Answer
2) Option C is correct.
The vertex of the quadratic function is at
x = 6, y = 8.
In coordinate form, the vertex = (6, 8)
4) Option A is correct.
-3 stretches the graph and reflects it about the x-axis.
Explanation
2) We are told to find the vertex of the quadratic function. The vertex of a quadratic function is the point at the base of the curve/graph of the function. It is the point where the value of the quadratic function changes sign.
The x-coordinate of this vertex is given as
x = (-b/2a)
The y-coordinate is then obtained from the value of the x-coordinate.
The quadratic function for the question is
y = -5 (x - 6)² + 8
We first need to put the quadratic function in the general form of
y = ax² + bx + c
So, we first simplify the expression
y = -5 (x - 6)² + 8
= -5 (x² - 12x + 36) + 8
= -5x² + 60x - 180 + 8
y = -5x² + 60x - 172
So,
a = -5
b = 60
c = -172
For the vertex
x = (-b/2a)
= [-60/(2×-5)]
= [-60/-10]
= 6
So, if x = 6.
y = -5x² + 60x - 172
y = -5(6²) + 60(6) - 172
y = -5(36) + 360 - 172
y = -180 + 360 - 172
y = 8
So, the vertex of the quadratic function is at
x = 6, y = 8.
In coordinate form, the vertex = (6, 8)
Option C is correct.
4) y = -3(x²)
The graph of x² is a parabola, but multiplying the function x² by -3 transforms the graph.
The 3, because it is greater than 1, stretches or enlarges the graph.
And the minus sign in front of the 3, ,that is, -3 reflects the graph about the x-axis.
So, altogether, -3 stretches the graph and reflects it about the x-axis.
Option A is correct.
Hope this Helps!!!
which of the following statements about the function f(x)=x2-2x-2 is true
Express f (x) = x^2 -2x in the form f(x) = (x - h ) ^2 - k
x^2 - 2x = +2
h = -b/2a and k = h^2
a = 1 , b= -2
h= -(-2)/ 2(1) = 2/2 = 1
k = h^2 = 1^2 = 1
So, x^2 - 2x = (x-1) ^2 - 1
To rewrite the complete equation
f(x) = (x - 1)^2 - 1 - 2
f(x) = (x - 1)^2 - 3,
[tex]f(x)=(x-1)^2-3[/tex]The minimum value is is -3
Option D is the answer
Graph J(2,-1), K(4,-5), and L(3,1) and reflect across the x=-1. Please draw the line of reflection.
You have the following points:
J(2,-1), K(4,-5), L(3,1)
To reflect the previous points around the line x = -1, consider the horizontal distance of each point to the given line x=-1. The reflected point is obtained by using the same distance to the line but in the other side.
You proceed as follow:
J(2,-1)
the distance of the previous point to the line x=-1 is 2-(-1) = 3. You subtract this value to x = -1. Thus, the x-coordinate of the new point is:
-1-3 = -4
and the new point is:
J(2,-1) => J'(-4,-1)
For K(4,-5) you have:
distance to the line x=-1 is 4-(-1) = 5. Subtract this value to the line.
-1-5 = -6
and the new point is:
K(4,-5) => K'(-6,-5)
For L(3,1):
distance to the line x=-1 is 3-(-1) = 4.
-1-4 = -5
and the new point is:
L(3,1) => L'(-5,1)
A plot of the original and reflected points is given below:
where the figure with black lines is the original figure and the figure with blue lines is the reflected one.
B. The perimeter of this rectangle is 20 centimeters. What is the value of X
Statement Problem: Find the value of x in the diagram below, given the perimeter of a rectangle as 20centimeters.
Solution:
The perimeter of a rectangle is;
[tex]P=2(l+w)[/tex]Where the length and width of the given rectangle is;
[tex]\begin{gathered} l=(x+3)cm \\ w=(x+1)cm \end{gathered}[/tex]Thus, the value of x is;
[tex]\begin{gathered} 2(l+w)=20 \\ 2(x+3+x+1)=20 \\ \text{Divide both sides by 2},\text{ we have;} \\ \frac{2\mleft(x+3+x+1\mright)}{2}=\frac{20}{2} \\ x+3+x+1=10 \\ \text{Collect like terms, we have;} \\ 2x+4=10 \\ \end{gathered}[/tex]Then, we subtract 4 from both sides of the equation, we have;
[tex]\begin{gathered} 2x+4-4=10-4 \\ 2x=6 \\ \text{Divide both sides by 2, we have;} \\ \frac{2x}{2}=\frac{6}{2} \\ x=3 \end{gathered}[/tex]The value of x is 3
The product of two positive consecutive odd integers is 195. Create and solve an equation to find the value of the integers. What is the sum of the two integers?
Let's define the next variables:
x: the first odd integer
y: the next odd integer
Since they are consecutive:
x + 2 = y
The product of them is 195, then:
x*y = 195
Replacing the y from the first equation into the second one:
x*(x + 2) = 195
x*x + x*2 - 195 = 0
x² + 2x - 195 = 0
Solving with help of the quadratic formula:
[tex]\begin{gathered} x_{1,2}=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a} \\ x_{1,2}=\frac{-2\pm\sqrt[]{2^2-4\cdot1\cdot(-195)}}{2\cdot1} \\ x_{1,2}=\frac{-2\pm\sqrt[]{784}}{2} \\ x_1=\frac{-2+28}{2}=13 \\ x_2=\frac{-2-28}{2}=-15 \end{gathered}[/tex]Given that we are only interested in positive integers, the solution x = -15 is discarded.
Therefore, the integers are 13 and 15
The sum of them is 13 + 15 = 28
what is 11.77 hr converted to hours and min
We are asked to convert 11.77 hours into hours and minutes. The first step is to divide the whole number from its decimal part, that is:
[tex]11.77\text{ hours=11 hours+ 0.77 hours}[/tex]Now we convert the decimal hours into minutes. To do that we use the conversion factor 1h = 60 minutes. We get:
[tex]0.77\text{hour}\frac{60\min}{1hour}=46.2\min [/tex]Therefore, 11.77 hours is approximately 11 hours and 46 minutes.
Graph the function by starting with the graph of the basic function and then using the techniques of shifting, compressing, stretching, and/or reflecting.f(x) = -2(x + 1)2 + 2
Okay, here we have this:
Considering the provided function, we are going to graph it, so we obtain the following:
First, we are going to start with the graph of the basic function, so the graph so far is:
Now, we are going to add 1 inside the square, which is reflected in a shift of the graph one unit to the left, so we have:
We continue, multiplying the entire function by -2, which results in a compression of the graph, and in a reflection on the x-axis due to the change of sign, now we have:
And finally we add two units to the whole function, therefore it means that the graph will move up two units, leaving the following graph:
Finally we obtain that the correct answer is the first answer choice.
QuestionFind the equation of the line with slope -2 which goes through the point (7, -1).Give your answer in slope-intercept form y = mx +b.Provide your answer below:
From the question, we can deduce the following:
• slope, m = -2
,• The line goes through the points: (7, -1)
Let's find the equation of the line in slope-intercept form.
Apply the slope-intercept formula of a line:
y = mx + b
Where m is the slope and b is the y-intercept.
Here, the slope, m is = -2
We also have the point:
(x, y) ==> (7, -1)
Substitue -2 form, 7 for x, and -1 for y to find b.
y = mx + b
-1 = -2(7) + b
-1 = -14 + b
Add 14 to both sides:
-1 + 14 = -1 + 14 + b
13 = b
The y-intercept is 13.
Therefore, the equation of the line in slope-intercept form is:
y = -2x + 13
ANSWER:
y = -2x + 13
Find a_1 for the geometric sequence with the given terms. a_3 = 54 and a_5 = 486
ANSWER
[tex]6[/tex]EXPLANATION
We want to find the first term of the sequence.
The general equation for the nth term a geometric sequence is written as:
[tex]a_n=ar^{n-1}[/tex]where a = first term; r = common ratio
Let us use this to write the equations for the third term and the fifth term.
For the third term, n = 3:
[tex]\begin{gathered} a_3=ar^2 \\ \Rightarrow54=ar^2 \end{gathered}[/tex]For the fifth term, n = 5:
[tex]\begin{gathered} a_5=ar^4 \\ \Rightarrow486=ar^4 \end{gathered}[/tex]Let us make a the subject of both formula:
[tex]\begin{gathered} 54=ar^2_{} \\ \Rightarrow a=\frac{54}{r^2} \end{gathered}[/tex]and:
[tex]\begin{gathered} 486_{}=ar^4 \\ a=\frac{486}{r^4} \end{gathered}[/tex]Now, equate both equations above and solve for r:
[tex]\begin{gathered} \frac{54}{r^2}=\frac{486}{r^4} \\ \Rightarrow\frac{r^4}{r^2}=\frac{486}{54} \\ \Rightarrow r^{4-2}=9 \\ \Rightarrow r^2=9 \\ \Rightarrow r=\sqrt[]{9} \\ r=3 \end{gathered}[/tex]Now that we have the common ratio, we can solve for a using the first equation for a:
[tex]\begin{gathered} a=\frac{54}{r^2} \\ \Rightarrow a=\frac{54}{3^2}=\frac{54}{9} \\ a=6 \end{gathered}[/tex]That is the first term.
help.par1. What is the product of 2/10 and 4/9?A. 6/19B. 4/45c. 9/20D. 11/14
To perform a product of fractions, we multiply the numerator with denominator and denominator with denominator:
[tex]\frac{2}{10}\cdot\frac{4}{9}=\frac{2\cdot4}{10\cdot9}[/tex]And solve:
[tex]\frac{2\cdot4}{10\cdot9}=\frac{8}{90}[/tex]And simplify:
[tex]\frac{8}{90}=\frac{4}{45}[/tex]The answer is option B. 4/45
Find the area of the figure. zyd 13 / yd The area of the figure is yd?
The area of the given parallelogram is:
A = b·h
b: base = 13 1/5 = (65 + 1)/5 = 66/5 = 13.2 yd
h: height = 27 1/2 = 13.5 yd
A = (13.2 yd)(13.5 yd) = 178.2 yd²
Car X weighs 136 pounds more than car Z. Car Y weighs 117 pounds more than car Z. The total weight of all three cars is 9439 pounds. How much does each car weigh?
Let x, y and z denote the weighs of car X, car Y and car Z, respectively.
We know that car X weighs 136 more than car Z, this can be express by the equation:
[tex]x=z+136[/tex]We also know that Y weighs 117 pounds more than car Z, this can be express as:
[tex]y=z+117[/tex]Finally, we know that the total weight of all the cars is 9439, then we have:
[tex]x+y+z=9439[/tex]Hence, we have the system of the equations:
[tex]\begin{gathered} x=z+136 \\ y=z+117 \\ z+y+z=9439 \end{gathered}[/tex]To solve the system we can plug the values of x and y, given in the first two equations, in the last equation; then we have:
[tex]\begin{gathered} z+136+z+117+z=9439 \\ 3z=9439-136-117 \\ 3z=9186 \\ z=\frac{9186}{3} \\ z=3062 \end{gathered}[/tex]Now that we have the value of z we plug it in the first two equations to find x and y:
[tex]\begin{gathered} x=3062+136=3198 \\ y=3062+117=3179 \end{gathered}[/tex]Therefore, car X weighs 3198 pound, car Y weighs 3179 pounds and car Z weighs 3062 pounds.
There are as many even counting numbers as there are counting numbers. Is this true or false?
The sets of even and odd counting numbers are both infinite in size (number of elements). However, we can map each even counting number to each odd counting number as follows:
[tex]\begin{gathered} 2\rightarrow1 \\ 4\rightarrow3 \\ 6\rightarrow5 \\ 8\rightarrow7 \\ \text{And so on}\ldots \end{gathered}[/tex]So we have the mapping rule:
[tex]\begin{gathered} 2n\rightarrow2n-1 \\ \text{Where }n=1,2,3,\ldots \end{gathered}[/tex]Then, we can say that there are as many even counting numbers as there are counting numbers, or equivalently, that both sets have the same cardinality.
Answer: True
Write the slope-intercept form of the equation of the line describedthrough: (-5, 4), perp. to y=3/4x+4
Answer:
To find the equation of the line passes through (-5,4) and perpendicular to the line y=3/4 x +4
The slope of the line y=3x/4 +4 is 3/4
Let m be the slope of the required equation of the line,
Since these two lines are perpendicular to each other.
we know that,
Product of the slopes of the perpendicular lines is -1
That is,
[tex]\frac{3}{4}\times m=-1[/tex][tex]m=-\frac{4}{3}[/tex]Slope of the required line is -4/3
Passes through the point (-5,4)
The equation of the line passes through the point (x1,y1) and m as its slope is
[tex](y-y1)=m(x-x1)[/tex]Substitute x1=-5, y1=4 and m=-4/3
we get,
[tex](y-4)=-\frac{4}{3}(x+5)[/tex][tex]y-4=-\frac{4}{3}x-\frac{20}{3}[/tex][tex]y=-\frac{4}{3}x-\frac{20}{3}+4[/tex][tex]y=-\frac{4}{3}x+\frac{(-20)+12}{3}[/tex][tex]y=-\frac{4}{3}x-\frac{8}{3}[/tex]The required slope-intercept form of the equation of the line is y=-4/3 x-8/3.
How to find slope & y interceptAnd solve for Y-6x+2y=10
Slope intercept form
y= mx + b
Where:
m= slope
b= y-intercept
So, first, we have to solve for y:
-6x +2y = 10
2y = 6x + 10
y = (6x + 10) /2
y = 3x + 5
Slope = 3
y-intercept = 5
I need help to solve. This is my daily practice assignment
The scenario formed a right triangle with an adjacent side of 24.2 ft. and included an angle of 37°.
First, let's recall the three main trigonometric functions.
[tex]\text{ Sine }\Theta\text{ = }\frac{Opposite\text{ Side}}{\text{Hypotenuse}}[/tex][tex]\text{ Cosine }\Theta\text{ = }\frac{Adjacent\text{ Side}}{Hypotenuse}[/tex][tex]\text{ Tangent }\Theta\text{ = }\frac{Opposite\text{ Side}}{Adjacent\text{ Side}}[/tex]In the scenario, the height of the flagpole appears to be the Opposite Side of the right triangle formed.
Since the function that we will be equating involves the Opposite Side and Adjacent Side of a right triangle, we will be applying the Tangent Function to find the height of the flagpole.
We get,
[tex]\text{ Tangent }\Theta\text{ = }\frac{Opposite\text{ Side}}{Adjacent\text{ Side}}[/tex][tex]Tangent(37^{\circ})\text{ = }\frac{x}{24.2}[/tex][tex]\text{ Tangent (37}^{\circ})\text{ x 24.2 = x}[/tex][tex]\text{ 18.23600801249 = x}[/tex][tex]\text{ 18.2 ft. }\approx\text{ x}[/tex]Therefore, the height of the flagpole is 18.2 ft.
(x²y³z)^1/5Rewrite the rational exponent expression as a radical expression.
Given the following rational exponent:
(x²y³z)^1/5
Putting it into a radical expression will be:
[tex]\text{ \lparen x}^2y^3z)^{\frac{1}{5}}\text{ = }\sqrt[5]{x^2y^3z^}[/tex]Makayla was scuba diving. She started at at-80 5/9 meters below the surface. She then swam up 20 2/9 meters from her storting location for a break. Alwhat location did she stop for her break compared to sea level?
Answer:
543/9 meters below the surface.
Explanation:
First, we need to transform the mixed numbers into fractions, so 80 5/9 meter and 20 2/9 meters are equivalent to:
[tex]\begin{gathered} A\frac{b}{c}=\frac{A\cdot c+b}{c} \\ 80\frac{5}{9}=\frac{80\cdot9+5}{9}=\frac{725}{9} \\ 20\frac{2}{9}=\frac{20\cdot9+2}{9}=\frac{182}{9} \end{gathered}[/tex]Now, to calculate the location where she stops for a break, we need to take 182/9 and subtract it from 725/9. So:
[tex]\frac{725}{9}-\frac{182}{9}=\frac{725-182}{9}=\frac{543}{9}[/tex]Therefore, Makayla stops for a break at 543/9 meters below the surface.
Hi. I think I am over thinking this question. Can you show me how this works step by step?
We know that:
MN = 7.3
DC = 8.7
M and N are midpoints of AD and BC respectively.
Since DC - MN = 8.7 - 7.3 = 1.4 and M and N are midpoints, we must have:
AB = 7.3 - 1.4 = 5.9
using exponential growthif the starting population of 5 rabbits grow at 200% each year, how many will there be in 20 years?
grow at 200% each year
so, population becomes twice in each year, then after 20 years:
[tex]\text{population}=5\times2^{20}=5\times1048576=5242880[/tex]answer: population is 5,242,880 after 20 years
Which line is parallel to this one: y=2/3x-9A.y=3/2x+8B.y=2/3x-9C.y=2/3x-1D.y=-3/2x+7
to find the line parallel to th egiven line:
[tex]y=\frac{2}{3}x-9[/tex]the line parallel to the given equation is
[tex]y=\frac{2}{3}x-1[/tex]The graph is,
For which value of x does p(x)=-4 in the graph below
You have to identify which dot in the graph corresponds to p(x)=-4
p(x)=-4 → this expression indicates that the value of the "output" is -4, in the graph, it will correspond to the dot that has y-coordinate= -4
The dots in the graph have the following coordinates:
The coordinates are always given in the following order (x,y), the first coordinate corresponds to the value of x (input) and the second coordinate corresponds to the value of y (output)
From the dots, the only one that has the y-coordinate -4 is the one located in the fourth quadrant with coordinates (2,-4)
I need help to graph the line this is a study guide check point it gives you the answer if you do not know it but I don’t want just the answer on how to do it I want the explanation of it being worked out
Given:
[tex]y=2x[/tex]To graph the given equation, we can plug in any values for x to get values for y as shown below:
Example 1:
Let x= 0
We plug in x= 0 into y=2x:
[tex]\begin{gathered} y=2x \\ y=2(0) \\ \text{Simplify} \\ y=0 \end{gathered}[/tex]Based on the above values of x and y, our point is (0,0).
Example 2:
We let x =2:
[tex]\begin{gathered} y=2x \\ y=2(2) \\ \text{Simplify} \\ y=4 \end{gathered}[/tex]It means that the point is (2,4).
Hence, the graph of y=2x is:
A data set has a mean of 58 and a standard deviation of 17. All of the data values are within three standard deviations of the mean. Which of the following could be the minimum and the maximumvalues of the data set?Minimum 5: Maximum 106Minimum 5; Maximum 111Minimum 8: Maximum 111Minimum 2, Maximum 109
To answer this question, we can use the standard normal distribution, and use the z-scores for finding the minimum and maximum values in the distribution.
The z-score is given by:
[tex]z=\frac{x-\mu}{\sigma}[/tex]We have that the maximum and minimum are within three standard deviations. The z-scores are a measure of the standard deviations from the population mean. Then, the values are for minimum, z = -3, and for maximum, z = 3.
The population's mean is equal to 58 (mu), and the standard deviation is equal to 17.
We are going to find the raw score, x, for the minimum and maximum values 3 standard deviations below and above the mean. Then, we have:
Minimum
[tex]-3=\frac{x-58}{17}\Rightarrow-3\cdot17=x-58\Rightarrow x=-51+58\Rightarrow x=7[/tex]Maximum
[tex]undefined[/tex]The functions f and g are defined as follows.g(x) = 4x-2-Xf(x)=-3x-1Find f (5) and g(-3).Simplify your answers as much as possible.f(s) = 0:Х?&(-3) = 0
We need to find f(5) and g(-3)
First, we will solve f(5), for this, we have the following function:
[tex]\begin{gathered} f(x)=-3x-1 \\ f(5)=-3\cdot(5)-1 \\ f(5)=-15-1 \\ f(5)=-16 \end{gathered}[/tex]Second, we will solve g(-3), for this, we have the following function:
[tex]\begin{gathered} g(x)=4x^2-x \\ g(-3)=4(-3)^2-(-3) \\ g(-3)=4\cdot9+3 \\ g(-3)=36+3 \\ g(-3)=39 \end{gathered}[/tex]In conclusion, f(5) = -16 and g(-3) = 39
if D equals 6x - 7 + 8 equals 4x + 9 find the DB
If DE = 6x - 7 and AE = 4x + 9, find DB.
DE and AE are line segments. We need to equal DE = AE
[tex]DE=AE[/tex][tex]6x-7=4x+9[/tex][tex]6x-4x=9+7[/tex][tex]2x=16[/tex][tex]x=8[/tex]Now, DB = DE + EB
[tex]DB=6x-7+4x+9[/tex][tex]DB=10x+2[/tex]x = 8
[tex]DB=10(8)+2\rightarrow DB=82[/tex]DB is 82
In a normal distribution, what percentage of the data falls within 2 standarddeviations of the mean?
In a normal distribution, 68% of data will fall within two standard deviations of the mean
Demetrius is single but would like to take care of his mother in the event of his death. He would like to replace her income of $25,000 a year with a multiple of 30. How much life insurance should he buy?
Answer:
750000
Step-by-step explanation:
The point (-2, 1) is translated 3 units down and 2 units to the left. What are the coordinates of the newpoint?Use the grid below if it helps.-65-43-214-4-3-2-1130-1--2-3-4-5-6(2,-4)o(-4,-2)(0, -4)(0,4)
Given that the point (-2, 1) is translated 3 units down and 2 units to left. The coordinate of the new point would follow the translation rule
[tex]T(x,y)\to(x-2,y-3)[/tex]Therefore, the coordinates of the new point following the translation rule are:
[tex]\begin{gathered} T(-2,1)\to_{}(-2-2,1-3) \\ =(-4,-2) \end{gathered}[/tex]The correct answer is (-4,-2)
A casting director wishes to find one male and one female to cast in his play. If he plans to audition 10 males and 14 females, in how many different ways can this be done?
There are 10 males and 14 females.
So the order doesn't matter and we cant repeat a person.
Given the before information, we are going to use combinations
[tex]c=\frac{n!}{r!(n-r)!}=[/tex]Where n is the total of people and r the election, so:
For males:
n=10
r=1
[tex]c=\frac{10!}{1!(10-1)!}=10[/tex]For females:
n=14
r=1
[tex]C=\frac{14!}{1!(14-1)!}=14[/tex]Finally, multiply both results:
10* 14 = 140
Therefore, there are 140 ways that the casting can be done.
Consider the equation below.x3 – 3x2 – 4 = 1/x-1+ 5The solutions to the equation are approximately x=and x=
Question:
Solution:
Consider the following equation:
[tex]x^3-3x^2-4=\frac{1}{x-1}+5[/tex]this is equivalent to:
[tex]x^3-3x^2=\frac{1}{x-1}+5+4[/tex]that is:
[tex]x^3-3x^2=\frac{1}{x-1}+9[/tex]Multiplying both sides by (x-1), we obtain:
[tex](x-1)(x^3-3x^2)=1+9(x-1)[/tex]this is equivalent to:
[tex](x-1)x^3-3x^2(x-1)=1+9(x-1)[/tex]solving for x, we obtain that the correct solutions are:
[tex]x\text{ }\approx0.90672[/tex]and
[tex]x\approx\: 3.68875[/tex]