Mathematics College

PLS HELP!!!!
Question 2
The vertex form of the equation of a horizontal parabola is given by
, where (h, k) is the vertex of the parabola and the absolute value of p is the distance from the vertex to the focus, which is also the distance from the vertex to the directrix. You will use GeoGebra to create a horizontal parabola and write the vertex form of its equation. Open GeoGebra, and complete each step below.

Part A
Mark the focus of the parabola you are going to create at F(-5, 2). Draw a vertical line that is 8 units to the right of the focus. This line will be the directrix of your parabola. What is the equation of the line?
















Part B
Construct the line that is perpendicular to the directrix and passes through the focus. This line will be the axis of symmetry of the parabola. What are the coordinates of the point of intersection, A, of the axis of symmetry and the directrix of the parabola?















Part C
Explain how you can locate the vertex, V, of the parabola with the given focus and directrix. Write the coordinates of the vertex.
















Part D
Which way will the parabola open? Explain.
















Part E
How can you find the value of p? Is the value of p for your parabola positive or negative? Explain.
















Part F
What is the value of p for your parabola?















Part G
Based on your responses to parts C and E above, write the equation of the parabola in vertex form. Show your work.















Part H
Construct the parabola using the parabola tool in GeoGebra. Take a screenshot of your work, save it, and insert the image below.
















Part I
Once you have constructed the parabola, use GeoGebra to display its equation. In the space below, rearrange the equation of the parabola shown in GeoGebra, and check whether it matches the equation in the vertex form that you wrote in part G. Show your work.
















Part J
To practice writing the equations of horizontal parabolas, write the equations of these two parabolas in vertex form:

focus at (4, 3), and directrix x = 2
focus at (2, -1), and directrix x = 8

Answers

Answer 1

Answers:

Part A

The directrix is a vertical line that is 8 units to the right of the focus F(-5, 2). So, we add 8 to the x-coordinate of the focus to get the equation of the directrix. The equation of a vertical line is of the form x = a, where a is the x-coordinate of any point on the line. So, the equation of the directrix is x = -5 + 8 = 3.

Part B

The line that is perpendicular to the directrix and passes through the focus is a horizontal line with the equation y = k, where k is the y-coordinate of the focus. So, the equation of the axis of symmetry is y = 2. The point of intersection, A, of the axis of symmetry and the directrix is the point where x = 3 and y = 2. So, A = (3, 2).

Part C

The vertex, V, of the parabola is the midpoint of the line segment from the focus to the directrix. Since the focus and the directrix are 8 units apart, the vertex is 4 units to the right of the focus. So, the coordinates of the vertex are V = (-5 + 4, 2) = (-1, 2).

Part D

The parabola will open to the right because the focus is to the left of the directrix.

Part E

The value of p is the distance from the vertex to the focus or the directrix. Since the vertex is 4 units to the right of the focus, p = 4. The value of p is positive when the parabola opens to the right and negative when it opens to the left. So, for this parabola, p is positive.

Part F

The value of p for this parabola is 4.

Part G

The vertex form of the equation of a horizontal parabola is [tex]\( (y - k)^2 = 4p(x - h) \)[/tex], where (h, k) is the vertex and p is the distance from the vertex to the focus. Substituting the coordinates of the vertex (-1, 2) and the value of p (4) into the equation, we get [tex]\( (y - 2)^2 = 4*4(x + 1) \), or \( (y - 2)^2 = 16(x + 1) \)[/tex].

Part H

*Look at Attachment*

Part I

*Look at Attachment*

Part J

1. For the parabola with focus at (4, 3) and directrix x = 2, the vertex is the midpoint between the focus and the directrix, which is (3, 3). The parabola opens to the right because the focus is to the right of the directrix, so p is positive. The distance from the vertex to the focus or the directrix is 1, so p = 1. The equation of the parabola is [tex]\( (y - 3)^2 = 4(x - 3) \)[/tex].

2. For the parabola with focus at

The vertex of the parabola with focus at (2, -1) and directrix x = 8 is (5, -1). The parabola opens to the left because the focus is to the left of the directrix, so p is negative. The distance from the vertex to the focus or the directrix is 3, so p = -3. The equation of the parabola is [tex]\( (y + 1)^2 = 4*(-3)*(x - 5) \), or \( (y + 1)^2 = -12(x - 5) \)[/tex].

PLS HELP!!!! Question 2The Vertex Form Of The Equation Of A Horizontal Parabola Is Given By , Where (h,
PLS HELP!!!! Question 2The Vertex Form Of The Equation Of A Horizontal Parabola Is Given By , Where (h,

Related Questions

can someone help answer this and explain how you did it

Answers

Answer:

  2

Step-by-step explanation:

You want the slope of segment DC given points A, B, C, D are collinear and the rise between B and A is 2 units, while the run is 1 unit.

Slope of a line

The slope of a line is the same everywhere on the line. It is the same for segment DC as for segment BA on the same line.

  slope = rise/run = 2/1 = 2

The slope of DC is 2.

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Answer:  SLOPE of DC    =    2

        Thanks for reposting the pertinent question:

        Therefore the SLOPE of DC:

         DC  =   2

Step-by-step explanation:  Cheers to the person who has explained and answered the question correctly as well.Make a Plan:  FORMULA FOR SLOPE OF A LINE:   m = rise/run =  y1  - y2 / x2 - x1

        POINTS:  D, C, B, and A  are COLLINEAR

Now, We can FIND That:

       SLOPE of:  DC  is  Equal (=)  To the SLOPE  of: AB

So, Now, The SLOPE of AB:

       AB  =   2/1   =  2

Now, we conclude that:

       Therefore the SLOPE of DC:

       DC  =   2

I hope this helps you!

Sample Responses Label axes according to input
and output variables. Plot the ordered pair of the
independent and dependent variable on the
coordinate plane. Identify if there is a relationship
Which of the following did you include in your
response?
O Label the x-axis the input variable, age.
O Label the y-axis the output variable, texting speed.
O Plot the points according to age and texting speed.
O Identify a relationship between the change in
texting speed as age increases.

Answers

The following elements were included in the response: labeling x-axis as age, labeling y-axis as texting speed, plotting points according to age and texting speed, and identifying a relationship between texting speed and age.

In the response, the following elements were included:

1. Labeling the x-axis as the input variable, age: This is important to indicate the independent variable being plotted.

2. Labeling the y-axis as the output variable, texting speed: This is crucial to indicate the dependent variable being represented.

3  Plotting the points according to age and texting speed: This involves placing the ordered pairs (age, texting speed) on the coordinate plane, with age values on the x-axis and texting speed values on the y-axis.

4. Identifying a relationship between the change in texting speed as age increases: By analyzing the plotted points and examining the pattern or trend, one can determine if there is a relationship between age and texting speed. For example, if the texting speed generally increases as age increases or if there is a linear or nonlinear relationship between the two variables.

Including all of these elements allows for a comprehensive analysis of the relationship between age and texting speed, providing a visual representation and enabling the identification of any patterns or trends.

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Please look at the photo for the question. Thank you!

Answers

The zeros with each multiplicity are given as follows:

Multiplicity one: x = 6.Multiplicity two: x = 11.Multiplicity three: x = -6 and x = -5.

How to obtain the multiplicities?

The factor theorem is used to define the functions, which states that the function is defined as a product of it's linear factors, if x = a is a root, then x - a is a linear factor of the function.

Considering the linear factors of the function in this problem, the zeros are given as follows:

(x + 6)³ -> zero at x = -6 with multiplicity of 3.(x - 11)² -> zero at x = 11 with multiplicity of 2.x - 6 -> zero at x = 6 with multiplicity of 1.(x + 5)³ -> zero at x = -5 with multiplicity of 3.

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Tyquan bought a medium pizza pie, which had 6 slices. He came
home, ate two slices, and went to do his homework. His brother
Kareem walked into the kitchen, saw the pizza, but because he wasn't
that hungry, he only ate half a slice. A minute later, their baby sister
ran in and demanded something to eat. Kareem knew she did not have a
big appetite, so he cut the half of a slice he had left in half again, and
gave his sister the small piece. What fraction of the pizza did the
three children end up eating? How much pizza is left for their
parents?

Answers

The three children ended up eating 5/12 of the pizza, and 7/12 of the pizza is left for their parents.

To determine the fraction of the pizza that the three children ate, we can add up the portions consumed. Tyquan ate 2 out of 6 slices, which is 2/6 or 1/3 of the pizza. Kareem ate half of a slice, which is 1/6 of the pizza. Kareem's sister then received half of Kareem's remaining half-slice, which is 1/2 of 1/6 or 1/12 of the pizza.

Adding up these fractions, the three children consumed 1/3 + 1/6 + 1/12 = 5/12 of the pizza. To determine how much pizza is left for their parents, we subtract the fraction consumed by the children from the whole pizza. The fraction of pizza left for the parents is 1 - 5/12 = 7/12 of the pizza.

Therefore, the three children ended up eating 5/12 of the pizza, while 7/12 of the pizza is left for their parents.

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Variables t and N are such that when lgN is plotted against lg t, a straight line graph passed through the points (0.45,1.2) and (1, 3.4) is obtained

(i) Express the equation of the straight line graph in the form lgN = m lg t + lg c, where m and c are constants to be found.

(ii) Hence express N in terms of t

Answers

The expression for N in terms of t is N = k tn.

Given that the variables t and N are such that when lgN is plotted against lg t, a straight line graph passed through the points (0.45,1.2) and (1, 3.4) is obtained. The question requires us to express N in terms of t.
A straight line graph represents an equation in the form y = mx + c, where y represents the dependent variable, m represents the slope of the line, x represents the independent variable, and c represents the y-intercept. Thus, we can write the equation of the line obtained as:
lg N = m lg t + c (1)
To find the value of m and c, we can use the two points (0.45, 1.2) and (1, 3.4) that the line passes through. Substituting the values of x and y into equation (1), we get:
lg N1 = m lg t1 + c   ...(2)
lg N2 = m lg t2 + c   ...(3)
where N1 = antilog(1.2), t1 = antilog(0.45), N2 = antilog(3.4), and t2 = antilog(1).
Taking the logarithm of both sides of equation (2) gives:
lg lg N1 = lg(m lg t1 + c)
Plotting a graph of lg lg N1 against lg t1 gives a straight line with a slope of m and a y-intercept of lg c. Similarly, from equation (3), we can plot a graph of lg lg N2 against lg t2 to obtain another straight line with a slope of m and a y-intercept of lg c.
Substituting the values of N1, t1, N2, and t2 into equations (2) and (3), we get:
1.2 = mt1 + c   ...(4)
3.4 = mt2 + c   ...(5)
Subtracting equation (4) from equation (5) eliminates c and gives:
2.2 = m(t2 - t1)
Simplifying the above equation gives:
m = 2.2 / (t2 - t1)
Substituting the value of m in equation (4) gives:
1.2 = [2.2 / (t2 - t1)]t1 + c
Simplifying the above equation gives:
c = 1.2 - [2.2t1 / (t2 - t1)]
Therefore, the equation of the line is:
lg N = [2.2 / (t2 - t1)] lg t + [1.2 - 2.2t1 / (t2 - t1)] (6)
We need to express N in terms of t. Taking antilogarithm of both sides of equation (6) gives:
N = at[2.2 / (t2 - t1)] x at[1.2 - 2.2t1 / (t2 - t1)]
Simplifying the above equation gives:
N = k tn
where k = at[1.2 - 2.2t1 / (t2 - t1)] and n = 2.2 / (t2 - t1).

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