I know this is alot to ask but could someone please check to see if I did this right and help me with the transformation. I was given this transformation problem. The cross section of the roof of a house is modelled by the function y= -5\12|x-12|+5, where y>0 or y=0. I need to find the slope,height,and lengths of each side including the base. I found the height to be 10units so on the graph the peak is (0,10)Did I do this right a couple of my friends said the peak was at (0,12) Then I used a^2 + b^2=c^2 and sides are 26 units, base 48 units, slope 5/12 but I'm not sure how to expalin the transformation applied to the graph of y = |x| to obtain the graph of y= -5\12|x-12|+5, where y>0 or y=0. Thank you so very much.

Yes, you did the calculations correctly. To explain the transformation applied to the graph of y = |x| to obtain the graph of y= -5\12|x-12|+5, where y>0 or y=0, you can say that the graph of y = |x| has been shifted 12 units to the right and 5 units up, and then multiplied by -5/12. This results in the graph of y= -5\12|x-12|+5, where y>0 or y=0.

To determine if you did the calculations correctly, let's break down the steps:

1. Finding the height: You correctly determined that the height is 10 units. Since the vertex of the graph is at (0, 10), you are correct, and your friends are mistaken.

2. Finding the lengths of each side, including the base: You used the Pythagorean theorem (a^2 + b^2 = c^2) correctly. If you apply this theorem to the triangle formed by the peak of the roof (0, 10), and the points where the roof intersects the x-axis (12, 0) and (-12, 0), you will get the correct lengths.

The distance between the peak (0, 10) and the x-intercept (12, 0) is equal to the distance between the peak and the other x-intercept (-12, 0). This means the base is symmetric, resulting in a total length of 48 units.

Applying the Pythagorean theorem:

- One side of the triangle can be calculated as the distance from (0, 10) to (12, 0):
a^2 + b^2 = c^2
12^2 + 10^2 = c^2
144 + 100 = c^2
244 = c^2
c ≈ 15.62 units (rounded to two decimal places)

- The other side of the triangle is the same length, so it will also be approximately 15.62 units.

Hence, the lengths of each side will be approximately: 15.62 units, 15.62 units, and 48 units (base).

3. Finding the slope: To find the slope, it is helpful to consider the symmetry of the graph. Since the roof is symmetrical, the slope on either side of the peak will be the same.

Recall that the equation of the roof is y = -5/12|x-12| + 5.

To find the slope, we can differentiate this equation. However, since the roof is symmetric, we can simplify this process.

The slope of the roof, represented by m, will be the change in y divided by the change in x for any two points on the roof. Let's take the points (0, 10) and (12, 0) as an example.

m = (change in y) / (change in x)
m = (10 - 0) / (0 - 12)
m = 10 / -12
m = -5/6

Therefore, the slope of the roof is -5/6.

4. Transformation explanation: The original equation, y = |x|, represents the graph of a V-shaped absolute value function centered at the origin (0, 0).

By applying the transformation y = -5/12|x-12| + 5, we can observe the following changes:
- The 5/12 coefficient stretches the graph vertically, making it flatter compared to the original graph.
- The |x-12| moves the graph 12 units to the right, shifting the peak to (12, 0).
- Lastly, the +5 moves the entire graph 5 units up, resulting in the peak being located at (12, 5).

Overall, you did most of the calculations and explanations correctly. The only discrepancy was the height, which you correctly determined as 10 units, while your friends suggested it was 12 units. However, the correct value is indeed 10 units.

To check if you have done the calculations correctly, let's go through each part of the problem step by step.

First, let's find the slope of the roof. The slope can be determined by looking at the coefficient in front of the absolute value in the equation. In this case, the coefficient is -5/12, which means the roof is sloping downwards. So, you are correct in finding the slope as -5/12.

Next, let's find the height of the roof. To do this, we need to find the maximum value of y in the equation. You correctly identified the peak of the roof as (0, 10). So, the height of the roof is indeed 10 units. Your friends might have made a mistake in their calculation.

Now, let's find the lengths of the sides and the base. To do this, we need to consider the x-values that correspond to the points where y=0. In other words, we need to find the roots or zeros of the equation.

Setting y=0 in the equation, we get:
0 = -5/12|x-12| + 5

To remove the fraction, we can multiply both sides of the equation by 12:
0 = -5|x-12| + 60

To isolate the absolute value, we can subtract 60 from both sides:
-60 = -5|x-12|

Dividing by -5, we get:
12 = |x-12|

The equation |x-12| = 12 represents two separate linear equations: x-12 = 12 and x-12 = -12. Solving these equations will give us the x-values.

For x-12 = 12, we get x = 24.
For x-12 = -12, we get x = 0.

Thus, the base of the roof has a length of 24 - 0 = 24 units.

Finally, to find the lengths of the sides, we can subtract the base from the total length of the roof. The total length can be calculated by finding the distance between the two x-values we obtained earlier.

The total length is 24 units (base), and the x-values are 0 and 24. So, the lengths of the sides are 24 - 0 = 24 units and 24 - 24 = 0 units.

In summary:
- The slope of the roof is -5/12.
- The height of the roof is 10 units.
- The base of the roof is 24 units.
- The lengths of the sides are 24 units and 0 units.

Now, let's explain the transformation applied to the graph of y = |x| to obtain the graph of y = -5/12|x-12| + 5, where y > 0 or y = 0.

The original equation y = |x| represents a V-shaped graph with its vertex at the origin. This graph has a slope of 1.

To transform this graph into y = -5/12|x-12| + 5, the following steps were applied:

1. Horizontal shift: The graph of y = |x| was shifted 12 units to the right. This changed the vertex from (0, 0) to (12, 0).

2. Vertical stretch and reflection: The graph was stretched vertically by a factor of 5/12. This made the entire graph narrower and flipped it upside down.

3. Vertical shift: The entire graph was shifted 5 units upwards. This changed the vertex from (12, 0) to (12, 5).

By applying these transformations, we obtained the graph of y = -5/12|x-12| + 5, where y > 0 or y = 0, which represents the cross section of the roof of a house.

I hope this explanation helps you understand the problem and the steps involved in finding the slope, height, lengths of each side, and the transformation applied to the graph. If you have any further questions, feel free to ask!