(SAT Prep) Find the value of x in each of the following exercises:
Answer:
The value of x is 120°
Step-by-step explanation:
To solve for x, you need to introduce a third line that is parallel to the two parallel lines.
This line should divide the angle at x into two as shown in the attachment.
By the alternate interior angle property, m=70°
and
n=50°
This implies that, x=50+70=120°
Calculate the surface area of the following triangular prism:
A. 156 cm^2
B. 240 cm^2
C. 336 cm^2
D. 288 cm^2
Answer:
B- 240 cm²
Step-by-step explanation:
Hope it can help you lovelots
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round the fraction 3^3/2 to the nearest whole number
Evaluating 3^3/2 gives us 5.196152423, so we can round this to 5.
kristy earns money based on the number of hours worked, with the same amount paid for each hour. she earned $110.50 after working 13 hours
Answer:
She earned $8.5 per hour.
Step-by-step explanation:
110.5/13=8.5
Every jump a game piece makes measures 8 9 . The piece starts at point A = 7 and jumps to the right. As soon as the piece jumps over B = 24, it switches direction and jumps to the left. The piece then stops at point A. How many jumps did the game piece take?
Answer:
The number of jumps that the game piece took, was 40.
Step-by-step explanation:
Every jump a game piece makes measures [tex]\frac{8}{9} = 0.889[/tex].
Now, the piece starts at point A = 7 and jumps to the right.
So, the number of jumps required by the piece to jump over B = 24 will be
[tex]\frac{24 - 7}{0.889} = 19.122[/tex] jumps ≈ 20 jumps {As the piece crosses the point B}
Now, as soon as the piece jumps over B = 24, it switches direction and jumps to the left. And the piece then stops at point A.
Therefore, the number of jumps that the game piece took, was (20 × 2) = 40. (Answer)
Final answer:
Calculating the jumps based on the jump size of 8 / 9 units, the piece makes 20 jumps to go from point A past point B, and then 23 jumps to return to point A, for a total of 43 jumps.
Explanation:
To solve the question, we must calculate the total number of jumps the game piece takes from the starting point A at 7, over point B at 24, and back to A. Each jump measures 8 / 9 units. To calculate the number of jumps to reach just past point B (24 units), we can set up a division problem to find the number of jumps it would take to first reach or pass 24 starting from 7. The formula for this is (B - A) / jump size.
So, the calculation for jumps to point B is: (24 - 7) / 8 / 9 = 17 × 9 / 8 = 153 / 8 = 19.125. Since the game piece cannot make a partial jump, we must round up to 20 jumps to reach beyond 24 units.
After the piece switches direction, we perform a similar calculation to find how many jumps it takes to go back to point A (7 units).
Here, since the piece is already beyond 24, it only needs to jump back to the next integer value before 7 which is 6.
Now, the game piece is at (20 × 8 / 9) + 7 = approximately 24.222 + 7 = approximately 31.222 units along the path.
To calculate the jumps back to 6, we have (31.222 - 6) / 8 / 9 which is approximately 22.556 jumps, and rounding up gives us 23.
Therefore, the total number of jumps made is the sum of the jumps to point B and back to point A,
which is 20 + 23 = 43 jumps.
((7,-1) and (21,-5) what is the equation in slope intercept form?
Answer:
y=-2/7x+1
Step-by-step explanation:
m=(y2-y1)/(x2-x1)
m=(-5-(-1))/(21-7)
m=(-5+1)/14
m=-4/14
simplify
m=-2/7
y-y1=m(x-x1)
y-(-1)=-2/7(x-7)
y+1=-2/7(x-7)
y=-2/7x+2-1
y=-2/7x+1
The equation of the line in slope-intercept form is: [tex]\[ \boxed{y = -\frac{2}{7}x + 1} \][/tex]
To find the equation of the line in slope-intercept form y = mx + b that passes through the points 7, -1 and 21, -5 we need to follow these steps:
1. Find the slope m of the line.
2. Use the slope and one of the points to find the y-intercept b
3. Write the equation in the form y = mx + b
Step 1: Find the slope m
The formula for the slope between two points [tex]\((x_1, y_1)\) and \((x_2, y_2)\) is:[/tex]
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
For the points 7, -1 and 21, -5
[tex]\[ x_1 = 7, y_1 = -1 \][/tex]
[tex]\[ x_2 = 21, y_2 = -5 \][/tex]
Substitute these values into the slope formula:
[tex]\[ m = \frac{-5 - (-1)}{21 - 7} = \frac{-5 + 1}{21 - 7} = \frac{-4}{14} = -\frac{2}{7} \][/tex]
Step 2: Find the y-intercept b
Use the slope[tex]\( m = -\frac{2}{7} \)[/tex] and one of the points
[tex]\[ -1 = -\frac{2}{7}(7) + b \][/tex]
b = 1
Step 3: Write the equation
Now that we have the slope [tex]\( m = -\frac{2}{7} \)[/tex] and the y-intercept b = 1 we can write the equation in slope-intercept form:
[tex]\[ y = -\frac{2}{7}x + 1 \][/tex]
Thus, the equation of the line in slope-intercept form is:
[tex]\[ \boxed{y = -\frac{2}{7}x + 1} \][/tex]
HELP PLEASE, I REALLY DONT UNDERSTAND THIS!
Answer: Since 3−5+2=0, then 2
is the additive inverse of 3−5 is 2
Since 5−3−2=0, then −2is the additive inverse of 5−3. is -2
Step-by-step explanation: additive inverse
The additive inverse of any number
x
is the number that gives zero when added to
x
. Example: the additive inverse of
5 is −5.
The sum of the square of a positive number and the square of 3 more than the number is 89. What is the number?
The number is 5 which sum of the square of a positive number and the square of 3 more than the number 89.
What is a quadratic function?The quadratic function is defined as a function containing the highest power of a variable is two.
We have to determine the number which sum of the square of a positive number and the square of 3 more than the number 89.
Let the first number would be x
So second number = x + 3
According to the given condition,
⇒ x² + (x + 3)² = 89
⇒ x² + x² + 6x + 9 = 89
⇒ 2x² + 6x + 9 = 89
⇒ 2x² + 6x - 80 = 0
⇒ x² + 3x - 40 = 0
⇒ (x + 8)(x - 5) = 0
⇒ x + 8 = 0 or x - 5 = 0
⇒ x = -8 or x = 5
The answer is 5 because we are looking for a positive number.
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Final answer:
The positive number where the sum of its square and the square of 3 more than the number equals 89 is 5.
Explanation:
To determine the positive number where the sum of squares of the number and the square of 3 more than the number equals 89, we set up the following equation:
Let x be the positive number. Then:
x² + (x + 3)² = 89
Expanding the equation:
x² + x² + 6x + 9 = 89
Combining like terms, this becomes:
2x² + 6x + 9 = 89
Subtract 89 from both sides:
2x² + 6x - 80 = 0
We now have a quadratic equation. Dividing by 2 for simplicity:
x² + 3x - 40 = 0
Factor the quadratic equation:
(x + 8)(x - 5) = 0
Setting each factor equal to zero gives us two possible solutions for x:
x + 8 = 0, so x = -8 (which we can disregard as we are looking for a positive number)x - 5 = 0, so x = 5 (which is the positive number we're looking for)Therefore, the positive number is 5.
In the skateboard design, VW bisects XY at point T , and XT =39.9 cm. Find XY
Answer:
The measure of XY is 79.8 cm.
Step-by-step explanation:
An image of the question is attached here.
Given:
Measure of XT = 39.9
VW is the bisector.
So T is the midpoint of XY .
Note: Bisector divides the line in two equal parts.
According to the question:
⇒ [tex]XT = TY[/tex] ...as T is the midpoint.
⇒ [tex]XY=XT +TY[/tex] ...Segment addition Postulate
⇒ [tex]XY=XT+XT[/tex]
⇒ [tex]XY=2(XT)[/tex]
⇒ [tex]XY=2(39.9)[/tex]
⇒ [tex]XY=79.8[/tex] cm
The measure of XY in the skateboard design is 79.8 centimeter.
The box plots show the average wind speeds, in miles per hour, for various cities in two different countries. Average Wind Speeds of Cities in Country A 2 box plots. The number line goes from 1 to 11. For the average wind speeds of cities in country A, the whiskers range from 1 to 9.5, and the box ranges from 3 to 7. A line divides the box at 4. For the average wind speeds of cities in country B, the whiskers range from 1.2 to 11, and the box ranges from 4 to 9. A line divides the box at 6. Average Wind Speeds of Cities in Country B Which statement describes the symmetry of the data in the two box plots? The data in country A are more symmetric than the data in country B. The data in country B are more symmetric than the data in country A. The data in both countries have about the same symmetry. The symmetry of the data cannot be determined by looking at the box plots.
Answer:
"B"
Step-by-step explanation:
"the median wind speed for country B is greater than the median wind speed for county A"
got it right on the quiz, and good luck ;)
The median wind speed for country B is greater than the median wind speed for county A.
The answer is option B.
What is a median?A median is a number in the middle of a list of numbers that are sorted, ascended, or descended and can define a set of data. The median is sometimes used in contrast to the definition where there are outsiders in a sequence that may distort the value ratio.
Why is median important?The importance of median value is that it provides an idea about the distribution of data. If the definition and median of the data set are the same, then the data is evenly distributed from the smallest to the highest values.
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Find the 22nd term of the following sequence:
5, 8, 11, ...
63
71
14
68
68 is the 22nd term of the following sequence.
Step-by-step explanation:
The given sequence is with the same common difference between the two consecutive number in the series thus it is said to be the Arithmetic progression ( AP). For finding the nth term in the AP we have a formula tn = a + (n-1) × dHere a is the first term , n is the number of the term to be founded and d is the common difference between the two consecutive number in the series.Thus here tn = 5 + ( 22 - 1 ) × 3.On subtracting we get tn = 5 + (21 ) × 3 On multiplying we get tn = 5 + 63After adding we get tn = 68. It is the 22nd term in the given series.the twenty second (22) term of the sequence 5, 8, 11, ...... is: D. 68.
Given the following data:
First (1st) term = 5Second term = 8Third term = 11To find the twenty second (22) term of the sequence:
Mathematically, the [tex]n^{th}[/tex] term of a sequence is calculated by using the following formula;
[tex]a_n = a + (n - 1)d[/tex]
Where:
a is the first term.n is the term number.d is the common difference.First of all, we would determine the common difference.
[tex]d = 2^{nd} \; term - 1^{st}\;term\\\\d = 8 - 5 \\\\d = 3[/tex]
Substituting the given parameters into the formula, we have;
[tex]a_{22} = 5 + (22 - 1)3\\\\a_{22} = 5 + (21)3\\\\a_{22} = 5 + 63\\\\a_{22} = 68[/tex]
Therefore, the twenty second (22) term of the sequence is 68.
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You can access your funds easier if your account has_
liquidity.
A. more
B. less
Answer:
if it as more
Step-by-step explanation:
no need to thank me just add me on snap wgilpenn
If you do this you are officially skilled: (SAT Prep) In the figure, if PN = LN, NP Is parallel to MQ, and QL bisects ∠PQM, what is value of x?
Answer:
Try the suggested solution, shown on the picture attached
Step-by-step explanation:
Note, 'm(MNO)' means m(∠MNO), and for issues 2, 4 ,5 - the described angles are angles inside the declared triangle.
Answer:
67º
Step-by-step explanation:
There are 28 boys in the band
The 28 boys are 7/10 and the girls are 3/10 of the students
What is the total number of students in the marching band
Answer:
t = 40
Step-by-step explanation:
Start by breaking the equation down to a simpler form. To do this you would need to divide 28 by 7.
28 / 7 = x
x = 4
Now we can create a new equation.
4x = t
T would represent the total number of students in the band. X would be the any number from 1 to 10. This is because 10 is the highest faction to make the fraction a whole. Since we a looking for the total we would put ten in place for x.
4 x 10 = t
t = 40
There you go.
Hope this helped.
What number can be written as 400,000 + 8,000 + 400 + 70 + 1?
180.417
480.417
057 8047208.17
408,471
480.471
408,417
Answer:
it's 408,471
Step-by-step explanation:
[tex]400000 \\ + 8000 \\ \: + 400 \\ \: \: + 70 \\ \: \: \: + 1 \\ = 408471[/tex]
Cube Root Function Question! 15 points!
The graph of h(x) is a translation of f (x) = Root Index 3
Which equation represents h(x)?
Answer: Second option.
Step-by-step explanation:
Below are some transformations for a function f(x) :
1. If [tex]f(x)+k[/tex], the function is shifted "k" units up.
2. If [tex]f(x)-k[/tex], the function is shifted "k" units down.
3. If [tex]f(x-k)[/tex], the function is shifted "k" units right.
4. If [tex]f(x+k)[/tex], the function is shifted "k" units left.
The Cube root parent function is:
[tex]f(x)=\sqrt[3]{x}[/tex]
By definition, the graph of this function passes through the origin, as you can observe in the picture attached.
In this case, you need to analize the graph given in the exercise. You can see that the the graph of the function h(x) is like the parent function f(x), but shifted 2 units left.
Therefore, based on the transformations explained above, you can determine that the equation of the function h(x) is the following:
[tex]h(x)=f(x+2)\\\\h(x)=\sqrt[3]{x+2}[/tex]
3. The product of 7/10 and another factor
is greater than 7/10. Which could be
the other factor?
A. 4/3
B. 5/9
C. 10/12
D. 7/7
Answer:
C
Step-by-step explanation:
product of 7 and 10 is 70
product of 10 and 12 is 120
a = 12
b =45
c =120
d =49
Wendy has $60 to buy seed for her birdfeeders. Each bag of seed costs $8.
How many bags of seed can she buy?
Answer:7 bags
Step-by-step explanation:
Divide 60 by 8 because she has a total of $60 and each bag costs $8. The answer would actually be 7.5 bags but since you can’t buy half a bag your answer would be 7 bags.
A girl 160 cm tall stands 360 cm from a lamp post at night. Her shadow from the light is 90 cm long. How high is the lamp post?
The answer is.
The Lamp post is 800cm
x = height of lamp post
160/90 = x/450
450 is the length from the base of the lamp post to the end of the shadow (360+90)
cross multiple
90x = 72000
x =800cm
Have a great day!
The lamp post is 800 cm high
From the diagram in the attachment below,
The height of the girl is /AG/ = 160cm
The length of her shadow is /SG/ = 90cm
The height of the lamp post is /LP/ = /LM/ + /MP/ = x + 160cm
To determine the height of the lamp post, we will calculate the value of x
Consider ΔASG
Determine <ASG = θ
From
[tex]tan\theta = \frac{opposite}{adjacent}[/tex]
Opposite = /AG/ = 160 cm
Adjacent = /SG/ = 90 cm
∴ [tex]tan\theta = \frac{160}{90}[/tex]
Also, Consider ΔLAM
[tex]tan\theta = \frac{/LM/}{/AM/}[/tex]
NOTE: <LAM = θ (Corresponding angles) since line AM is parallel to line SP
∴[tex]\frac{160}{90} = \frac{/LM/}{/AM/}[/tex]
But, /LM/ = x and /AM/ = 360 cm
∴[tex]\frac{160}{90} = \frac{x}{360}[/tex]
[tex]90x = 360 \times 160[/tex]
[tex]x = \frac{360 \times 160}{90}[/tex]
[tex]x = 4 \times 160 \\[/tex]
[tex]x = 640 cm[/tex]
x = 640 cm
Recall that, the height of the lamp post is /LP/ = /LM/ + /MP/ = x + 160cm
∴ The height of the lamp post is 640 cm + 160cm
The height of the lamp post is 800 cm
Hence, the lamp post is 800 cm high
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if 1/5 of the remaining blueberries is used to make muffins, how many pounds of blueberries are left in the container
To determine how many pounds of blueberries are left in the container after using 1/5 for muffins, calculate 4/5 of the initial amount, as this represents the remaining 80%. The ability to work with fractions and percentages is crucial here.
Explanation:The question involves understanding percentages and unit conversion. If 1/5 of the remaining blueberries is used to make muffins, that means 80% (or 4/5) of the blueberries are left in the container. To find out how many pounds of blueberries are left, we first need to know the total quantity before using any for muffins. Assuming we had a specific weight to begin with, we would calculate 4/5 of that weight to determine what remains.
For example, if there were 10 pounds of blueberries initially, 2 pounds (1/5) would be used for muffins, leaving 8 pounds (4/5) of blueberries in the container. It's essential to be comfortable with fractions and percentage calculations when dealing with such problems in mathematics.
a. Each bowl contains 1/12 of the total blueberries.
b. There were 72 ounces of blueberries in the full container.
c. There are 3 pounds of blueberries left in the container.
a. What fraction of the blueberries is in each bowl?
Initially, 1/6 of the blueberries are poured equally into two bowls. Therefore, each bowl receives 1/6 * 1/2 = 1/12 of the total blueberries.
b. If each bowl has 6 ounces of blueberries in it, how many ounces of blueberries were in the full container?
Since each bowl has 6 ounces of blueberries, and there are two bowls, the total amount of blueberries poured into the bowls is 6 * 2 = 12 ounces. Since this represents 1/6 of the total blueberries, we can set up the equation:
Total blueberries = 12 * 6 = 72 ounces
c. If 1/5 of the remaining blueberries are used to make muffins, how many pounds of blueberries are left in the container?
After pouring 1/6 of the blueberries into the bowls, 5/6 of the blueberries remain. Then, 1/5 of these remaining blueberries are used to make muffins. So, the fraction of blueberries remaining after making muffins is 5/6 * 4/5 = 20/30 = 2/3.
To convert the remaining blueberries to pounds, since 1 pound equals 16 ounces, we divide the total remaining ounces by 16:
Remaining blueberries in pounds = (72 ounces * 2/3) / 16 ounces/pound = (72 * 2) / (3 * 16) = 144 / 48 = 3 pounds
So, there are 3 pounds of blueberries left in the container.
The probable question maybe:
A container is filled with blueberries. 1/6 of the blueberries are poured equally into two bowls.
a. What fraction of the blueberries is in each bowl?
b. If each bowl has 6 ounces of blueberries in it, how many ounces of blueberries were in the full container?
c. If 1/5 of the remaining blueberries are used to make muffins, how many pounds of blueberries are left in the container?
PLEASE HELP... BRAINLIEST... PLEASE..
17. Determine the scale factor for each dilation. Determine whether the dilation is an enlargement, reduction or isometry dilation.
The given dilation is an isometry dilation.
Step-by-step explanation:
Step 1; First, we need to compare the dimensions of the two figures. We check to see if the side lengths are the same. If the parameters are the same, the dilation could be an enlargement or a reduction. Whereas if the parameters are the same, it could be an isometric dilation or just a reflection.
Step 2; The first shape has a side length of approximately 2.5 units. We compare this to the same side length as the second shape. The second shape has the same side length.
The first shapes side length of the / same side length of the second shape = 2.5 / 2.5 = 1,
So the scale factor is 1. As the parameters do not change, it could either be a reflection or an isometric dilation.
Step 3; The base side in shape 1 is BC whereas the same base side in shape 2 is [tex]A^{1}[/tex][tex]B^{1}[/tex]. So shape ABCDE has rotated to form the shape the dilation is isometric and not a reflection with a scale factor of 1.
Answer:
1: Isometry
Step-by-step explanation:
As both the shapes are having equal area hence the scale factor remains 1.
As scale factor remains 1this is neither enlargement or reduction.
It is the isometry that the distance between two pints remains the same.
if theres 56 docks and 20 of them join and then 5 of them leave how many ducks are left
Answer:
Step-by-step explanation:
Answer:
71 ducks
Step-by-step explanation:
1. 56 + 20 = 76
2. 76 - 5 = 71
Simplify:
2(d + 3) + 3(d – 3)
A. –5d – 1
B. 5d – 3
C. 5d + 15
D. 6d – 3
Answer:
I would say it is c but im not sure.
Step-by-step explanation:
Well you use distributive property to solve this to get a simplified answer.
[tex]b. \: 5d - 3 \\ \\ 1. \: 2d + 6 + 3(d - 3) \\ 2. \: 2d + 6 + 3d - 9 \\ 3. \: (2d + 3d) + (6 - 9) \\ 4. \: 5d - 3[/tex]
Which number is irrational?
A. 3/17
B. Square root of 25
C. 0.666
D. Square root of 33
Answer:
B. Square root of 25
Step-by-step explanation:
It is a whole number.
Thus, it is not an irrational number.
The square root of 25 is equal to 5
Therefore
An irrational number is a number that cannot be written as a ratio of two integers. It is a non-terminating and non-repeating decimal.
WILL MARK BRAINLEIST!!!!!!
Suppose that it takes Calvin 3 hours to wax a car if he works alone and it takes Alvin 5 hours to wax a car if he works alone. How long does it take them to wax a car if they work together? Write an equation and solve for the unknown. Show your work.
Answer:
4 hours
Step-by-step explanation:
I am not sure, what I did was, if Calvin and Alvin both waxed half the car, their original time would split in half, then I added those values together, which is also the average.
y=mx+b is a pretty safe form to follow
C=3
A=5
if you are doing variables
I tried my best.
How do you write 10+3x and 4x on a graph?
See the graph below
Explanation:First of all, let's rename those expressions as follows:
[tex]\left\{ \begin{array}{c}y=10+3x\\y=4x\end{array}\right[/tex]
So we we have two lines written in slope-intercept form. In order to graph those lines, let's find two points for each:
First line:
[tex]For \ x=0: \\ \\ y=10+3(0) \\ \\ y=10 \\ \\ \\ For \ x=-10/3: \\ \\ y=10+3(-10/3) \\ \\ y=10-10 \\ \\ y=0 \\ \\ \\ So \ the \ line \ passes \ through \ (0,10) \ and \ (-10/3,0)[/tex]
This line is the green one shown below.
Second line:
[tex]For \ x=0: \\ \\ y=4(0) \\ \\ y=0 \\ \\ \\ For \ x=1: \\ \\ y=4(1) \\ \\ y=4 \\ \\ \\ So \ the \ line \ passes \ through \ (0,0) \ and \ (1,4)[/tex]
This line is the red one shown below.
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1/3x+1/2y=10
1/5x-3y=-3/5
Step-by-step explanation:
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Diicinrhcurnje8f77 8eun3if7 hh8
Describe the error made in subtracting the two
rational expressions shown
1 1
x-
2x+1
x + 1
x-2
(x-2)(x+1) (x - 2)(x +1)
(x-2)(x+1)
Answer:
(This is in my own words)
ALL terms of the numerator must be subtracted out, not just the first term. -2 should be subtracted out to get a numerator of x+1-x+2. Thus, the difference of the numerator should be 3, and not –1. This is when simplified correctly.
On solving the expression correctly, we get -
3/(x - 2)(x + 1)
What is Equation Modelling?
Equation modelling is the process of writing a mathematical verbal expression in the form of a mathematical expression for correct analysis, observations and results of the given problem.
Given is the following equation -
1/(x - 2) - 1/(x + 1)
We have -
1/(x - 2) - 1/(x + 1)
[(x + 1) - (x - 2)]/(x - 2)(x + 1)
3/(x - 2)(x + 1)
Therefore, on solving the expression correctly, we get -
3/(x - 2)(x + 1)
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Elena ate 2/8 of a pizza and Dylan ate 1/5. estimate how much of the pizza did they eat?
Pls Help need for tomorrow if u answer then i will give 10+ points and brainliest answer
Answer:
[tex]\frac{9}{20}[/tex]
Step-by-step explanation:
Given:
Elena ate 2/8 of a pizza.
Dylan ate 1/5 of a pizza.
Question asked:
How much of the pizza did they eat ?
Solution:
By simply adding.
Elena ate of a pizza. = [tex]\frac{2}{8}[/tex]
Dylan ate of a pizza = [tex]\frac{1}{5}[/tex]
Total of a pizza they ate = [tex]\frac{2}{8}[/tex] + [tex]\frac{1}{5}[/tex]
By taking LCM of 8 and 5 ; 40
[tex]\frac{10 + 8}{40} = \frac{18}{40} = \frac{9}{20}[/tex]
Thus, [tex]\frac{9}{20}[/tex] f the pizza they eat.
The focus of a parabola is (0, - 2) The directrix of the parabola is the line y = - 3 What is the equation of the parabola
Answer:
Option B: [tex]$ \textbf{y} \hspace{1mm} \textbf{=} \hspace{1mm} \frac{\textbf{-1}}{\textbf{2}} \textbf{x}^{\textbf{2}} \hspace{1mm} \textbf{-} \hspace{1mm} \frac{\textbf{5}}{\textbf{2}} $[/tex]
Step-by-step explanation:
When the focus (h, k) of a parabola and the equation of the directrix y = c are given, the equation of the parabola is given by:
[tex]$ \textbf{(x - h)}^{\textbf{2}} \hspace{1mm} \textbf{+} \hspace{1mm} \textbf{k}^{\textbf{2}} \hspace{1mm} \textbf{-} \hspace{1mm} \textbf{c}^{\textbf{2}} \hspace{1mm} \textbf{=} \hspace{1mm} \textbf{2(k - c)y}} $[/tex]
Here, we are given the focus: (h, k) = (0, -2)
Directrix: y = c = -3.
We substitute in the formula to get the equation of the parabola.
[tex]$ (x - 0)^2 + (-2)^2 - (-3)^2 = 2(-2 - (-3))y $[/tex]
[tex]$ \implies x^2 + 4 - 9 = 2(- 2 + 3)y $[/tex]
[tex]$ \implies x^2 - 5 = 2(1) y$[/tex]
[tex]$ \implies 2y = x^2 - 5 $[/tex]
Dividing by 2, throughout we get:
[tex]$ \textbf{y} \hspace{1mm} \textbf{=} \hspace{1mm} \frac{\textbf{-1}}{\textbf{2}} \textbf{x}^{\textbf{2}} \hspace{1mm} \textbf{-} \hspace{1mm} \frac{\textbf{5}}{\textbf{2}} $[/tex] which is the required answer.