The equation of the parabola with a vertex at the origin and a directrix at; y = 5 as described in the task content is; x² = -20y.
How to write equation of a parabola with a vertex at the origin?It follows from the task content that the vertex of the parabola is at the origin while it's directrix is the line; y = 5.
Therefore, it follows that the focus of the parabola is; f = -5, since; (f - k) = (k - 5) where, k = 0.
Ultimately, a = 1/4(f-k).
a = 1/4(-5-0) = -1/20.
Recall, the vertex form equation of a parabola takes the form;
y = a(x - h)² + k; where (h, k) is the vertex.
Therefore, since the vertex is at the origin; we have;
y = (-1/20) (x -0)² + 0
y = -x²/20
20y = -x²
Ultimately, the required equation of the parabola is;
x² = -20y.
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Answer:
[tex]y = -\frac{1}{20} x^{2}[/tex]
Step-by-step explanation:
[tex]y = \frac{1}{4(f-k)}(x - h)^{2} + k[/tex] Use the equation of the parabola. The vertex is at the origin, then (h, k) is (0, 0). Substitute 0 for h and k in the equation.
[tex]y = \frac{1}{4(f-0)} (x-0)^{2} + 0[/tex] Simplify.
[tex]y = \frac{1}{4f} x^{2}[/tex]
The directrix is given [tex]y = 5[/tex]. The distance from the focus to the vertex equals the distance from the vertex to the directrix, so
[tex]f - k = k - 5\\f = -5[/tex] Substitute -5 for f in the equation of the parabola above.
[tex]y = \frac{1}{4 * - 5} x^{2}[/tex] Simplify.
[tex]y = - \frac{1}{20} x^{2}[/tex]
The graph below plots the values of y for different values of x: plot the ordered pairs 1, 3 and 2, 4 and 3, 9 and 4, 7 and 5, 2 and 6, 18 What does a correlation coefficient of 0.25 say about this graph? (1 point) x and y have a strong, positive correlation x and y have a weak, positive correlation x and y have a strong, negative correlation x and y have a weak, negative correlation
The presented data for x and y show a significant, positive correlation with 0.78 serving as the coefficient of correlation.
The intensity and direction of a relationship between two variables are indicated by a correlation coefficient, which is a number between -1 and 1. In other words, it shows how comparable two or more variables' measurements are across a dataset. The other variables shift in the same direction when one changes.
The intensity and direction of a relationship between two variables are indicated by a correlation coefficient, which is a number between -1 and 1. In other words, it shows how comparable two or more variables' measurements are across a dataset.
Given x and y values in this case
x 1 2 3 4 5 6
y 3 4 9 7 2 18
xy 3 8 27 28 10 108
x² 1 4 9 16 25 36
y² 9 16 81 49 4 324
∑x = 21,
∑y = 43,
∑xy = 194,
∑x² = 91,
∑y² = 483,
n = 6
the relationship between x and y now:
r = (n∑xy - ∑x.∑y)/ √[{n∑x²- (∑x)²}{n∑y² - (∑y)²}]
r = ( 6 X 194 - 21 X 43 ) / √[ {6 X 91 - (21)²}{ 6 X 483 - (43)²}]
r = ( 1164 - 903) / √( 546 - 441)(2898 - 1849)
r = 261 / √ 105 X 1049
r = 261 / √110145
r = 261 / 331.88
r = 0.78
As a result, the x and y data's correlation coefficient is 0.78.
The strong, positive association between x and y
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Which four groups make up nearly one half of the male population?
We are given the percentages for males on the left side and the percentages for females on the right side.
We are trying to find four groups that make up nearly 1/2 of the male population, so we want the percentages for these groups to add up to nearly 50%.
Let's take a look at the lowest age groups of the male population, which in turn make up the highest proportion of the male population.
We have the age group 0-4, which makes up roughly 12%.
We have the age group 5-9, which makes up roughly 11%.
We have the age group 10-14, which makes up roughly 11%.
We have the age group 15-19, which makes up roughly 12.5%.
If we add all of these percentages together, we get:
12% + 11% + 11% + 12.5% = 46.5%
Since we cannot go any smaller than 46.5% because this is the closest percentage to 50%, the four groups that make up nearly one half of the male population must be:
Ages 0-4, 5-9, 10-14, and 15-19
Find a degree 3 polynomial having zeros -8, 2 and 8 and the coefficient of x3 equal 1.
The degree 3 polynomial is "x³-2x²-64x+128".
We have to find a polynomial of degree 3.We are given that the polynomial has -8, 2, and 8 as its roots.We are also given that the coefficient of x³ equals 1.If x = -8 is a root, then:(x+8) is a factor of the polynomial.If x = 2 is a root, then:(x-2) is a factor of the polynomial.If x = 8 is a root, then:(x-8) is a factor of the polynomial.All the three roots are of the same polynomial.So, the polynomial is the product of these factors.(x+8)(x-8)(x-2)(x²-64)(x-2)x³-2x²-64x+128The required polynomial is "x³-2x²-64x+128".To learn more about polynomials, visit :
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micrometers, T is measured in seconds, and D accounts for the weakening of the earthquake due to the distance from the epicenter.
If an earthquake occurred for 4 seconds and D = 2, which graph would model the correct amount on the Richter scale?
The graph that can be used to model the correct amount on the Richter scale is r = log(a/4) + 2.
How to illustrate the information?It should be noted that the magnitude of an earthquake is gotten by using the equation:
r = log(a / t) + d.
where
r = magnitude of an earthquake
a = amplitude
t = time
d = distance.
In this case, the earthquake occurred for 4 seconds and D = 2. We'll substitute the value into the equation and this will be r = log(a/4) + 2.
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PLEASEEE HELPP
Megan is in a musical production. She was at practice for 2 hours yesterday and 3 hours today. Altogether she spent of her practice hours dancing. How many total hours did Megan spend dancing yesterday and today?
Answer:
2.0833
Step-by-step explanation:
2 3/4 + 3 1/2 =6.25
6.25(1/3)=2.0833
Find the rates of change in population for both parks between 2009 and 2014, and determine which park showed faster population growthduring those years.
The table and graph given show the information about two national parks.
It is required to find the rate of change of the population of the two parks between 2009 and 2014, and then determine the park that shows faster population growth during those years.
Notice that the variable x represents the number of years after 2005.
Hence, the years 2009 and 2014 represent, x=4 and x=9, respectively.
The formula for the rate of change of a function f(x) between x=a and x=b is given by the formula:
[tex]\frac{f(b)-f(a)}{b-a}[/tex]Calculate the rate of change for park A:
Substitute a=4 and b=9 into the formula:
[tex]\frac{f(9)-f(4)}{9-4}[/tex]Substitute f(9)=2870 and f(4)=2370 from the table for park A into the expression for the rate of change:
[tex]\frac{2870-2370}{9-4}=\frac{500}{5}=100\text{ swallows per year}[/tex]Calculate the rate of change for park B:
[tex]\frac{f(9)-f(4)}{9-4}[/tex]Substitute f(9)=2800 and f(4)=2200 from the graph of park B:
[tex]\frac{2800-2200}{9-4}=\frac{600}{5}=120\text{ swallows per year}[/tex]Notice that the rate of change in population for park B is higher than that of park A.
Hence, park B showed faster population growth during those years.
Answers:
Rate of change for park A= 100 swallows per year.
Rate of change for park B= 120 swallows per year.
Park B showed faster population growth during those years.
Oscar drew a scale drawing of a house. The garage, which is 8 meters long in real life, is 4centimeters long in the drawing. What is the drawing's scale factor?Simplify your answer and write it as a fraction.
Answer:
1/200
Explanation:
Length of the garage in the drawing = 4 centimeters
Length of the garage in real life = 8 meters
The ratio of the lengths = 4 cm : 8m
Divide both sides by 4
[tex]\begin{gathered} \frac{4\operatorname{cm}}{4}\colon\frac{8m}{4} \\ 1\operatorname{cm}\colon2m \end{gathered}[/tex]1 m =100cm
2m= 2 x 100 =200cm
Converting both to the same unit, we have:
[tex]\begin{gathered} 1\operatorname{cm}\colon200\operatorname{cm} \\ =\frac{1}{200} \end{gathered}[/tex]The drawing's scale factor in fraction form is 1/200.
A student incorrectly simplifies an expression. The expression and work is shown below.
-(-6)(-3) - 2/3 (22 - 7)
3/5
Step 1: -(-6) (-3) - 2/3 (15)
3/5
Step 2: 18 - 2/3 (15)
3/5
Step 3: 18 - 10
3/5
Step 4: 8_3_5
Answer they got: 4 4/5
Idetnfiy which step is incorrect.
.
PLSSS HELP MEEEEEE
Answer:
so in the second step you see how the 18 is a positive
its supposed to be a negative
in the step 1, the part is -(-6)(-3), right?
so the negative symbol outside the negative 6 cancels out the negative, making it a positive number
it becomes 6(-3), which them multiplies into -18
so, i think the explanation is "the person forgot to change the 18 to a negative in the second step"
yea lol i think thats it
8. Tabitha made 12 out of 23 free throw attempts in the first half of practice and 20 out
of 27 attempts in the second half of practice. What is her free throw average for the
whole practice, written as a decimal?
Answer: 0.64 ...........
Drop down options for 1 2 and 4 are Forest A or Forest B
Given :
[tex]A(t)=107(1.015)^t[/tex][tex]B(t)=88(1.025)^t[/tex]1)
Set t=1 we get
[tex]A(t)=107(1.015)^t[/tex]Graph the image of ∆KLM after a dilation with a scale factor of 4, centered at the origin.
The red triangle is the original figure
The green triangle is the image
Scale factor is 4
A sequence is defined recursively using the formula f(n+1) =-0.5f(n) . If the first term of the sequence is 120, what is f(5)?
−15
−7.5
7.5
15
Suppose we have the recursive formula of sequence, [tex]\displaystyle{f(n+1)=0.5f(n)}[/tex]. From this formula, we know that:
[tex]\displaystyle{f(2)=0.5f(1)}\\\\\displaystyle{f(3)=0.5f(2)}\\\\\displaystyle{f(4)=0.5f(3)}\\\\\displaystyle{f(5)=0.5f(4)}[/tex]
Since the first term of sequence is 120. Therefore, [tex]\displaystyle{f(1)=120}[/tex]. Since [tex]\displaystyle{f(4)=0.5f(3)}[/tex] then substitute in [tex]\displaystyle f(5)[/tex]:
[tex]\displaystyle{f(5)=0.5\cdot 0.5f(3)}[/tex]
Then substitute f(3) down to f(1):
[tex]\displaystyle{f(5)=0.5\cdot 0.5 \cdot 0.5 \cdot 0.5 \cdot 120}\\\\\displaystyle{f(5)=(0.5)^4\cdot 120}\\\\\displaystyle{f(5)=7.5}[/tex]
Therefore, f(5) = 7.5
in the figure shown Line O is parallel to line P and line M is Parallel to linea N
we have that
In this problem angle x and angle of 62 degrees are supplementary angles
that means
x+62=180
solve for x
x=180-62
x=118 degreesThree cubes of side 10 cm are joined end to end to form a cuboid as given in the figure. Find is total surface area.
Solution:
The cuboid is a solid-shaped figure formed by six faces. A cuboid is a simple figure. It has three dimensions - width, length, and height. Thus, the cuboid is a parallelepiped. Now, the surface area of the parallelepiped is the sum of the areas of all sides, that is:
[tex]S\text{ =2(}lw+lh+wh\text{)}[/tex]where
l is the lenght
w is the width
and
h is the height
According to the figure given in the problem, we have that:
l = 30
w = 10
h = 10
thus, the surface area of the given cuboid would be:
[tex]\begin{gathered} S\text{ =2(}lw+lh+wh\text{)} \\ \text{ = 2((}30\cdot10\text{)+(30}\cdot10\text{)+(10}\cdot10\text{))=}1400 \end{gathered}[/tex]So that, we can conclude that the correct answer is:
[tex]1400[/tex]
Points B and C are on parallel lines with s and T, respectively. The size of the angle BAC is 96 degrees. Find the acute angle between straight AC and straight t if it is known that it is three times the size of the acute angle between straight AB and straight S.
Answer:
72°Step-by-step explanation:
Let the acute angles be B and C.
If we add a parallel line through the point A, we'll see that:
m∠BAC = B + CAnd we know that:
m∠BAC = 96C = 3BPlug in the known parameters to get:
B + 3B = 964B = 96B = 24We are looking for the value of C:
C = 3*24 = 72In 5 minutes, a conveyor belt moves 100 pounds of recyclable aluminum from the delivery truck to a storage area. A smaller belt moves the same quantity of cans thesame distance in 9 minutes. If both belts are used, find how long it takes to move the cans to the storage area.The conveyor belts can move the 100 pounds of recyclable aluminum from the delivery truck to a storage area in minutes.(Simplify your answer. Type an integer, fraction, or a mixed number.)
we know that
a greater conveyor belt moves 100 pounds ------------> 5 minutes
so
Applying proportion
1-minute ---------> move 100/5=20 pounds
a smaller conveyor belt moves 100 pounds ------> 9 minutes
so
Applying proportion
1-minute --------> move 100/9 pounds
therefore
If both belts are used
then
1-minute --------> move 20+(100/9)=(180+100)/9=280/9 pounds
Applying proportion
Find out how long it takes to move 100 pounds
[tex]\begin{gathered} \frac{1}{\frac{280}{9}}=\frac{x}{100} \\ \\ solve\text{ for x} \\ x=\frac{100}{\frac{280}{9}}=\frac{900}{280} \end{gathered}[/tex]Simplify the fraction
45/14
Convert to a mixed number
45/14=42/14+3/14=3 3/14 minutes
The answer is 3 3/14 minutesI need help with this practice Please read below ‼️‼️Use pencil and paper to graph the function, if you can’t, please use a drawing/writing tool that is *NOT* a graphing tool. If you cannot do this let me know
Answer:
Step-by-step explanation:
The trigonometric functions are represented by the following function form:
[tex]\begin{gathered} f(x)=\text{Atrig(Bx-C)}+D \\ \text{where,} \\ A=\text{ amplitud} \\ B,C=\text{ phase shift} \\ D=\text{ vertical shift} \end{gathered}[/tex]Then, for the following function:
[tex]f(x)=-\cot (x+\frac{\pi}{6})[/tex]Since it is an arctan function reflected the y-axis:
A journal on how to determine a quadratic equation given the roots
Answer:
this is your answer hope this helps you
The roots of an equation is "solutions" of the equation. Solutions are the numerical values equal to the variable after solving it.
To determine the sum and product of the roots of a quadratic equation, the equation has to be written in the form ax²+bx+c = 0
From that form, the sum of the roots is given by -b/c and the product is c/a
For example, we want to find the sum and product of the roots of the quadratic equation 3x²-x = 2
Notice that the equation is not written in the form described above. So, it has to be written as 3x²-x - 2 = 0
From this form, we see that and a = 3, b = -1 and c = -2
Therefore, the sum is -b/a = 1/3 while the product is c/a = -2/3 = -2/3
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Analyzing Graphs of Functions
Which graph shows a function where f(2)= 4?
21
12
12y
12ty
8
8
8
8
4
4
4
х
4 8 12
-12-8 -4
4 8 12
-12-8-4
4 8 12
12-48-4
4 8 12
4
-12-84
8-44
-8
-12
8
-8
-12
-12 V
Intro
Done
From the graphs we can conclude:
[tex]undefined[/tex]Questions 24-25
The scores of a standardized IQ test are normally distributed with a mean score of 100 and a standard deviation of 15.
Question 24 2
Find the probability that a randomly selected person has an IQ score higher than 105.
Question options:
0.9522
0.3694
-1.15
0.6306
Question 25
A random sample of 55 people is selected from this population. What is the probability that the mean IQ score of the sample is greater than 105?
Question options:
0.0067
0.3694
0.6306
0.9933
Using the normal distribution and the central limit theorem, it is found that the probabilities are given as follows:
24. Single person has an IQ score above 105: 0.3694.
25. Sample mean (55 people) above 105: 0.0067.
Normal Probability DistributionThe z-score of a measure X of a variable that has mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex] is given by the rule presented as follows:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The z-score measures how many standard deviations the measure X is above or below the mean of the distribution, depending if the z-score is positive or negative.From the z-score table, the p-value associated with the z-score is found, and it represents the percentile of the measure X in the distribution.By the Central Limit Theorem, the sampling distribution of sample means of size n has standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].The mean and the standard deviation of IQ scores are given as follows:
[tex]\mu = 100, \sigma = 15[/tex]
The probability that a single person has an IQ score higher than 105 is one subtracted by the p-value of Z when X = 105, hence:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
Z = (105 - 100)/15
Z = 0.33
Z = 0.33 has a p-value of 0.6306
1 - 0.6303 = 0.3694.
For the sample of 55, the standard error is:
[tex]s = \frac{15}{\sqrt{55}} = 2.02[/tex]
Hence:
[tex]Z = \frac{X - \mu}{s}[/tex]
Z = (105 - 100)/2.02
Z = 2.48
Z = 2.48 has a p-value of 0.9933.
1 - 0.9933 = 0.0067.
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Please explain and help me get the correct answer. Thank you. Practice work that is not graded.
First, from the formula given we solve the equation for D:
[tex]\begin{gathered} 4PD=D^2LN\pi, \\ \frac{4PD}{LN\pi}=D^2, \\ \sqrt{\frac{4PD}{LN\pi}}=D^{}. \end{gathered}[/tex]Now, substituting the given data:
(a) PD=405 cu in, L=4.7 in, and N=7,
in the above equation we get:
[tex]D=\sqrt[]{\frac{4\times405}{4.7\times7\times\pi}}in=\sqrt[]{15.6736175}in\approx3.96\text{ in}[/tex](b) PD=399.4 cu in, L=2 in, and N=6,
in the above equation we get:
[tex]D=\sqrt[]{\frac{4\times399.4}{2\times6\times\pi}}in=\sqrt[]{42.37765618}in\approx6.51\text{ in}[/tex]Answer:
(a) 3.96 in.
(b) 6.51 in.
find volume of hemisphere who’s great circle has an area of 12.5 ft squared
The surface area of a hemisphere is given as 12.5 square feet. It is required to find the volume of the hemisphere.
To do this, equate the given surface area to the formula, find the radius, and then substitute the radius into the volume formula to find the volume.
The surface area of a hemisphere radius, r is given as:
[tex]S=3\pi r^2[/tex]Substitute S=12.5 into the formula and solve for r in the resulting equation:
[tex]\begin{gathered} 12.5=3\pi r^2 \\ \Rightarrow3\pi r^2=12.5 \\ \Rightarrow\frac{3\pi r^2}{3\pi}=\frac{12.5}{3\pi} \\ \Rightarrow r^2=\frac{25}{6\pi} \\ \Rightarrow r=\sqrt{\frac{25}{6\pi}} \end{gathered}[/tex]The volume of a hemisphere is given as:
[tex]V=\frac{2}{3}\pi r^3[/tex]Substitute the calculated value of r into the volume formula:
[tex]\begin{gathered} V=\frac{2}{3}\pi\left(\sqrt{\frac{25}{6\pi}}\right)^3 \\ \text{ Substitute }\pi=3.14\text{ into the equation:} \\ \Rightarrow V=\frac{2}{3}(3.14)\left(\sqrt{\frac{25}{6(3.14)}}\right)^3\approx3.2\text{ square feet} \end{gathered}[/tex]The required volume is about 3.2 square feet.
A rectangular garden has one side with a length of x + 7 and another with a length 2x + 3. Find the perimeter of the garden.
A rectangular garden has one side with a length of x + 7 another with a length 2x + 3
We are asked to find the perimeter of the garden
Let me draw this rectangular garden to better understand the problem
Recall that the perimeter of a rectangular shape is given by
[tex]P=2(W+L)[/tex]Where W is the width and L is the length of the rectangular garden
Let us substitute the given values into the above formula
[tex]P=2(x+7+2x+3)[/tex]Now simplify the equation
[tex]\begin{gathered} P=2(x+2x+7+3) \\ P=2(3x+10) \\ P=6x+20 \end{gathered}[/tex]Therefore, the perimeter of the rectangular garden is equal to 6x + 20.
What is the magnitude (size) of -7.5?_______________________________O A. -7.5, because |-7.5| = 7.5O B. 7.5, because |-7.5 = 7.5O C. 7/5, because |-7.5| = 7/5O D. 7.5, because | -7.5| = - 7.5
it is given that the expression is -7.5
the magnitude of -7.5 is,
I-7.5I = 7.5
thus, the answer is option A
This figure shows a circle with a radius of 3.
A circle with its radius labeled as 3.
What is the circumference of the circle?
The circumference of the circle has a value of 18.86 units
How to determine the circumference of the circle?From the question, we have the following parameters:
Radius, r = 3 units
The circumference of the circle can be calculated using the following circumference formula
C = 2πr
Where
π = 22/7 (a constant)
r = 3
Substitute the known values in the above equation
So, we have the following equation
C = 2 * 22/7 * 3
Evaluate the products
So, we have the following equation
C = 18.86
Hence, the circumference is 18.86 units
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The figure shows a circle inscribed into a regular pentagon.Cis the center of the circle and the regular pentagon.G and H are on the edge of both the circle and the regular pentagon.The radius of the circle is 3 inches.GHCPart A. Find the area of the dark shaded region. Show your work.Part B. Find the area of the light shaded region. Show your work.
Solution
Part A: The area of the dark shaded region = S1,
where
The radius of the circle is 3 inches.
[tex]\begin{gathered} S_1=\frac{4}{5}\pi r^2 \\ =\frac{4}{5}\pi.3^2 \\ S_1=\frac{36}{5}\pi in^2 \end{gathered}[/tex]Part B: The area of the light shaded region = S,
[tex]\begin{gathered} S=S_2-S_1 \\ S=5\times\frac{1}{2}r.rtan36 \\ S=\frac{45}{2}\sqrt{5-2\sqrt{5}} \\ S=\frac{45}{2}\sqrt{5-2\sqrt{5}}-\frac{36}{5}\pi in^2 \end{gathered}[/tex]What is the image of ( − 8 , 0 ) after a dilation by a scale factor of 1/4 centered at the origin?
Step-by-step explanation:
it is then simply scanned (multiplied).
the new point is
(-8 × 1/4, 0 × 1/4) = (-2, 0)
Given the following table with selected values of the linear functions g(x) and h(x), determine the x-intercept of g(h(x)).
The x–intercept of g(h(x)) is the option;
[tex] \displaystyle {-\frac{2}{3}}[/tex]
What is a The x–intercept of a function?The x–intercept of a function is given by the point where the function intersects the x–axis.
From the given table, the slope of the graph of g(x), [tex] m_1 [/tex] is given by the equation;
[tex] \displaystyle{ m_1 = \frac{ - 4 - ( - 8)}{ - 4 - ( - 6))} = \frac{4}{2} = 2}[/tex]
The equation of g(x) in point slope form is therefore;
g(x) - (-4) = 2×(x - (-4)) = 2•x + 8
g(x) = 2•x + 8 - 4 = 2•x + 4
g(x) = 2•x + 4
The slope of the function h(x), [tex] m_2 [/tex] is found using the equation;
[tex] \displaystyle{ m_2 = \frac{ 8 - 14}{ - 4 - ( - 6))} = \frac{ - 6}{2} = - 3}[/tex]
h(x) presented in point and slope form is therefore;
h(x) - 8 = (-3)×(x - (-4)) = -3•x - 12
h(x) = -3•x - 12 + 8 = -3•x - 4
h(x) = -3•x - 4
g(h(x)) = 2•h(x) + 4
Plugging in the value of h(x) gives,;
g(h(x)) = 2×(-3•x - 4) + 4 = -6•x - 8 + 4
g(h(x)) = -6•x - 4
The x–intercept of g(h(x)) is given by the point where g(h(x)) = 0, which gives;
At the x–intercept, g(h(x)) = 0 = -6•x - 4
[tex] \displaystyle{ x = \frac{4}{-6} = -\frac{ 2}{3} }[/tex]
The x–intercept is [tex] \displaystyle{ -\frac{ 2}{3} }[/tex]
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Triangle BCD is similar to triangle EFG. Find the measure of side EF. Round youranswer to the nearest tenth.
Triangle BCD is similar to triangle EFG.
IS THIS A FUNCTION YES OR NO?? (ALGEBRA 1)
Answer: Yes
Step-by-step explanation:
Yes, the numbers in the table are a function. The equation for the equation would be
y = 2x