The graph of a function contains the points (6, 7) and (10, 13). Which of these statements is correct based on this information? A. If the graph also contains the point (4, 4) then the function must not be linear.
B. If the graph also contains the point (12, 16) then the function must not be linear.
C. If the graph also contains the point (1, 2) then the function could be linear.
D. If the graph also contains the point (8, 10) then the function could be linear.

The true statements are:

B and D.

Step-by-step explanation:

First, a linear function can be written as:

y = a*x + b

Such that a is the slope, and b is the y-intercept.

If we know that the graph of the function passes through the points (x₁, y₁) and (x₂, y₂) then the slope is:

a = (y₂ - y₁)/(x₂ - x₁)

Here we have a function that passes through the points (6, 7) and (10, 13), then if this function is linear, the slope is:

a = (13 - 7)/(10 - 6) = 6/4 = 3/2

Then the function will be something like:

f(x) = (3/2)*x + b

Knowing that the function passes through (6, 7), this means that:

f(6) = 7

Then:

(3/2)*6 + b = 7

3*3 + b = 7

9 + b = 7

b = 7 - 9 = -2

The function, if linear, is:

f(x) = (3/2)*x  - 2

Now let's read the statements:

A)  If the graph also contains the point (4, 4) then the function must not be linear.

If the point is a solution of the equation, then the function is linear.

If the point is not a solution of the equation, then the function is not linear.

f(4) = 4 = (3/2)*4  - 2

        4 = 3*2 - 2

        4 = 6 - 2 = 4

Then (4, 4) is a solution, so if the graph also contains the point (4, 4) the function can be linear. Then the statement "A" is false.

Now let's do the same analysis for the other statements:

B)  If the graph also contains the point (12, 16) then the function must not be linear.

F(12) = 6 = (3/2)*12 - 2 = 3*6 - 2 = 18 - 2 = 16

           6 = 16

(12, 16) is not a solution, so if the graph contains this point, the function is not a linear function, the statement is true.

C) If the graph also contains the point (1, 2) then the function could be linear.

f(1) = 2 = (3/2)*1 - 2 = 3/2 - 2 = -1/2

The point (1, 2) is not a solution, so if the graph contains this point, then the function can't be linear, so the statement is false.

D) If the graph also contains the point (8, 10) then the function could be linear.

f(8) = 10 = (3/2)*8 - 2

        10 = 3*4 - 2 = 12 - 2

        10 = 10

The point (8, 10) is a solution of the linear equation, so if the graph contains this point, the function could be linear, so the statement is true.

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