1. The UV index on a sunny day can be modelled by the function y= -0.15(x-13)^2+7.6 , where x represents the time of day on a 24-h clock and f(x) represents the UV index. At what time(s) was the UV index 7?
2. Determine if each system has one, two, or no solutions. (Show your work - 3 marks each)
a)y=2x^2-2x+1 and 3x-5
b)y=x^2+3x-16 and y= -x^2-8x-18
#1 well just solve
-0.15(x-13)^2+7.6 = 7
just the usual quadratic stuff.
#2
The discriminant gives this information.
b^2-4ac:
<0 no roots
=0 1 root
>0 two roots
205-495-0134
1. To find the time(s) when the UV index is 7, we need to set y equal to 7 and solve for x.
Given the function y = -0.15(x-13)^2 + 7.6, we set y = 7:
7 = -0.15(x-13)^2 + 7.6
Now, let's solve for x:
0 = -0.15(x-13)^2 - 0.6
0.6 = -0.15(x-13)^2
Divide both sides by -0.15:
-4 = (x-13)^2
Taking the square root of both sides:
±2 = x - 13
Adding 13 to both sides:
x = 13 ± 2
So, the UV index is 7 at 11:00 AM (13 + 2) and 3:00 PM (13 - 2).
2.
a) For the system of equations:
y = 2x^2 - 2x + 1
3x - 5
To determine the number of solutions, we need to analyze the quadratic equation y = 2x^2 - 2x + 1.
Since this is a quadratic equation, it can have two solutions, one solution, or no solution depending on the discriminant.
The discriminant is given by: b^2 - 4ac
For the quadratic equation 2x^2 - 2x + 1:
a = 2, b = -2, c = 1
The discriminant is:
(-2)^2 - 4(2)(1) = 4 - 8 = -4
Since the discriminant is negative (-4 < 0), the quadratic equation has no real solutions.
Therefore, the system of equations has no solutions.
b) For the system of equations:
y = x^2 + 3x - 16
y = -x^2 - 8x - 18
To determine the number of solutions, we need to analyze the quadratic equations y = x^2 + 3x - 16 and y = -x^2 - 8x - 18.
Again, we can determine the number of solutions based on the discriminant.
For the quadratic equation x^2 + 3x - 16:
a = 1, b = 3, c = -16
The discriminant is:
(3)^2 - 4(1)(-16) = 9 + 64 = 73
Since the discriminant is positive (73 > 0), the quadratic equation has two distinct real solutions.
For the quadratic equation -x^2 - 8x - 18:
a = -1, b = -8, c = -18
The discriminant is:
(-8)^2 - 4(-1)(-18) = 64 - 72 = -8
Since the discriminant is negative (-8 < 0), the quadratic equation has no real solutions.
Thus, the system of equations has two distinct real solutions.
1. To find the time(s) when the UV index was 7, we need to solve the equation y = 7 for x.
The given UV index function is y = -0.15(x-13)^2 + 7.6.
Substituting y = 7, we have:
7 = -0.15(x-13)^2 + 7.6
Now, we can solve for x by isolating the variable:
0.15(x-13)^2 = 7.6 - 7
0.15(x-13)^2 = 0.6
Divide both sides by 0.15:
(x-13)^2 = 0.6 / 0.15
(x-13)^2 = 4
Now, take the square root of both sides:
x-13 = ±√4
x-13 = ±2
Solve for x in both cases:
x = 13 + 2 = 15
x = 13 - 2 = 11
Therefore, the UV index was 7 at 11:00 AM and 3:00 PM.
2. Let's solve each system of equations separately:
a) y = 2x^2 - 2x + 1 and 3x - 5
To determine the number of solutions, we need to determine if the two equations intersect. We can do this by comparing their slopes.
The first equation is a quadratic function, and the second equation is a linear function.
The quadratic function has a positive leading coefficient (2), which means the parabola opens upward. The linear function has a positive slope (3). Hence, the two graphs will intersect at some point, resulting in exactly one solution.
b) y = x^2 + 3x - 16 and y = -x^2 - 8x - 18
Again, we compare the slopes of the two equations.
The first equation is a quadratic function, while the second equation is also a quadratic function with a negative leading coefficient. This means the parabola opens downward, but it is shifted down by 2 units due to the constant term.
As both equations are quadratic with different concavities, they may intersect at most twice.
To determine the number of solutions, we need to analyze the discriminant (b^2 - 4ac) of the quadratic equations. If the discriminant is positive, there are two real solutions. If it is zero, there is exactly one real solution. If it is negative, there are no real solutions.
For equation b):
Discriminant = (-8)^2 - 4(1)(-18) = 64 + 72 = 136
Since the discriminant is positive, there are two real solutions.
Therefore, the systems have:
a) exactly one solution
b) two solutions