To find the linear function with the given properties, we need to find the slope and y-intercept of the function.
The slope of a linear function is given by the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are two points on the line.
Using the two given points, we can find the slope of the function as:
m = (2 - 5) / (1 - (-7)) = -3/8
The y-intercept of a linear function is the value of the function when x = 0. To find the y-intercept, we can use the point-slope form of a linear equation:
y - y1 = m(x - x1)
where (x1, y1) is one of the given points and m is the slope we just calculated.
Using the point (1, 2), we can write the equation as:
y - 2 = (-3/8)(x - 1)
Simplifying, we get:
y = (-3/8)x + 19/8
Therefore, the linear function with the given properties is:
f(x) = (-3/8)x + 19/8
We can verify that this function satisfies the given properties by checking that:
f(-7) = (-3/8)(-7) + 19/8 = 5
f(1) = (-3/8)(1) + 19/8 = 2
So the function f(x) = (-3/8)x + 19/8 satisfies the given properties.