
Explanation:
A perfect square trinomial is of the form
(m+n)^2 = m^2 + 2mn + n^2
We can see why this works through use of the FOIL rule.
So the first and last terms must be perfect squares, and they must be positive. This rules out choices B and D since we have -4 and -9 as those last terms being negative.
Choice C is ruled out because the middle coefficient -9 isn't correct. The middle coefficient must be some even number since 2 is a factor of 2mn.
This leaves choice A. We can do a bit of trial and error to find that
(x-2)^2 = x^2 - 4x + 4
showing that choice A is a perfect square trinomial.
Answer:
A) x² - 4x + 4
Step-by-step explanation:
Given the following special factoring formulas on perfect square trinomials:
u² + 2uv + v² = (u + v)²
u² - 2uv + v² = (u - v)²
Apply these concepts on the given set of equations to see which special factoring formulas apply on the given options. I will only work on one of the given options, and you could apply these same techniques on the other ones to compare the results with your calculations.
To find the binomial factors of quadratic equations, you must find factors with product uv, and sum v.
In Option A: x² - 4x + 4
Right off the bat, it seems like the difference of squares may apply to this equation. You must find the factors that will product of 4 and a sum of -4.
The possible factors are:
product uv: -2 × -2 = 4,
sum v : -2 + -2 = -4
Hence, the binomial factor of x² - 4x + 4 = (x - 2)²
If you expand this binomial through FOIL method:
(x - 2)(x - 2) = x² - 2x - 2x + 4 = x² - 4x + 4
This means that the trinomial x² - 4x + 4 produces a perfect square binomial factors of (x - 2)².
Therefore, the correct answer is Option A) x² - 4x + 4.
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Write the prime factorization of 21. Use exponents when appropriate and order the factors from least to greatest (for example, 2235).
Answers
The prime factorization of the number 21 is:
21 = 3*7
How to write the prime factorization?We want to write the prime factorization of 21.
To do so, we just need to divide the number by prime numbers.
The first prime number we can try is 2, if we divide by 2 we get:
21/2 = 10.5
This is not an integer, so 2 is not a factor.
The next one is 3:
21/3 = 7
Now we can rewrite:
21 = 3*7
Where 3 and 7 are prime numbers, so that is the prime factorization.
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If you buy lean ground beef and it is listed as 85% lean, how many grams of fat are in 0.75lb of this product? ( 1lb=454 g)
Answers
In 0.75lb (340g) of 85% lean ground beef, there are approximately 72 grams of fat.
To calculate the grams of fat in 0.75lb of 85% lean ground beef, we need to determine the fat content based on the percentage given. The term "85% lean" indicates that 85% of the weight of the ground beef is lean meat, while the remaining 15% is fat.
First, we convert 0.75lb to grams. Since 1lb is equal to 454g, multiplying 0.75lb by 454g/lb gives us 340g.
Next, we calculate the fat content by multiplying the weight of the ground beef (340g) by the percentage of fat (15%).
340g * 0.15 = 51g
Therefore, in 0.75lb (340g) of 85% lean ground beef, there are approximately 51 grams of fat.
However, the question asks for the fat content, so we subtract this value from the total weight to find the grams of fat:
340g - 51g = 289g
Therefore, there are approximately 72 grams of fat in 0.75lb (340g) of 85% lean ground beef.
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The correct answer in the box. Linear function g is shown in the graph.
Write the slope-intercept form of the equation representing this function.
Answers
Answer:
this is right on the test
Step-by-step explanation:
Answer:
The correct answer is
g(x)=2x+1
Step-by-step explanation:
I got it right on the Edmentum test.
8 less than three-fourths of x is y
PLEASE HELPPPP
Answers
Answer:
y=3/4x-8
Step-by-step explanation:
Answer:
The answer to 3/4x - 8 = y is the following:
x= 4y+32 over 3
y= 3 over 4 x−8
Step-by-step explanation:
Brainliest Plzzz!!!
EX24) 29 du Use the chain rule to find the indicated derivative. og, where du g(u, v) = f(x(u, v),y(u, v)), f(x,y) = 7x³y³.x(u, v) = ucosv, y(u, v) = usiny = 56u² cos v sin³ v
Answers
∂g/∂u is equal to 21u⁵cos⁴(v)sin⁴(v)(cos(v) + u³cos⁴(v)sin²(v)sin(v)).
To find the indicated derivative, we need to use the chain rule. Let's differentiate step by step:
Given:
g(u, v) = f(x(u, v), y(u, v))
f(x, y) = 7x³y³
x(u, v) = ucos(v)
y(u, v) = usin(v)
To find ∂g/∂u, we differentiate g(u, v) with respect to u while treating v as a constant:
∂g/∂u = (∂f/∂x) * (∂x/∂u) + (∂f/∂y) * (∂y/∂u)
To find ∂f/∂x, we differentiate f(x, y) with respect to x:
∂f/∂x = 21x²y³
To find ∂x/∂u, we differentiate x(u, v) with respect to u:
∂x/∂u = cos(v)
To find ∂f/∂y, we differentiate f(x, y) with respect to y:
∂f/∂y = 21x³y²
To find ∂y/∂u, we differentiate y(u, v) with respect to u:
∂y/∂u = sin(v)
Now, we can substitute these partial derivatives into the equation for ∂g/∂u:
∂g/∂u = (21x²y³) * (cos(v)) + (21x³y²) * (sin(v))
To find the simplified form, we substitute the given values of x(u, v) and y(u, v) into the equation:
x(u, v) = ucos(v) = u * cos(v)
y(u, v) = usin(v) = u * sin(v)
∂g/∂u = (21(u * cos(v))²(u * sin(v))³) * (cos(v)) + (21(u * cos(v))³(u * sin(v))²) * (sin(v))
Simplifying further, we get:
∂g/∂u = 21u⁵cos⁴(v)sin⁴(v)(cos(v) + u³cos⁴(v)sin²(v)sin(v))
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If there are 1000 grams in a kg and 455 grabs in a pound, how many pounds are there per kg
Answers
Answer:
The answer is 0.45359237.
Step-by-step explanation:
I assume you are converting between kilogram and pound.
Answer:
2.20462
Step-by-step explanation:
1000 / 455 = 2.20462
When you code a SELECT statement, you must code the four main clauses in the following order
SELECT, WHERE, ORDER BY, FROM
SELECT, ORDER BY, FROM, WHERE
SELECT, FROM, WHERE, ORDER BY
SELECT, FROM, ORDER BY, WHERE
Answers
When coding a SELECT statement in SQL, it is crucial to follow a specific order for the four main clauses to ensure accurate and efficient retrieval of data. The correct order to code the clauses is c. SELECT, FROM, WHERE, ORDER BY.
Firstly, the SELECT clause specifies the columns that you want to retrieve data from. This clause is always coded at the beginning of the statement. Next, the FROM clause identifies the table or tables from which you want to retrieve data. It is important to note that the tables in the FROM clause must be listed before the WHERE clause.
The WHERE clause is used to filter the data based on certain criteria or conditions. This clause is typically coded after the FROM clause. Lastly, the ORDER BY clause sorts the retrieved data in ascending or descending order based on a specified column. This clause is always coded at the end of the statement.
Therefore, the correct order for coding the four main clauses in a SELECT statement is SELECT, FROM, WHERE, and ORDER BY. By following this order, you can ensure that your query runs smoothly and returns the correct results.
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Find g(1) if g(x) = x^2 + 1.
2 3
Answers
Answer:
\(y = {1}^{2} + 1 = 1 + 1 = 2\)
Jennifer has a $20 bill to buy school supplies. She wants to buy 4 notebooks that cost $2.55 each and 3 cases of pencils that cost $1.99 each.
How much change will she get back after she pays for the items?
$3.83
$5.97
$10.20
$16.17
Answers
Answer:
$3.83
Step-by-step explanation:
Step 1- Multiply
4×2.55= 10.20
3×1.99= 5.97
Step 2- Add
10.20+5.97= 16.17
Step 3- Subtract
20-16.17= 3.83
She will get $3.83 back after she pays for the items.
sat math scores follow a normal distribution with a mean of 511 and a standard deviation of 110. suppose we choose a student at random. what is the probability that the student scores between 450 and 600?
Answers
The probability that a student scores between 450 and 600 on the SAT math section is approximately 0.4147 or 41.47%.
To find the probability that a student scores between 450 and 600 on the SAT math section, we need to use the properties of the normal distribution. We know that the mean is 511 and the standard deviation is 110.
First, we need to standardize the values of 450 and 600 using the formula:
z = (x - μ) / σ
where x is the value we want to standardize, μ is the mean, and σ is the standard deviation.
For 450:
z = (450 - 511) / 110 = -0.55
For 600:
z = (600 - 511) / 110 = 0.81
Next, we need to find the area under the normal curve between these two standardized values. We can use a table or a calculator to find that the area between z = -0.55 and z = 0.81 is approximately 0.4147.
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A hypothesis can be differentiated from a theory because it is...
a. a specific prediction arising from the theory. c. it talks about how one specific variable affects d. all of the above.
Answers
A hypothesis can be differentiated from a theory because it is a specific prediction arising from the theory. The statement that is true is, a hypothesis can be differentiated from a theory because it is a specific prediction arising from the theory. A hypothesis is a suggested explanation of a phenomenon or observed data that is testable.
Hypotheses are more detailed, and are written to explain precisely what you think is going to occur in your research and the reason for that prediction. A hypothesis will be rejected if it does not fit the data, whereas a theory will be modified to fit the data.
A hypothesis is more like a forecast, and a theory is more like a law that explains how certain events operate. Thus, we can conclude that a hypothesis is a specific prediction arising from the theory.
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help me plz! question shown in pic
Answers
Answer:
x=17
y=10
Step-by-step explanation:
hope this helps you
What is the equation of the line that passes through the point (5, -6) and has a
slope of -3/5?
Answers
find the circumference of a circle if the area is 81pi m^2 use 3..14 for pi and round your answer to the nearest tenth
Answers
Answer: C=2πr
Step-by-step explanation:
Answer:
9 cm.
Step-by-step explanation:
We are given that the area of a circle is 81π cm2. 81 π c m 2 . Hence the radius of the given circle is 9 cm.
You finish washing 34 of your windows in an hour and a half. At that rate, how much longer will it take you to finish washing all the windows? in return i will help u with Physics
Answers
Answer:
2 hours
Step-by-step explanation:
\(\dfrac{\frac{3}{4}}{\frac{3}{2}}=\dfrac{1}{x} \\\\\\\dfrac{1}{2}=\dfrac{1}{x} \\\\\\x=2\)
Hope this helps!
Help with this question, not sure what the answer is
Answers
Answer:
3rd option
Step-by-step explanation:
The zeros ( where the graph crosses the x- axis ) are x = 3 and x = 7
The corresponding factors are then (x - 3) and (x - 7)
The equation is then
f(x) = a(x - 3)(x - 7) ← a is a multiplier
To find a substitute any point on the graph into the equation.
Using (8, 15 ) , then
15 = a(8 - 3)(8 - 7) = a(5)(1) = 5a ( divide both sides by 5 )
3 = a
Thus
f(x) = 3(x - 3)(x - 7)
calculate the ground distance if the map distance is 20 cm write your answer in kilometers
Answers
The ground distance, given the map distance and the scale, would be 5 kilometers.
How to find the distance?If the scale on a map is 1:25,000, it means that one unit of distance on the map represents 25,000 units of distance on the ground. If the map distance is 20 cm, we can find the ground distance as follows:
Convert the map distance from centimeters to kilometers:
20 cm = 0.2 m = 0. 0002 km
Find the ground distance using the scale:
Ground distance = Map distance / Scale
Ground distance = 0. 0002 km / ( 1 / 25,000 )
Ground distance = 0.0002 km x 25,000
Ground distance = 5 km
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The full question is:
The scale on a map is 1:25,000. Calculate the ground distance if the map distance is 20 cm write your answer in kilometers
a poll of $100$ eighth-grade students was conducted to determine the number of students who had a dog, a cat or a fish. the data showed that $50$ students had a dog, $40$ students had a cat, and $20$ students had a fish. further, $19$ students had only a dog and cat, $2$ students had only a cat and a fish, $3$ students had only a dog and a fish, and $12$ students had only a fish. how many students had none of these pets?
Answers
14 students had none of the pets mentioned in the problem.
Given Question is related to Sets and Function
By using the principle of inclusion-exclusion,
D = set of students who had a dog
C = set of students who had a cat
F = set of students who had a fish
Let's determine the sizes of the sets and their intersections:
|D| = 50
|C| = 40
|F| = 20
|D ∩ C| = 19
|C ∩ F| = 2
|D ∩ F| = 3
|D ∩ C ∩ F| = 0
|D ∪ C ∪ F| = ?
The size of the union of the sets as follows:
|D ∪ C ∪F| = |D| + |C| + |F| - |D ∩ C| - |C ∩ F| - |D ∩ F| + |D ∩ C ∩ F|
|D ∪ C ∪ F| = 50 + 40 + 20 - 19 - 2 - 3 + 0 = 86
Therefore, there were 86 students who had at least one of these pets.
To find the number of students who had none of these pets, we can subtract this number from the total number of students:
100 - 86 = 14
Therefore, 14 students had none of the pets mentioned in the problem.
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Please find angle P as a whole value
Answers
The measure of the angle P is 14 degrees
How to determine the angleTo determine the value of the angle, we have that the six trigonometric identities are;
sinetangentcotangentsecantcosecantcosineFrom the information given, we have that;
Opposite side of angle P = 24
Adjacent x = 10
Using the tangent identity, we have that;
tan = opposite/adjacent
substitute the values, we get;
tan P = 24/10
Divide the values, we have;
tan P = 0. 24
Find the inverse tangent, we get;
P = 14 degrees
Then, the measure of the angle is 14 degrees
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in a class of 12 boys and 9 girls, the teacher selects three students at random to write problems on the board. what is the probability that all the students selected are boys? (round your answer to four decimal places.)
Answers
in a class of 12 boys and 9 girls, the teacher selects three students at random to write problems on the board then the probability that all the students selected are boys is 0.165
given that a class consists of 12 boys and 9 girls. The teacher picks three students to present their work to the rest of the class and says that the three students are being selected at random
Here selecting any 3 students from the group of total 21 students is combination because order does not matter.
Total no of ways of selecting any 3 from total 21 = 21C3 = 1330
No of ways of selecting only 3 boys = No of ways of selecting random 3 boys from total 12 boys
= 12C3 =220
the probability be that all three students
selected are boys=220/1330 = 0.165
Hence , the probability that all the students selected are boys is 0.165
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when a variable follows a normal distribution, what percent of observations are contained within 1.75 standard deviations of the mean?
Answers
Using a normal distribution table or calculator, we can find that approximately 88.8% of the observations will fall within this range. This means that if a variable follows a normal distribution, approximately 88.8% of the observations will fall within 1.75 standard deviations of the mean.
When a variable follows a normal distribution, it is often assumed that the distribution is symmetrical around the mean, with 50% of the observations falling above the mean and 50% falling below. However, we can use standard deviations to better understand the distribution of the data.
If a variable follows a normal distribution, approximately 68% of the observations will fall within one standard deviation of the mean. This means that if the mean is 100 and the standard deviation is 10, approximately 68% of the observations will fall between 90 and 110.
When we move to 1.75 standard deviations away from the mean, we can use a normal distribution table or calculator to find the percentage of observations falling within that range. Using the same example as before, if the mean is 100 and the standard deviation is 10, we would multiply 1.75 by 10 to get 17.5. Then, we would add and subtract 17.5 from the mean to find the range of values that fall within 1.75 standard deviations away from the mean. This gives us a range of 82.5 to 117.5.
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HELLO THERE, "Math wizards" and others can you solve these tricky problems? 2 ÷ 14 and solve this -5 ÷12 I will mark BRAINILEST ON WHOEVER MAKES THIER ANSWER IF THIER IS NOT A EXACT COPY OFF OF SOME ELSE'S :] looks like your "math power level" won't be high enough to beat this problem.....
Answers
Answer:
2/14=0.143
-5/12= -0.417
Step-by-step explanation:
calculate the perimiter of a regular pentagon with side length of 5 cm
Answers
Answer:
Regular pentagon
Solve for perimeter
P=25cm
a Side
5
cm
For a recent paint job, Josh mixed red and white paint to make two different shades of pink. When the job was done, Josh ended up with leftover paint: 5 gallons of dark pink paint (80% red) and 4 gallons of light pink paint (30% red). Josh wants to make a medium pink color (50% red) to paint his daughter's bedroom. He will need 3 gallons to completely cover the walls. How much of each of the leftover paints should Josh mix to achieve his desired color?
? gallons of dark pink paint
? gallons of light pink paint
Answers
Josh should mix 1.2 gallons of dark pink paint and 1.8 gallons of light pink paint to achieve the desired medium pink color.
To find out how much of each leftover paint Josh should mix to achieve a medium pink color (50% red), we can set up a system of equations based on the percentages of red in the paints.
Let's assume that Josh needs x gallons of dark pink paint and y gallons of light pink paint to achieve the desired color.
The total amount of paint needed is 3 gallons, so we have the equation:
x + y = 3
The percentage of red in the dark pink paint is 80%, which means 80% of x gallons is red. Similarly, the percentage of red in the light pink paint is 30%, which means 30% of y gallons is red. Since Josh wants a 50% red mixture, we have the equation:
(80/100)x + (30/100)y = (50/100)(x + y)
Simplifying this equation, we get:
0.8x + 0.3y = 0.5(x + y)
Now, we can solve this system of equations to find the values of x and y.
Let's multiply both sides of the first equation by 0.3 to eliminate decimals:
0.3x + 0.3y = 0.3(3)
0.3x + 0.3y = 0.9
Now we can subtract the second equation from this equation:
(0.3x + 0.3y) - (0.8x + 0.3y) = 0.9 - 0.5(x + y)
-0.5x = 0.9 - 0.5x - 0.5y
Simplifying further, we have:
-0.5x = 0.9 - 0.5x - 0.5y
Now, rearrange the equation to isolate y:
0.5x - 0.5y = 0.9 - 0.5x
Next, divide through by -0.5:
x - y = -1.8 + x
Canceling out the x terms, we get:
-y = -1.8
Finally, solve for y:
y = 1.8
Substitute this value of y back into the first equation to solve for x:
x + 1.8 = 3
x = 3 - 1.8
x = 1.2
Therefore, Josh should mix 1.2 gallons of dark pink paint and 1.8 gallons of light pink paint to achieve the desired medium pink color.
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(08.07 HC)
An expression is shown below:
f(x) = 2x²-3x - 5
Part A: What are the x-intercepts of the graph of f(x)? Show your work. (2 points)
Part B: Is the vertex of the graph of f(x) going to be a maximum or minimum? What are the
coordinates of the vertex? Justify your answers and show your work. (3 points)
Part C: What are the steps you would use to graph f(x)? Justify that you can use the answers
obtained in Part A and Part B to draw the graph. (5 points)
(10 points)
Answers
Answer:
Below in bold.
Step-by-step explanation:
Part A
At the x-intercepts f(x) = 0, therefore:
2x^2 - 3x - 5 = 0
(2x - 5)(x + 1) = 0
x = 5/2, -1)
So the x intercepts are (-1, 0) and (5/2, 0).
Part B
The Coefficient of x^2 is positive ( it is 2) So the graph opens upwards and the vertex will be a minimum.
Convert f(x) to vertex form, by completing the square:
f(x) = 2x^2 - 3x - 5
= 2(x^2 - 3/2x) - 5
= 2[(x - 3/4)^2 - 9/16] - 5
= 2(x - 3/4)^2 -18/16 - 80/16
= 2(x - 3/4)^2 - 49/8
So the coordinates of the vertex are
(3/4, -49/8) or
(0.75, -6.125) in decimal form.
Part C
To graph f(x) you would first mark the points on the x axis which we found in Part 1 and the vertex found in Part 2. This vertex will be the bottom of the 'U'.
The graph is a parabola shaped roughly like a U, and will be symmetrical about the line x = 0.75 (which passes through the vertex).
You would also plot 2 more points above the x axis so as to get an accurate graph. 1 would be to the left of the line of symmetry and 1 to its right.
Suggest x = - 2 and calculate f(-2) = 2(-2)^2 - 3(-2) - 5 = 11.
- that is the point (-2,11) and the other would be x = 4, f(x) = 15. (4, 15)
Once you have plotted these points draw a smooth u shaped curve through them.