Answer:
a) maximum; the parabola opens downward
b) positive; it must lie above the x-axis
c) x = 1.5
Step-by-step explanation:
The x-intercepts of a function are the points where the graph of the function crosses the x-axis. The y-values there are zero.
The "differences" of a function are related to the average slope between adjacent points. Second differences are related to the rate of change of the slope of the function. When <em>second differences are negative</em>, as here, the slope of the quadratic function is decreasing, becoming more negative. We say the <em>curvature</em> of the function is <em>negatve</em>, and that it <em>opens downward</em>.
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<h3>a, b.</h3>
If the graph of the parabola opens downward, and it crosses the x-axis, it must have a <em>maximum</em> that is a <em>positive value of y</em>.
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<h3>c.</h3>
The graph of a parabola is symmetrical about its vertex. That means points on the same horizontal line are the same distance from the line of symmetry, which must go through the vertex. The x-coordinate of the vertex will be the x-coordinate of the midpoint between the two x-intercepts:
x = (-2 +5)/2 = 3/2
The x-coordinate of the vertex is x = 1.5.
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<em>Additional comment</em>
The attachment shows a table with three evenly-spaced points on the curve. The calculations show first differences (d1) and second differences (d2). You can see that the sign of the second diffference is negative, in agreement with the given conditions.
Answer:
(f + g)(x) = 3x^2 + 3x/2 - 9
Step-by-step explanation:
In order to find a composite function through addition, you simply add the two equations together.
f(x) + g(x)
x/2 - 3 + 3x^2 + x - 6
3x^2 + 3x/2 - 9
Class A: 6v + 8b = 202
Class B: 12v + 16b = 284
Solve using the elimination method:
since 6v and 12v are perfect for elimination, multiply the class A equation by 2 so that the van variable cancels out:
12v + 16b = 404
12v + 10b = 284
Then subtract the bottom equation from the top:
6b = 120
b = 20
Now you know that each bus can hold 20 students.
Just plug this into one of the original equations to solve for vans:
6v + 8(20) = 202
6v + 160 = 202
6v = 42
v=7
So then you know that each van can hold 7 students.
Check:
12 (7) + 10 (20) = 284
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If
is a number that is both divisible by 4 and 5, then

4 and 5 are coprime, so we can use the Chinese remainder theorem to solve this system and find that
is a solution to the system, where
is any integer. Simply put, any multiple of 20 fits the bill.
Now, there are 11 numbers between 100 and 300 that are divisible by 20 (100, 120, 140, and so on). We have
when
, so the sum we want to compute is
