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A lattice point is a point in the plane with integer coordinates. How many lattice points are inside the region bounded by the graphs of $y = x^2$ and $y = 7$?

 Jun 21, 2023
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To find the number of lattice points inside the region bounded by the graphs of y = x^2 and y = 7, we need to count the number of lattice points that satisfy both conditions: y = x^2 and y ≤ 7.

Let's analyze the given conditions:

Condition 1: y = x^2

Since y = x^2, it implies that y must be a perfect square to satisfy this condition. Therefore, the possible values of y are 0, 1, 4, 9, 16, 25, 36, 49, etc.

Condition 2: y ≤ 7

Since y must be less than or equal to 7, the possible values of y satisfying this condition are 0, 1, 4, and 7.

Now, we need to determine the number of lattice points that satisfy both conditions. From the above analysis, we can see that there are three possible values for y: 0, 1, and 4.

For y = 0, there is only one corresponding lattice point: (0, 0).

For y = 1, there are two corresponding lattice points: (1, 1) and (-1, 1).

For y = 4, there are five corresponding lattice points: (2, 4), (-2, 4), (1, 4), (-1, 4), and (0, 4).

Therefore, there are a total of 1 + 2 + 5 = 8 lattice points inside the region bounded by the graphs of y = x^2 and y = 7.

 Jun 21, 2023

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