After a dreary day of rain, the sun peeks through the clouds and a rainbow forms. you notice the rainbow is the shape of a parabola.

1. the equation for this parabola is y = -x2 + 36.

2. create a table of values for a linear function. a drone is in the distance, flying upward in a straight line. it intersects the rainbow at two points. choose the points where your drone intersects the parabola and create a table of at least four values for the function. remember to include the two points of intersection in your table.
3. analyze the two functions. answer the following reflection questions in complete sentences.
what is the domain and range of the rainbow? explain what the domain and range represent. do all of the values make sense in this situation? why or why not?
what are the x- and y-intercepts of the rainbow? explain what each intercept represents.
is the linear function you created positive or negative? explain.

what are the solutions or solution to the system of equations created? explain what it or they represent.

Answers

Remember that a quadratic with two real zeroes can be written as a(x - r_1)(x - r_2), where a is a constant and r_1 and r_2 are the zeroes (or roots) of the function. Since the graph shows that the two zeroes are at -6 and 6, the equation has to be of the form

y = a(x - ({-6}))(x - 6), or
y = a(x + 6)(x - 6)

To solve for a, let's use the point at the vertex (0, 36) and plug that in:

36 = a(0 + 6)(0 - 6)
36 = {-36}a
a = {-1}
(It makes sense that a is negative since the parabola opens down.)

So, the equation of the parabola is

y = -(x + 6)(x - 6), or
\bf y = -x^2 + 36

Now for the second part, just pick any two points with which we can draw a line with a positive slope. I'll use x = -2 and 1:

y = -({-2})^2 + 36 = {-4} + 36 = 32
y = -(1)^2 + 36 = {-1} + 36 = 35

So, our two points are (-2, 32) and (1, 35). To find the equation of the linear function that goes through these two points, let's use slope-intercept form, which is f(x) = mx + b. The slope m is given by \frac{y_2 - y_1}{x_2 - x_1}, so

m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{35 - 32}{1 - ({-2})} = 1
So, the equation of the linear function so far is just f(x) = x + b, and we can find b by plugging in one of the points on the line:

35 = 1 + b
b = 34

Thus, the equation of the linear function is

\bf f(x) = x + 34

And you can find more points on the line simply by plugging other values of x, such as (0, 34) and (5, 39).
Remember that a quadratic with two real zeroes can be written as a(x - r_1)(x - r_2), where a is a constant and r_1 and r_2 are the zeroes (or roots) of the function. Since the graph shows that the two zeroes are at -6 and 6, the equation has to be of the form

y = a(x - ({-6}))(x - 6), or
y = a(x + 6)(x - 6)

To solve for a, let's use the point at the vertex (0, 36) and plug that in:

36 = a(0 + 6)(0 - 6)
36 = {-36}a
a = {-1}
(It makes sense that a is negative since the parabola opens down.)

So, the equation of the parabola is

y = -(x + 6)(x - 6), or
\bf y = -x^2 + 36

Now for the second part, just pick any two points with which we can draw a line with a positive slope. I'll use x = -2 and 1:

y = -({-2})^2 + 36 = {-4} + 36 = 32
y = -(1)^2 + 36 = {-1} + 36 = 35

So, our two points are (-2, 32) and (1, 35). To find the equation of the linear function that goes through these two points, let's use slope-intercept form, which is f(x) = mx + b. The slope m is given by \frac{y_2 - y_1}{x_2 - x_1}, so

m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{35 - 32}{1 - ({-2})} = 1
So, the equation of the linear function so far is just f(x) = x + b, and we can find b by plugging in one of the points on the line:

35 = 1 + b
b = 34

Thus, the equation of the linear function is

\bf f(x) = x + 34

And you can find more points on the line simply by plugging other values of x, such as (0, 34) and (5, 39).
Okay assuming that this is y=-x^2+36.

Vertex form: y=-x^2+36
Intercept form: y=(-x-6)(x-6)
Vertex: (0,36)
Focus: (0,143/4)=(0,35.75)
Directrix: y=145/4=36.25
Axis of Symmetry: x=0
Focal parameter (distance between the directrix and focus): 1/2=0.5
x-intercepts: (-6,0), (6,0)
y-intercepts: (0,36)

I hope this helps. 
For this question, the points of intersection that I chose for this problem were (-4, 20) and (2, 32). Keep in mind that the linear function can have any two points that cross the parabola but the right point HAS to be lower than the left. This is shown by the question "flying upwards in a straight line." Anyway, now that you have two points, you need to find the y-intercept of the line and the slope. For the points that I have chosen my y-intercept was (0, 28) and the slope was 2/1. Now, make the equation. y = 2x + 28 (remember the graph goes up at intervals of 2) Now, you need to make a table of values. For mine, the x values increased by 1, starting at -4, and the y values increased by 2, starting at 20. That should answer the first part of the question. The second part was a little tougher. For the first bullet point, the domain (x) is all real numbers because the graph expands infinitely. The range (y) is any number less than 36 because 36 is the graph's highest point and it continually expands downward. For the second bullet point, the x-intercepts (you should be able to identify those) are where the rainbow touches the horizon. The y-intercept is where the rainbow reaches its highest point. For the third bullet point, it depends on your linear function. Mine happens to be positive because as the x-values increase, so do the y. If it were negative, the y-values would decrease as the x-increase. For the final bullet point, the solutions of the functions would be where your points of intersection are. To show this, plug the values into both equations and find that you will get the same numbers each time. Hope this helps!
Based on the docx you showed me, the equation for the parabola is y = x^2 + 36 and you want a table of values for a linear equation that intersects the parabola at (5, 6) and (-2, 34).

If you use these two points to create a line we get the equation:

y - 6 = \frac{34 - 6}{-2 - 5}(x - 5) (I just used point slope form)

This can be simplified to:

y = \frac{40}{-7}x + \frac{242}{7}

Now we just need to create a table of points on this line. We already have the points you gave and we can also use the y-intercept: (0, \frac{242}{7}) and the x-intercept: (\frac{121}{20}, 0).

So our table of value can be:

x          | y
______|________
-2         | 34
0          | 242 / 7
5          | 6
121/20 | 0
Based on the docx you showed me, the equation for the parabola is y = x^2 + 36 and you want a table of values for a linear equation that intersects the parabola at (5, 6) and (-2, 34).

If you use these two points to create a line we get the equation:

y - 6 = \frac{34 - 6}{-2 - 5}(x - 5) (I just used point slope form)

This can be simplified to:

y = \frac{40}{-7}x + \frac{242}{7}

Now we just need to create a table of points on this line. We already have the points you gave and we can also use the y-intercept: (0, \frac{242}{7}) and the x-intercept: (\frac{121}{20}, 0).

So our table of value can be:

x          | y
______|________
-2         | 34
0          | 242 / 7
5          | 6
121/20 | 0

We are given that,

The equation of the rainbow is represented by the parabola,

y=-x^2+36

Now, we are required to find a linear equation which cuts the graph of the parabola at two points.

Let us consider the equation joining the points (-6,0) and (0,36), given by y=6x+36.

So, the corresponding table for the linear equation is given by,

x             y=6x+36

-6                      0

0                       36

1                        42

6                       72

Now, we will answer the questions corresponding the functions.

1. Domain and Range of the rainbow.

Since, the equation of the rainbow is y=-x^2+36

So, from the figure, we get that,

Domain is the set of all real numbers.

Range is the set \{ y|y\leq 36 \}

Here, domain represents the points which are used to plot the path of the rainbow and range represents the points which are form the rainbow.

Not all points make sense in the range as the parabola is opening downwards having maximum point as (0,36).

2. X and Y-intercepts of the rainbow.

As, the 'x and y-intercepts are the points where the graph of the function cuts x-axis and y-axis respectively i.e. where y=0 and x=0 receptively'.

We see that from the figure below,

X-intercepts are (-6,0) and (6,0) and the Y-intercept is (0,36)

Here, these intercepts represents the point where the parabola intersects the individual axis.

3. Is the linear function positive or negative.

As the linear function is y=6x+36 represented by the upward flight of the drone.

So, the linear function is a positive function.

4. The solution of the system of equations is the intersection points of their graphs.

So, from the figure, we see that the equations intersect at the points (-6,0) and (0,36).

Thus, the solution represents the position when both the drone and rainbow intersect each other.


After a dreary day of rain, the sun peeks through the clouds and rainbow forms. you notice the rainb

In the picture attached, both the plot of the parabola and of the straight line are shown. The parabola represents the rainbow and the line, the path of the drone.  

Drone path

xy

-511

-415

031

135

What is the domain and range of the rainbow? Explain what the domain and range represent.  

The domain of a parabola is all real numbers, the range of this parabola is [-infinite, 36].   The domain represents all possible values x-variable can take. The range represents all possible values y-variable can take.

Do all of the values make sense in this situation? Why or why not?

No, it doesn't. Only values between [0, 36] of the range make sense, because negative values are below the horizon, where there is no rainbow

What are the x- and y-intercepts of the rainbow? Explain what each intercept represents.

x-intercepts: (-6, 0) and (0, 6). They represent the points at which the rainbow intercept the horizon.

y-intercepts: (0, 36). It represents the maximum height of the rainbow.

Is the linear function you created positive or negative? Explain.

The linear function created has a positive slope. As a consequence, the function is always increasing.

What are the solutions or solution to the system of equations created? Explain what it or they represent.

The solutions are the points at which the parabola and the line intercept. In this case they are points (-5, 11) and (1,35)


After a dreary day of rain, the sun peeks through the clouds and rainbow forms. you notice the rainb
1) Equation for the parabola y = - x² + 36.


2) A drone is in the distance, flying upward in a straight line. It intersects the rainbow at two points. Choose the points where your drone intersects the parabola and create a table of at least four values for the function.


The points of intersection will be: (0,36) and (4, 20)


Table of values for a linear function and the parabola

x        linear function      parabola y = - x² + 36

0        36                                  (0)² + 36 = 36

1        32                                  - (1)² + 36 = 35

2        28                                  - (2)² + 36 = 32

3        24                                   -3² + 36 = 27

4        20                                   -4² + 36 = 20

It is a linear function because the rate of change is constant: for each increase of 1 unit in x, there is a decrease of 4 units in y: ⇒ slope = - 4

Remember to include the two points of intersection in your table. Analyze the two functions; they are there (0,36) and (4, 20)

Answer the following reflection questions in complete sentences. What is the domain and range of the rainbow? 
Explain what the domain and range represent. Do all of the values make sense in this situation? Why or why not? What are the x- and y-intercepts of the rainbow? Explain what each intercept represents.

The domain is all the real values of x between - 6 and 6: - 6 ≤ x ≤ 6

The range is all the values between 0 and 36: 0 ≤ y ≤ 36

The domain represents the horizontal distance between the extremes of the rainbow at the ground.

The range represents: the altitude of the rainbow from the ground (y = 0). The height is at x = 0, y = 36.

The x-intercepts are - 6 and 6, that is the opening of the rainbow at the ground level.

The y-intercept is when x = 0, it is y = 36, and is the height of the rainbow.

Is the linear function you created positive or negative? Explain. What are the solutions or solution to the system of equations created? Explain what it or they represent.

The linear function is positive and decreasing. It is positive because it goes above the x-axis, this is y ≥ 0.

The solutions where already signaled: (0,6) and (4, 20). They are the points at which the drone intercepts the rainbow.
1 For this question, the points of intersection that I chose for this problem were (-4, 20) and (2, 32). Keep in mind that the linear function can have any two points that cross the parabola but the right point HAS to be lower than the left. This is shown by the question "flying upwards in a straight line." Anyway, now that you have two points, you need to find the y-intercept of the line and the slope. For the points that I have chosen my y-intercept was (0, 28) and the slope was 2/1. Now, make the equation. y = 2x + 28 (remember the graph goes up at intervals of 2) Now, you need to make a table of values. For mine, the x values increased by 1, starting at -4, and the y values increased by 2, starting at 20. That should answer the first part of the question. The second part was a little tougher. For the first bullet point, the domain (x) is all real numbers because the graph expands infinitely. The range (y) is any number less than 36 because 36 is the graph's highest point and it continually expands downward. For the second bullet point, the x-intercepts (you should be able to identify those) are where the rainbow touches the horizon. The y-intercept is where the rainbow reaches its highest point. For the third bullet point, it depends on your linear function. Mine happens to be positive because as the x-values increase, so do the y. If it were negative, the y-values would decrease as the x-increase. For the final bullet point, the solutions of the functions would be where your points of intersection are. To show this, plug the values into both equations and find that you will get the same numbers each time. Hope this helps


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