(08.07 HC)
An expression is shown below:
f(x) = 2x² - x - 10
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:
\(\textsf{A)} \quad x=-2, \:\:x=\dfrac{5}{2}\)
\(\textsf{B)} \quad \left(\dfrac{1}{4},-\dfrac{81}{8}\right)=(0.25,-10.125)\)
C) See attachment.
Step-by-step explanation:
Given function:
\(f(x)=2x^2-x-10\)
Part ATo factor a quadratic in the form \(ax^2+bx+c\) , find two numbers that multiply to \(ac\) and sum to \(b\) :
\(\implies ac=2 \cdot -10=-20\)
\(\implies b=-1\)
Therefore, the two numbers are -5 and 4.
Rewrite \(b\) as the sum of these two numbers:
\(\implies f(x)=2x^2-5x+4x-10\)
Factor the first two terms and the last two terms separately:
\(\implies f(x)=x(2x-5)+2(2x-5)\)
Factor out the common term (2x - 5):
\(\implies f(x)=(x+2)(2x-5)\)
The x-intercepts are when the curve crosses the x-axis, so when y = 0:
\(\implies (x+2)(2x-5)=0\)
Therefore:
\(\implies (x+2)=0 \implies x=-2\)
\(\implies (2x-5)=0 \implies x=\dfrac{5}{2}\)
So the x-intercepts are:
\(x=-2, \:\:x=\dfrac{5}{2}\)
Part B
The x-value of the vertex is:
\(\implies x=\dfrac{-b}{2a}\)
Therefore, the x-value of the vertex of the given function is:
\(\implies x=\dfrac{-(-1)}{2(2)}=\dfrac{1}{4}\)
To find the y-value of the vertex, substitute the found value of x into the function:
\(\implies f\left(\dfrac{1}{4}\right)=2\left(\dfrac{1}{4}\right)^2-\left(\dfrac{1}{4}\right)-10=-\dfrac{81}{8}\)
Therefore, the vertex of the function is:
\(\left(\dfrac{1}{4},-\dfrac{81}{8}\right)=(0.25,-10.125)\)
Part CPlot the x-intercepts found in Part A.
Plot the vertex found in Part B.
As the leading coefficient of the function is positive, the parabola will open upwards. This is confirmed as the vertex is a minimum point.
The axis of symmetry is the x-value of the vertex. Draw a line at x = ¹/₄ and use this to ensure the drawing of the parabola is symmetrical.
Draw a upwards opening parabola that has a minimum point at the vertex and that passes through the x-intercepts (see attachment).
Related Questions
I don’t understand this
7 yd 2 ft =_____ft
Answers
Answer:
23 ft
Step-by-step explanation:
of the $261,000 received from contributors, $240,000 came from four contributors, each giving $60,000; the other $21,000 came from small individual contributions.
Answers
In the total amount of $261,000 received from contributors, a majority of $240,000 came from four contributors, with each contributing $60,000. The remaining $21,000 was obtained from smaller individual contributions.
This statement describes the breakdown of contributions received from different sources. Out of the total amount of $261,000, the majority, which is $240,000, came from four specific contributors. Each of these contributors donated $60,000, resulting in a combined contribution of $240,000. The remaining $21,000 represents the sum of smaller individual contributions from various sources.
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What is the relative frequency of ages 65 to 69? round your answer to 4 decimal places
Answers
1. The percentage of CEOs who are 59 years or younger: 57.5% 2. The relative frequency for ages 65 to 69: 0.1096 3. The cumulative frequency for CEOs over 55 years in age: 51
To answer these questions, we need to calculate the total number of CEOs and perform some calculations based on the given data. Let's proceed step by step:
Step 1: Calculate the total number of CEOs.
The total number of CEOs is the sum of the frequencies for each age group:
Total CEOs = 4 + 3 + 15 + 20 + 21 + 8 + 2 = 73
Step 2: Calculate the percentage of CEOs who are 59 years or younger.
To determine the percentage, we need to find the cumulative frequency up to the age group of 59 years and divide it by the total number of CEOs:
Cumulative frequency for CEOs 59 years or younger = Frequency for age 40-44 + Frequency for age 45-49 + Frequency for age 50-54 + Frequency for age 55-59
= 4 + 3 + 15 + 20 = 42
Percentage of CEOs 59 years or younger = (Cumulative frequency for CEOs 59 years or younger / Total CEOs) * 100
= (42 / 73) * 100
≈ 57.53%
Rounded to the nearest tenth, the percentage of CEOs who are 59 years or younger is 57.5%.
Step 3: Calculate the relative frequency for ages 65 to 69.
To find the relative frequency, we need to divide the frequency for ages 65 to 69 by the total number of CEOs:
Relative frequency for ages 65 to 69 = Frequency for age 65-69 / Total CEOs
= 8 / 73
≈ 0.1096
Rounded to four decimal places, the relative frequency for ages 65 to 69 is approximately 0.1096.
Step 4: Calculate the cumulative frequency for CEOs over 55 years in age.
The cumulative frequency for CEOs over 55 years in age is the sum of the frequencies for the age groups 55-59, 60-64, 65-69, and 70-74:
Cumulative frequency for CEOs over 55 years = Frequency for age 55-59 + Frequency for age 60-64 + Frequency for age 65-69 + Frequency for age 70-74
= 20 + 21 + 8 + 2
= 51
The cumulative frequency for CEOs over 55 years in age is 51.
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The complete question is:
Forbes magazine published data on the best small firms in 2012. These were firms which had been publicly traded for at least a year, have a stock price of at least $5 per share, and have reported annual revenue between $5 million and $1 billion. The table below shows the ages of the chief executive officers for the first 73 ranked firms
Age:
40-44
45-49
50-54
55-59
60-64
65-69
70-74
Frequency:
4
3
15
20
21
8
2
1. What percentage of CEOs are 59 years or younger? Round your answer to the nearest tenth.
2. What is the relative frequency of ages 65 to 69? Round your answer to 4 decimal places.
3. What is the cumulative frequency for CEOs over 55 years in age? Round to a whole number. Do not include any decimals.
Can somebody plz help me out:)))
Answers
Answer:28 possibly? I know that you connect it with the angles use across from the angle I hope I helped a little
Step-by-step explanation:
Answer:
Sin 30°
= 0.5 appromaxit
Write an equation for the line in slope-intercept form
Answers
Answer:
Step-by-step explanation:
its -0 ,10
in a group of 10 people, each person shakes hands with everyone else once. how many handshakea re there
Answers
The possible number of handshakes in a group of 10 people, where each person shakes hands with everyone else once are equal to 45.
Number of people in the group = 10
Each person shakes hands with everyone else once. When n people shake hands with each other, the total number of handshakes = ⁿC₂
where C denotes the combination , is represent an arrangement of objects in different ways without care of order.
Total number of handshakes = ¹⁰C₂
= 10!/2!(10 - 2)! = 10!/2!×8!
= 10×9×8!/(2×1)8!
= (10×9)/2
= 45 handshakes.
In other words, when 10 people are shaking hands, each one will shake hands with 9 others. Then, the total possible number of handshakes in a group = 10 × 9 = 90
(but remember the point that , when A shakes hands with B, B is also shaking hands with A, i.e., we are counting one handshake twice, one from A's side and the other from B's side).
The total number of handshakes in actual way = 90/2 = 45 handshakes. Hence, the total number of handshakes in a group is 45.
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It’s this one and I’m done
Answers
Answer:
168 in
Step-by-step explanation:
Pythagoras says:
b² + 56² = 70²
so
b² = 70² - 56²
b = √1764 = 42
so the perimeter is 70+56+42 = 168
solve for w and y
Please help!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Answers
Answer:
w = 10√2
y = 20
Step-by-step explanation:
Sum of interior angles of a triangle = 180°
Therefore, the triangle on the left (with side length 10) is a right triangle,
since 180° - 45°- 45° = 90°
As this triangle is also an isosceles triangle (since it has 2 equal angles), then it will have 2 sides of the same length.
So the legs of the right triangle are both 10 units.
Using Pythagoras' Theorem to calculate the hypotenuse of this triangle:
⇒ 10² + 10² = c²
⇒ 200 = c²
⇒ c = √200
⇒ c = 10√2
The triangle with side w and base y is also isosceles triangle with base angles of 45°.
Therefore, the sides are equal in length.
So w = hypotenuse of the left triangle = 10√2
We know that the horizontal distance between the left base vertex and the top vertex of this triangle is 10, therefore y = 10 + 10 = 20
line y=kx-5 is a tangent to the curve of f. Find the values of k.
Answers
The values of k that make the line y=kx-5 a tangent to the curve of f are determined by the equation k = f'(k/2), where f'(x) is the derivative of f(x).
To find the values of k for which the line y=kx-5 is a tangent to the curve of f, we need to equate the slope of the line with the derivative of f at the point of tangency.
Let (a,b) be the point of tangency on the curve of f. Then we have:
k = f'(a) (the slope of the tangent line)
b = ka - 5 (the point (a,b) lies on the tangent line)
We need to solve for k. To do this, we first need to find f'(x), the derivative of f(x).
Once we have the derivative, we can equate it to k and solve for x to find the x-coordinate of the point of tangency. Then we can use the equation of the tangent line to find the corresponding y-coordinate.
For example, if f(x) = x^2, then f'(x) = 2x. Equating this to k, we have:
2x = k
Solving for x, we get:
x = k/2
Substituting this into the equation of the tangent line, we get:
b = ka - 5
= k(k/2) - 5
= (k^2)/2 - 5
So the point of tangency is (k/2, (k^2)/2 - 5).
Therefore, the values of k for which the line y=kx-5 is a tangent to the curve of f are given by:
k = f'(k/2)
where f'(x) is the derivative of f(x).
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Which value for x makes the sentence true?
12x = 8x + 12
Group of answer choices
x = 8
x = 16
x = 1
x = 3
Answers
Answer:
x=3
Step-by-step explanation:
12*3=36
8*3=24 +12= 36
36=36
Answer:
x= 3
Step-by-step explanation:
12x=8x+12
combine like terms
12x-8x=12
4x=12
x=
\( \frac{12}{4} \)
x=3
Check:
12x-8x=12
since x=3
12(3)-8(3)=12
36-24=12
If mCA = 50° and mER = 40⁰, what is the m∆CSA ?
Answers
Answer:
45°Step-by-step explanation:
Given:
mCA = 50° mER = 40⁰Angle formed by intersecting chords is the half the sum of arc measures formed by same chords:
m∠CSA = m∠ESR = 1/2(mCA + mER) = 1/2(50° + 40°) = 45°Let f(x)=3^x and g(x)=log3(x). Describe the relationship between f(x)and g(x). Be sure to reference key characteristics such as their domain, range, asymptotes and intercepts
Answers
The relationship between f(x) = 3^x and g(x) = log3(x) is that they are inverse functions. Inverse functions are pairs of functions that "undo" each other when composed.
In this case, f(x) raises 3 to the power of x, while g(x) takes the logarithm base 3 of x. The domain of f(x) is all real numbers, while the domain of g(x) is x > 0, as logarithms are only defined for positive numbers. The range of f(x) is all positive real numbers, while the range of g(x) is all real numbers.
The graphs of f(x) and g(x) exhibit interesting characteristics. f(x) is an exponential function that increases rapidly as x increases. It has a y-intercept at (0, 1) and no x-intercepts. There are no horizontal or vertical asymptotes. On the other hand, the graph of g(x) is a logarithmic function that grows slowly as x increases. It has a vertical asymptote at x = 0 and a y-intercept at (1, 0). The graph of g(x) reflects the exponential growth of f(x) across the line y = x, which is the identity line. In summary, f(x) = 3^x and g(x) = log3(x) are inverse functions with distinct characteristics and graphs that mirror each other across the line y = x.
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Help please, tank u :)
Answers
Answer:
16 ft
Step-by-step explanation:
Let the width of the rectangle be x ft
Therefore, length = (x + 2) ft
Area of the rectangle = 15 square ft
\(x(x + 2) = 15 \\ \\ {x}^{2} + 2x = 15 \\ \\ {x}^{2} + 2x - 15 = 0 \\ \\ {x}^{2} + 5x - 3x - 15 = 0 \\ \\ x(x + 5) - 3(x + 5) = 0 \\ \\ (x + 5)(x - 3) = 0 \\ \\ x + 5 = 0 \: or \: \: x - 3 = 0 \\ \\ x = - 5 \: \: or \: \: x = 3 \\ \\ \because \: x \: can \: not \: be \: - ve \\ \\ \therefore \: x = 3 \\ \\ \implies \: x + 2 = 3 + 2 = 5 \\ \\ perimeter \: of \: rectngle \\ = 2(5 + 3) \\ = 2 \times 5 \\ = 16 \: ft\)
hey what's the answer
show step by step explanation
if u don't know.u don't need to answer
Answers
Answer:
The 3rd one - I don't seem too sure with the answer, you know the 1st question's 2nd part, right?
ʰᵒᵖᵉ ⁱᵗ ʰᵉˡᵖˢ
# ꧁❣ RainbowSalt2²2² ࿐
Is anyone willing to do my ixl? I have loads of work to do and I need my ixl due tonight. It’s 8th grade math and I’d appreciate any help
Answers
Can anyone help me with these problems I would really appreciate it
Only reply if you serious want to help me please no trolling or spam please
Answers
Answer:
Starting from top left to right each row
1. 9
2. 17.5
3. 10.5
4. 22.5
5. 10
6. 16
7. 9
8. 19
9. 15
I hope that helps.
Step-by-step explanation:
Change 33.4 metres to centimetres.
Answers
The value of 33.4 meters to centimeters is 3340 centimeters
How to convert the units
Conversion of units is described as the conversion between different units of measurement for the same quantity, and it is done mostly through multiplicative conversion factors.
It also involves multiple steps of multiplication or division by a numerical factor or, a conversion factor.
From the information given, we have to convert meters to centimeters
Note that;
100 centimeter = 1 meter
Then x centimeters = 33.4 meters
cross multiply
x = 100(33.4)
multiply the values
x = 3340 centimeters
Hence, the value is 3340 centimeters
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if f(x) = 2x - 4 find f(k+1)
Answers
Answer: \(f(k+1)=2k-2\)
Step-by-step explanation:
Substituting \(k+1\) for \(x\),
\(f(k+1)=2(k+1)-4\\\\f(k+1)=2k+2-4\\\\f(k+1)=2k-2\)
Jameel Hardware sells door locksets for $74.99, including installation. The company hires
subcontractors to install the locksets. The subcontractors receive $22.00 per lockset installed.
Other overhead costs total $7.56 per set. If Jameel Hardware pays $32.55 for each lockset, what is
the company's net profit?
А
$6.44
$12.88
с
$20.74
D
$34.88
Answers
Answer:
12.88
Step-by-step explanation:
Answer:
its 12.88
Step-by-step explanation:
Taylor read 32 pages on Monday. On Tuesday, she read twice as much as she read on Monday. On Wednesday, she read 14 fewer pages than she did on Tuesday. How many total pages did Taylor read during the three days?
Answers
Answer:
Taylor read 146 pages during these three days.
Step-by-step explanation:
32 pages on Monday.
On Tuesday twice, as much as Monday. So 2*32 = 64 on Tuesday.
On Wednesday, 14 less pages than Tuesday. 64 - 14 = 50. So he read 50 pages on Wednesday.
How many total pages did Taylor read during these three days?
32 + 64 + 50 = 146
Taylor read 146 pages during these three days.
If f(x + 3) = x^2 + 4x − 7, find f(2).
Answers
The complete statement is that if the equation of the function f(x + 3) = x^2 + 4x − 7, the value of the function f(2) is -10
What are quadratic equations?Quadratic equations are second-order polynomial equations and they have the form y = ax^2 + bx + c or y = a(x - h)^2 + k
How to determine the solution to the quadratic equation?A quadratic equations can be split to several equations and it can be solved as a whole
In this case, we have
f(x + 3) = x^2 + 4x − 7
To find f(2), we make use of the following comparison
f(2) = f(x + 3)
This means that
2 = x + 3
Evaluate
x = -1
Substitute x = -1 in f(x + 3) = x^2 + 4x − 7
f(-1 + 3) = (-1)^2 + 4(-1) − 7
This gives
f(2) = -10
Hence, the complete statement is that if the equation of the function f(x + 3) = x^2 + 4x − 7, the value of the function f(2) is -10
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Calculate the average speed of an airplane, in miles per hour, if it travels 1,000 miles in 2
2 1 2
hours.
Answers
The average speed of the airplane which travels the given distance and time is 400mph.
SpeedSpeed is simply defined as distance traveled per unit of time. It is the change of position of a moving object per unit of time.
It is expressed as;
s = distance / time = d / t
Given the data in the question;
Distance travelled d = 1000miTime taken t = 2 and 1/2 hrs = 2.5hrsAverage speed; s = ?To determine the average speed of the airplane, we substitute our given values into the expression above.
s = d / t
s = 1000mi / 2.5hrs
s = 400mph
Therefore, the average speed of the airplane which travels the given distance and time is 400mph.
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jordan drove 378 miles in 7 hours. if he can keep the same pace, how long will it take him to drive 1,188 miles?
Answers
Jordan would take 22 hours to drive 1,188 miles at the same speed he drove 378 miles in 7 hours.
How to find the time that Jordan take to drive 1,188 miles?We can use the formula:
time = distance / speed
where speed is constant, to find the time it would take Jordan to drive 1,188 miles.
The speed at which Jordan drove can be calculated as:
speed = distance / time
where distance is 378 miles and time is 7 hours.
speed = 378 / 7 = 54 miles per hour
Now, we can use the speed to find the time it would take to drive 1,188 miles:
time = distance / speed = 1,188 / 54 = 22 hours
Therefore, if Jordan can maintain the same pace, it would take him 22 hours to drive 1,188 miles.
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one card is selected from a deck of 52 cards and placed in a second deck (also 52 cards). a card then is selected from the second deck. (a) what is the probability that the second card is an ace? (b) given that an ace was drawn from the second deck, what is the conditional probability that an ace was transferred?
Answers
The probability of drawing an ace from the second deck is 1/13. Given that an ace was drawn from the second deck, the conditional probability that an ace was transferred is 1/17.
Probability that the second card is an ace:
There are initially 4 aces in the second deck.
The total number of cards in the second deck is 52.
Therefore, the probability of drawing an ace is 4/52.
Probability = Number of favorable outcomes / Total number of outcomes
Probability = 4/52
Probability = 1/13
Conditional probability that an ace was transferred, given that an ace was drawn from the second deck:
Since an ace was drawn from the second deck, we know that the second card is one of the four aces.
There was initially one card transferred from the first deck to the second deck, so there are now 51 cards left in the second deck.
Out of these 51 cards, only 3 are aces (since one ace has already been drawn).
Conditional Probability = Number of favorable outcomes / Total number of remaining outcomes
Conditional Probability = 3/51
Conditional Probability = 1/17
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Find m k
Help please
Answers
Answer:
hope this helps, happy learning :).
How do you easily find the vertical asymptote of an equation?
Answers
To easily find the vertical asymptote of an equation, you can follow these steps:
1. Identify the denominator of the rational function. The vertical asymptote will occur where the denominator is equal to zero.
2. Set the denominator equal to zero and solve for the variable. This will give you the value of the variable where the vertical asymptote occurs.
3. Write the equation of the vertical asymptote in the form of x = value.
For example, if the equation is f(x) = (x + 2) / (x - 3), the denominator is (x - 3). Setting this equal to zero and solving for x gives us:
x - 3 = 0
x = 3 and so the vertical asymptote is at x = 3.
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find the exponential equation whose graph passes through the points (-1, 4 3 43 ) and (3,108)
Answers
To find the exponential equation whose graph passes through the points (-1, 4 3 43) and (3, 108), we can use the general form of an exponential equation:y = ab^xwhere y is the output (dependent variable), x is the input (independent variable), a is the initial value or y-intercept, and b is the base or growth factor.
To find the values of a and b, we can use the given points.Using the point (-1, 4 3 43):43 = ab^(-1)43/b = a
Using the point (3, 108):108 = ab^3108/b^3 = aSubstituting a into the first equation:43/b = 108/b^3b^3 = 108/43b ≈ 1.4764
Substituting b into the second equation:a = 108/b^3a ≈ 0.0301Therefore, the exponential equation whose graph passes through the points (-1, 4 3 43) and (3, 108) is:y = 0.0301(1.4764)^x
To make this a 250-word answer, we can explain in more detail what the different parts of the equation mean. The initial value a is the value of y when x is equal to zero.
In this case, a ≈ 0.0301 represents the y-intercept of the graph.The base b is the factor by which y changes when x increases by one unit. In this case, b ≈ 1.4764 represents the rate of growth of the function.
Since b is greater than 1, the function is an exponential growth function.
The value of y at any other value of x can be found by substituting that value of x into the equation and simplifying. For example, if x = 5:y = 0.0301(1.4764)^5 ≈ 0.1638( rounded to four decimal places).
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There are 5 positions available in the new school. Of the applicant, 12 are men and 8 are women. In how many ways can 3 men and 2 women be chosen if they are equally considered?
Answers
There are 3080 ways 3 men and 2 women can be chosen if they are equally considered, using the multiplication principle of counting
What is the multiplication principle of countingThe multiplication principle states that if there are m ways to perform one task and n ways to perform another task, then there are m x n ways to perform both tasks together.
To find the number of ways to choose 3 men from the 12 men, we can use the formula for combination, which is: ⁿCᵣ = n! / (r! (n-r)!).
where n is the total number of men and r is the number of men chosen
so, the number of ways to choose 3 men from the 12 men = ¹²C₃ = 1.
Similarly, we evaluate the number of ways to choose 2 women from the 8 women
as = ⁸C₂ = 14
Now, using the multiplication principle, we can find the total number of ways 3 men and 2 women be chosen if they are equally considered.
220 x 14 = 3080
Therefore, there are 3080 ways 3 men and 2 women can be chosen if they are equally considered, using the multiplication principle of counting
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Which equation represents a line with a slope
of 2 that passes through the point (1, 5)?
Answers
The volume of a rectangular prism with a cone shaped hole in it is approximately 163.22cm³ (as shown below).
✏️ What is the height of the cone?
Answers
Answer:
Height of the cone = 5 cm
Step-by-step explanation:
Volume of the rectangular prism with a cone shaped hole = Volume of the rectangular prism - Volume of the cone
Volume of the rectangular prism = Length × Width × Height
= 5 × 5 × 7
= 175 cm³
Volume of the cone = \(\frac{1}{3}\pi r^{2}h\)
Here, r = Radius of the cone
h = Height of the cone
Volume of the cone = \(\frac{1}{3}\pi (\frac{3}{2})^{2}(h)\)
= 0.75πh cm³
Volume of the rectangular prism with a hole = 163.22 cm³
175 - 0.75πh = 163.22
0.75πh = 175 - 163.22
0.75πh = 11.78
h = 5 cm