Neither option (a) nor option (b) has a horizontal asymptote at y = 2, there is no correct answer to this question.
What is rational function?A rational function is a mathematical function that can be expressed as a ratio of two polynomial functions, where the denominator is not equal to zero.
To have vertical asymptotes at x = 3 and x = –4, the denominator of the rational function must have factors of (x – 3) and (x + 4), respectively.
Option (a) has a denominator of (x² + x – 12), which can be factored as (x – 3)(x + 4), satisfying the conditions for vertical asymptotes.
Option (b) has a denominator of (x² – x – 12), which can also be factored as (x – 3)(x + 4), satisfying the conditions for vertical asymptotes.
Therefore, the answer is either option (a) or option (b).
To determine which of these options has a horizontal asymptote at y = 2, we can perform long division or use the fact that the leading term of the rational function will determine the horizontal asymptote.
Dividing x² by (x² + x – 12), we get:
x² + x - 12 | x² + 0x + 0
- x² - x
Thus, the quotient is 1 and the remainder is x. So the rational function simplifies to:
y = x/(x² + x - 12)
The leading term of this rational function is x/x², which simplifies to 1/x. Therefore, it does not have a horizontal asymptote at y = 2.
Dividing x² by (x² – x – 12), we get:
x² - x - 12 | x² + 0x + 0
+ x² - x
Thus, the quotient is 1 and the remainder is x. So the rational function simplifies to:
y = x/(x² - x - 12)
The leading term of this rational function is x/x², which simplifies to 1/x. Therefore, it does not have a horizontal asymptote at y = 2.
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The daily cost of production in a factory is calculated using ()= 200+ 16, where is the number of complete products manufactured. Which set of numbers best defines the domain of ()?
Integers
positive real numbers
positive rational numbers
whole numbers
The domain of the function is positive integers. The function () = 200 + 16 represents the daily cost of production in a factory, where x is the number of complete products manufactured.
Since the production of products cannot be negative or fractional, the domain of the function is restricted to non-negative integers.
Therefore, the domain of the function is the set of positive integers:
{1, 2, 3, 4, ...}
Any value outside this set would not make sense in the context of the problem, as a fraction or a negative number of products cannot be produced.
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Lanny got a short term job selling computers. He is paid on
commission. In order to impress customers, he bought a few nice suits. If he has
$20 000 in sales, he will lose $140. If he has$30 000 in sales, he will make a $90
profit. Determine the:
a. rate of change and initial value
b. the amount he needs to sell to break even
c. The amount he needs to sell in order to make $1000 profit
The rate of change and initial value are 0.023 and -600 respectively.
The amount he needs to sell to break even is $26,086.96.
The amount he needs to sell in order to make $1000 profit is $69,565.22.
What is the slope-intercept form?Mathematically, the slope-intercept form of the equation of a straight line is given by this mathematical expression;
y = mx + c
Where:
m represents the slope or rate of change.x and y are the points.c represents the y-intercept or initial value.Based on the information provided, a system of equations that models the situation is given by;
-140 = 20,000x + c ...equation 1.
90 = 10,000x + c ...equation 2.
By subtracting equation 1 from equation 2, we have:
230 = 10,000x
x = 0.023
For the initial value, we have:
90 = 10,000x + c
c = -140 - 20,000(0.023)
c = -600.
In order to break even, y must be equal to 0;
y = 0.023x - 600
0 = 0.023x - 600
x = 600/0.023
x = $26,086.96.
In order to make $1000 profit, Lanny must sell the following:
y = 0.023x - 600
1,000 = 0.023x - 600
x = 400/0.023
x = $69,565.22.
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During the summer, every student became 5% taller. Eric was x before summer, after summer he was 151.2 cm.
Let's assume Eric's height before summer is x cm. After summer, he became 5% taller, which means his new height is 1.05x cm. We also know that his new height is 151.2 cm. So we can set up an equation:
1.05x = 151.2
To solve for x, we can divide both sides by 1.05:
x = 151.2 / 1.05
x = 144 cm
Therefore, Eric's height before summer was 144 cm.
Find the missing side length and angles of △ABC given that
m∠B=42∘, a=10, and c=23.
Round each answer to the nearest tenth if necessary.
23.m∠C = 48°m∠A = 90°
Given that the triangle ABC, m∠B = 42∘, a = 10, and c = 23. Now we have to find the missing side length and angles of △ABC. Step-by-step explanation: The formula to find the missing side length of a triangle is given by the Pythagoras theorem. Hence, using the Pythagoras theorem, we get;b2 = c2 - a2 = 23² - 10²b² = 529b = √529b = 23Length of b is 23.We know that the sum of all three angles of a triangle is equal to 180°. Therefore, using this concept, we can find the remaining angles of the triangle as follows;m∠A + m∠B + m∠C = 180°m∠A + 42° + m∠C = 180°m∠A + m∠C = 138°Also, we know that the sum of the two opposite angles of a triangle is equal to the third angle. Therefore, we can use this concept to find one of the remaining angles.m∠A = 180° - m∠C - m∠Bm∠A = 180° - m∠C - 42°m∠A = 138° - m∠C/ Thus,m∠A + m∠C = 138°And,m∠A = 138° - m∠COn solving these two equations, we get,m∠C = 48°m∠A = 90°Using the Pythagoras theorem we find the missing side length of a triangle as given below;b² = c² - a² = 23² - 10²b² = 529b = √529b = 23The length of the missing side is 23.Therefore, the missing side length and angles of △ABC are:Length of b is 23.m∠C = 48°m∠A = 90°
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very confused please help explan the answer
Answer:
[tex]a_{n}[/tex] = 3 + 5(n - 1)
Step-by-step explanation:
there is a common difference between consecutive terms , that is
8 - 3 = 13 - 8 = 18 - 13 = 5
this indicates the sequence is arithmetic with explicit formula
[tex]a_{n}[/tex] = a₁ + d(n - 1)
where a₁ is the first term and d the common difference
here a₁ = 3 and d = 5 , then
[tex]a_{n}[/tex] = 3 + 5(n - 1) ← is the explicit formula
Express 0939 as a fraction.
According the given question 0.0939 as a Fraction equals 939/10000.
What is fractiοn?Fractiοns are referred tο as a whοle's cοnstituent parts in mathematics. A single item οr a cοllectiοn οf items can be the cοmplete. When we really cut a slice οf cake frοm the cοmplete cake, the cοmpοnent represents the percentage οf the cake. The wοrd "fractiοn" cοmes frοm Latin.
To write 0.09390 as a fraction you have to write .09390 as numerator and put 1 as the denominator. Now you multiply numerator and denominator by 10 as long as you get in numerator the whole number.
.09390 = .09390/1 = 0.939/10 = 9.39/100 = 93.9/1000 = 939/10000
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Complete Question:
Express 0.09390.0939 as a fraction.
There are 5 students in Mrs. Templeton's art class. Each student has 2 paintbrushes on his desk. Which of the following shows how to find the total number of paintbrushes in the classroom?
1. 5 divided by 2
2. 5 times 5
3. 2 times 2
4. 5 times 2
Answer:
D
Step-by-step explanation:
To find the total number of paintbrushes in the classroom, we need to multiply the number of paintbrushes on each student's desk by the number of students.
Since there are 5 students and each student has 2 paintbrushes, we can use the multiplication operation to find the total number of paintbrushes:
5 students × 2 paintbrushes per student = 10 paintbrushes
Therefore, the correct option is 4. 5 times 2.
Billy ran 52 yards. How many feet did he run?
Answer:
The answer to your problem is, 156
First lets find out how many feet or in one yard ?
Well 1 yard = 3 feet.
Now that we know that we can use our knowledge to answer the next question:
If Bill ran 52 yards, how many feet did he ran?
We can do math to solve our answer. :
1 x 52 = 52.
Now we need to multiply 52 x 3.
That is a little bit to much, so lets make the problem easier!
By breaking up the problem \/ to make it more easier.
50 x 3. ( 5 x 3 = 15, just add a zero! ) = 150
3 x 2 = 6.
Next add:
150 + 6 = 156.
Thus the answer to your problem is, 156
Peace's average mark on her 5 maths tests was 88. If her lowest score was dropped, her new average would be 90. What is her lowest mark?
Her average score on the four remaining tests is indeed 90, which confirms that her lowest score was 80.
Let's assume Peace's lowest score on the five math tests is x.
According to the problem statement, her average mark on all five tests is 88. This means that the sum of her scores on all five tests is:
5 * 88 = 440
If her lowest score was dropped, then the sum of her scores on the remaining four tests would be:
4 * 90 = 360
We know that the sum of her scores on all five tests is 440, so we can write an equation:
440 - x = 360
Solving for x, we get:
x = 80
Therefore, her lowest mark was 80. We can check this by finding her average score after dropping her lowest score:
(88 + 88 + 88 + 88 + 90) / 5 = 90
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Write the equation of a circle given a center of (-5,-12) and a radius of 7
Answer:
(x+5)^2 + (y+12)^2 = 49
Step-by-step explanation:
(x-h)^2 + (y-k)^2 = r ^2 is the general equation for circles, where r is the radius and h,k are the center's coordinates.
The effectiveness of a new antibacterial cream called Formulation NS is being tested. From previous research, it is known that without medication, the mean healing time (defined as the time for the scab to fall off) is 7. 6 days. A random sample of 250 college students apply Formulation NS to their wounds. The mean healing time for these students is 7. 5 days with a standard deviation of 0. 9 days. Test at the 5% significance level if applying Formulation NS speeds (takes less time) healing than foregoing treatment. Use the critical value approach
The t-statistic is less than the critical value, we fail to reject the null hypothesis
The hypothesis testing to be conducted here is a two-tailed test for the population mean, using the formula:
H0: μ = 7.6 days
Ha: μ ≠ 7.6 days
The critical value is ± 1.960, with a confidence level of 95%.
The sample mean is 7.5 days and the sample standard deviation is 0.9 days. Using the formula, we can calculate the t-statistic:
t = (7.5 - 7.6) / (0.9 / √250) = -1.105
Since the t-statistic is less than the critical value, we fail to reject the null hypothesis, suggesting that Formulation NS does not significantly speed up the healing process compared to the foregoing treatment.
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What can baby lizards do that baby snakes can’t
Baby lizards can run, climb and in some cases even change color to match their surroundings, while baby snakes are generally limited to crawling and slithering.
Baby lizards have developed legs, claws and a tail to help them navigate their environment, while baby snakes have lost their legs during evolution and have developed a long, slender body to help them move around.
Additionally, some baby lizards are born with a protective membrane around their eggs that allows them to move around freely, while baby snakes are usually born inside an egg that they must break out of before hatching. Overall, baby lizards have more mobility and flexibility than baby snakes, which gives them an advantage in many environments.
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please help me in this
Answer: At a greengrocer, two bananas and one apple cost $1.16 .
Than the equation becomes
2x + y = 1.16
one banana and one apple cost 0.71.
Than the equation becomes
x + y = 0.71
Subtracting x + y = 0.71 from 2x + y = 1.16
2x - x + y - y = 1.16 - 0.71
x = 0.45
Put in the equation x + y = 0.71
0.45 + y = 0.71
y = 0.71 - 0.45
y = 0.26
The cost of the one apple is 0.26
Step-by-step explanation:
25 POINTS
Where do the graphs of f of x equals cosine of the quantity x over 2 and g of x equals square root of 3 minus cosine of the quantity x over 2 intersect on the interval [0, 360°)?
300°
150° and 210°
60°
30° and 330°
The graphs of f of x equals cosine of the quantity x over 2 and g of x equals square root of 3 minus cosine of the quantity x over 2 intersect on the interval [0, 360°) at: D. 30° and 330°.
What is a point of intersection?In Mathematics and Geometry, a point of intersection simply refers to the location on a graph where two (2) lines intersect or cross each other, which is primarily represented as an ordered pair containing the point that corresponds to the x-coordinate (x-axis) and y-coordinate (y-axis) on a cartesian coordinate.
In this context, we can logically deduce that the point of intersection of the graphs would be at function f(x) = function g(x):
[tex]cos(\frac{x}{2} )=\sqrt{3} -cos(\frac{x}{2} )\\\\\sqrt{3}=cos(\frac{x}{2} ) + cos(\frac{x}{2} )\\\\\sqrt{3}=2cos(\frac{x}{2} )\\\\cos(\frac{x}{2} )=\frac{\sqrt{3}}{2} \\\\cos(\frac{x}{2} )=\frac{\sqrt{3}}{2}\times \frac{\sqrt{3}}{\sqrt{3}} \\\\cos(\frac{x}{2} )= \frac{3}{2\sqrt{3}}[/tex]
x/2 = 15
x = 15(2)
x = 30
x/2 = 165
x = 2(165)
x = 330.
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Which equation would help find the measure of ∠m? x 52 78 = 180 52 x = 180 78 180 52 = 78 x 180 x = 78 52
Answer:
A
Step-by-step explanation:
There are 6000 registered voters in Roseville and 3/4 of these voters are registered Democrats. A survey indicates that 2/3 of the registered Democrats are in favor of Bond Measure A and 3/5 of the other registered voters are in favor of this measure.
The total number of voters in favor of Bond Measure A is 3900 voters in favor of Bond Measure A.
How to find out how many voters are in favor of Bond Measure A.First we need to calculate the total number of voters who are registered Democrats and the total number of voters who are not registered Democrats, and then calculate the number of voters in favor of the measure for each group and add them together.
Total number of registered Democrats:
3/4 of 6000 registered voters = 4500 registered Democrats
Number of registered Democrats in favor of Bond Measure A:
2/3 of 4500 registered Democrats = 3000 registered Democrats in favor of the measure
Number of registered Democrats not in favor of Bond Measure A:
4500 - 3000 = 1500 registered Democrats not in favor of the measure
Total number of voters who are not registered Democrats:
6000 - 4500 = 1500 voters who are not registered Democrats
Number of voters who are not registered Democrats and in favor of Bond Measure A:
3/5 of 1500 voters who are not registered Democrats = 900 voters who are not registered Democrats and in favor of the measure
Therefore, the total number of voters in favor of Bond Measure A is:
3000 registered Democrats in favor + 900 other registered voters in favor = 3900 voters in favor of Bond Measure A.
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A Level III video editor takes 20 hours to add special effects to a movie. A Level II video editor takes 26 hours and a Level I video editor takes 32 hours. Find how long it would take them to add the special effects if they all work together.
It would take them about 8.4 hours to add the special effects if they all work together.
What is formula for work done?1/T = 1/T1 + 1/T2 + 1/T3 +... + 1/Tn, where T is the time it takes for all n persons to complete the task and T1, T2,..., Tn are the times it takes for each person to do the task individually, is the formula for addressing issues involving work done by several people. This equation is based on the premise that each person's contribution to the activity is inversely proportionate to the time it takes them to do it. In other words, a person contributes more effort to the entire task the faster they work. This method may be used to determine how long it will take all n persons working together to do the assignment in total.
The work done is calculated by:
1/T = 1/T1 + 1/T2 + 1/T3
Here, we have T1 = 20, T2 = 26 and T3 = 32 hours for work done separately.
Substituting the values:
1/T = 1/20 + 1/26 + 1/32
1/T = (32/640) + (24/640) + (20/640)
1/T = 76/640
T = 640/76
T ≈ 8.4 hours
Therefore, it would take them about 8.4 hours to add the special effects if they all work together.
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Write a formula for the number of seconds, in any number
of minute
Number of seconds = Number of minutes x 60
What is a minute?A minute is a unit of time measurement that is equal to 60 seconds. It is commonly used to measure short periods of time, such as the duration of a conversation or the time it takes to complete a task.
What is time measurement?Time measurement is the process of quantifying the duration of an event or the interval between two events. It involves measuring the elapsed time between a start point and an end point, or between two points in time.
To convert any number of minutes into seconds, we can use the formula:
Number of seconds = Number of minutes x 60
This formula works because there are 60 seconds in one minute. So, to convert minutes to seconds, we simply multiply the number of minutes by 60 to get the equivalent number of seconds.
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A microwave was originally sold for $137 and has been marked down to $66. What is the percentage decrease for the microwave? ____%
The percentage decrease in the price of the microwave is 51.82%.
What is the percentage decrease?Percentage is used to determine the relative value of a digit as a number out of a hundred. The sign that is used to represent percentages is %. In order to express a value as a percentage, multiply the number by 100. Percentage is a measure of frequency.
Percentage decrease = (change in price / initial price) x 100
Change in price = initial price - new price
$137 - $66 = $71
(71/137) x 100 = 51.82%
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HELP ME PLEASE!!!! THIS IS DUE TODAY!!!
Answer:
Yes, the triangles are similar.
Step-by-step explanation:
The triangles are similar because of the three angle measurements.
1) One of the measurements are labeled, and the other two are implied.
2) The measurement of angles HMG and JMK are equal because they are vertical angles.
3) Lastly, because two of the angles are the same, we know that the last angles, GJK and HGJ are the same because the three angles must have a sum of 180°.
For sample proportion 0.450 and sample size 31, what is the
standard error of the distribution of sample proportions? Enter
your answer with 3 decimal places.
The standard error of the distribution of sample proportions is approximately 0.090.
The formula to find the standard error of the distribution of sample proportions is given by:Standard error = √(p(1-p)/n)Where,p = Sample proportionn = Sample sizeGiven that the sample proportion is 0.450 and the sample size is 31, we can substitute the values into the formula to find the standard error.Standard error = √(0.450(1-0.450)/31) ≈ 0.090So, the standard error of the distribution of sample proportions is approximately 0.090.
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DW
Problem 2: Given are a segment AB and a ray CD. Use compass, straightedge, and pencil,
to construct a point X on CD such that
CX = 2 1/2 AB
m is parallel to AB, and passing through points C.
Therefore, m II AB.
In geometry, a line segment is a part of a line bounded by two distinct endpoints and contains every point of the line between its endpoints. The length of a line segment is given by the Euclidean distance between its endpoints. A closed segment includes two endpoints, an open segment does not include both endpoints; a half-open segment has exactly one end. In geometry, a line segment is usually represented using a line above the two end symbols (like AB).
We use the basic rules of construction to draw lines according to the problem.
Draw a straight line AB and take a point C outside of it. Using a ruler and compass to draw the AB line, follow the steps below:
Draw a line AB, and take a point C outside this line. Take any point P on AB.Connect C to P.Take P as the center of the circle, take a suitable radius, draw an arc and cut AB in D, and PC in E.With C as the center and the same radius as in the previous step, draw an arc FG, and PC to H.Adjust the compass to the length of DE. Without changing the compass aperture, draw an arc HG at point I with H as the center.Draw a line l connecting points C and I as shown in the figure.Therefore, the line l is parallel to the line AB.
Complete Question:
Draw a line, say AB, take a point C outside it. Through C, draw a line parallel to AB using ruler and compasses only.
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For what ranges of values, holding all else constant, could each of the objective function coefficients be changed without changing the optimal solution? (
The optimal solution of an objective function does not change when its coefficients are changed within a certain range of values. This range of values for each coefficient depends on the function itself and the value of other coefficients.
For example, if the objective function is 3x + 2y, then the range of values for the coefficient of x (3) can be changed from 2 to 4 without changing the optimal solution. Similarly, the coefficient of y (2) can be changed from 1 to 3 without changing the optimal solution.
Therefore, the ranges of values for each coefficient of the objective function, holding all else constant, can be changed without changing the optimal solution.
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The area of cross-section of a solid cylinder is 803. 84 ft2, and the height of the solid is 12. 25 ft. Find the volume of the solid
cylinder using the Cavalieri's Principle.
9800. 25 ft
9587. 15 f3
9847. 04 ft3
9807. 50 ft
V = π(√(803.84/π))^2(12.25) = 9587.15 ft^3. Hence, following Cavalieri's theory, the solid cylinder has a volume of around 9587.15 ft3.
According to Cavalieri's principle, if two solids have the same height and every plane section passing through one solid along the height has the same area as the corresponding plane section passing through the other solid, the two solids must also have the same volume.
A circle of radius r has the same cross-section as a solid cylinder. A = r2 calculates the circle's area.
Therefore, πr^2 = 803.84 r^2 = 803.84/π\sr = √(803.84/π)
The solid cylinder is 12.25 feet tall.
We may get the volume of the solid cylinder using the volume of a cylinder formula, V = r2h: V = π(√(803.84/π))^2(12.25) = 9587.15 ft^3
Hence, following Cavalieri's theory, the solid cylinder has a volume of around 9587.15 ft3.
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8) Is \( f(x)=6 x^{2}-\frac{4}{x} \) odd, even or neither? You must show all of your algebra. 10 points 9) Find the equation of the line that goes though \( (-1,3) \) and \( (2,-4) \). 10 points
This function is neither odd nor even. To show this, we can take the algebraic approach.
The definition of an odd function is that for every \( x \), \( f(-x) = -f(x) \).
Therefore, we can substitute \( -x \) for \( x \) and see if this statement is true.
\( f(-x) = 6(-x)^{2}-\frac{4}{-x} \)
\( f(-x) = -6 x^{2}+\frac{4}{x} \)
Since this is not equal to \( -f(x) \), which is \( -6 x^{2}-\frac{4}{x} \), this function is neither odd nor even.
9) The equation of the line that goes through the two points \( (-1,3) \) and \( (2,-4) \) is \( y = \frac{1}{3}x + \frac{17}{3} \).
To find the equation, we can use the point-slope formula. The slope of the line is \( \frac{-7}{3} \), which we can find using the two points.
\( m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{-4-3}{2-(-1)} = \frac{-7}{3} \)
The point-slope formula for the equation of the line is \( y - y_{1} = m(x - x_{1}) \). We can substitute in the slope and the point \( (-1,3) \) to get the equation.
\( y - 3 = \frac{-7}{3}(x - (-1)) \)
\( y - 3 = \frac{-7}{3}(x + 1) \)
\( y = \frac{-7}{3}x - \frac{7}{3} + 3 \)
\( y = \frac{1}{3}x + \frac{17}{3} \)
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Classify the quadrilateral by its most specific name. Explain your reasoning.
To classify a quadrilateral by its most specific name, we need to examine its sides and angles.
What is a quadrilateral?A quadrilateral is four-sided polygon with four angles. It is closed shape with straight lines that do not cross each other.
Here are the most common types of quadrilaterals and their defining characteristics:
Parallelogram - quadrilateral consist of two pairs of parallel sides.
Rectangle - parallelogram with four right angles.
Rhombus - parallelogram with four congruent sides.
Square - rectangle and a rhombus, with four right angles and four congruent sides.
Let's consider an example.Suppose we are given a quadrilateral with four sides of equal length and two pairs of opposite parallel sides. This means that it has the characteristics of a parallelogram and a rhombus.
Reasoning: The quadrilateral has four equal sides, which makes it a rhombus. Additionally, it has two pairs of parallel sides, which satisfies the criteria for a parallelogram. However, since a rhombus is a type of parallelogram with additional properties, it is the most specific name for this quadrilateral.
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11. F(x) = 2x2² - 12x +9
a. Concave
Narrower / Wider / Same
b. Axis of Symm.
(Show work here. )
C. Vertex
(Show work here. )
Width: (Circle One)
d. Domain:
_/2
10637910-y
Range:
/2
Inlo4 lemobib A
-/2
$
T
-3
t
m
$43
N
P
3
-5
e. Y-intercept & Reflection:
n
Additional Point & Reflection: _______
(Show work here. )
C
/3
/2
a. To establish whether the function is concave up or concave down, we must obtain the function's second derivative. F(x) = 2x2 - 12x + 9 F'(x) = 4x - 12 F"(x) = 4x - 12 F"(x) = 4.
The function is concave up because the second derivative has a constant value of 4, which is positive. Since the coefficient of x2 is more than one, the function is narrower than the conventional parabola y = x2. b. The equation x = -b/2a gives the parabola's axis of symmetry, where a and b are the coefficients of the x2 and x terms, respectively.A = 2 and b = -12 for the function F(x) = 2x2 - 12x + 9. As a result, the axis of symmetry is x = -12/(2*2) = 3. c. The point at which The coordinates (h, k) define the parabola, where h = -b/2a and k = f. (h). We already determined that the axis of symmetry for the function F(x) = 2x2 - 12x + 9 is x = 3. So, h = 3. To find k, enter h = 3 into the function: F(3) = 2(3)^2 - 12(3) + 9 = -9 As a result, the vertex is (3, -9). Width: The parabola's width is the distance between the x-intercepts, which may be calculated using the quadratic formula.
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Find the equation of a line that passes through the points (1,-3) and (3,-4).
[tex](\stackrel{x_1}{1}~,~\stackrel{y_1}{-3})\qquad (\stackrel{x_2}{3}~,~\stackrel{y_2}{-4}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{-4}-\stackrel{y1}{(-3)}}}{\underset{\textit{\large run}} {\underset{x_2}{3}-\underset{x_1}{1}}} \implies \cfrac{-4 +3}{2} \implies \cfrac{ -1 }{ 2 } \implies - \cfrac{ 1 }{ 2 }[/tex]
[tex]\begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{(-3)}=\stackrel{m}{- \cfrac{ 1 }{ 2 }}(x-\stackrel{x_1}{1}) \implies y +3 = - \cfrac{ 1 }{ 2 } ( x -1) \\\\\\ y+3=- \cfrac{ 1 }{ 2 }x+\cfrac{1}{2}\implies y=- \cfrac{ 1 }{ 2 }x+\cfrac{1}{2}-3\implies {\Large \begin{array}{llll} y=- \cfrac{ 1 }{ 2 }x-\cfrac{5}{2} \end{array}}[/tex]
Answer: First, let's find the slope of the line:
slope = (change in y) / (change in x)
slope = (-4 - (-3)) / (3 - 1)
slope = -1/2
Now, let's choose one of the points, say (1,-3), and use the point-slope formula:
y - y1 = m(x - x1)
where m is the slope and (x1, y1) are the coordinates of the point.
So, substituting in the values we get:
y - (-3) = (-1/2)(x - 1)
Simplifying this equation, we get:
y + 3 = (-1/2)x + 1/2
Subtracting 3 from both sides, we get:
y = (-1/2)x - 5/2
Therefore, the equation of the line that passes through the points (1,-3) and (3,-4) is y = (-1/2)x - 5/2.
Your welcome.
Step-by-step explanation:
how to find slope of x + y = 9
Answer:
m = -1
Step-by-step explanation:
Let's change the equation to y = mx + b
m = the slope
x + y = 9
Subtract x both sides
y = -x + 9
m = -1
So, the slope is -1
PLEASE SOMEBODY HELP: Generate a symbolic rule for locating the point that divides a line segment into two parts so that the ratio of the lengths is m: n, with the point closer to the left endpoint.
Step-by-step explanation:
Let the coordinates of the left endpoint of the line segment be (x1, y1) and the coordinates of the right endpoint be (x2, y2). Let the point dividing the line segment be (x, y). Then the distance between the left endpoint and (x, y) is mx/(m+n) and the distance between (x, y) and the right endpoint is nx/(m+n).
Using the distance formula, we have:
[tex]distance \: between \: (x1, y1) \: and \: (x, y) = \sqrt{((x - x1)^2 + (y - y1)^2)} = \frac{mx}{(m+n)} [/tex]
[tex]distance \: between \: (x, y) \: and \: (x2, y2) = \sqrt{((x2 - x)^2 + (y2 - y)^2)} = \frac{nx}{(m+n)} [/tex]
Squaring both equations, we get:
[tex](x - x1)^2 + (y - y1)^2 = \frac{mx}{(m+n)^2}[/tex]
[tex](x2 - x)^2 + (y2 - y)^2 = \frac{nx}{(m+n)^2}[/tex]
Expanding the squares, we get:
[tex]x^2 - 2x1x + x1^2 + y^2 - 2y1y + y1^2 = \frac{mx}{(m+n)^2} [/tex]
[tex]x^2 - 2x2x + x2^2 + y^2 - 2y2y + y2^2 = \frac{nx}{(m+n)^2}[/tex]
Rearranging and simplifying, we get:
[tex]x = \frac{(mx2 + nx1)}{(m+n)} \: and \: y = \frac{(my2 + ny1)}{(m+n)}[/tex]
Therefore, the point that divides the line segment into two parts so that the ratio of the lengths is m: n and the point is closer to the left endpoint is:
[tex](x, y) = \frac{(mx2 + nx1)}{(m+n)}, \frac{(my2 + ny1)}{(m+n)}[/tex]