Final answer:
None of the options (a, b, c, d) match the conditions f(-4) = 2 and f(6) = -3 for a linear function. Through calculations, we find the correct linear function satisfying these conditions is f(x) = -0.5x.
Step-by-step explanation:
To find the linear function f with f(-4) = 2 and f(6) = -3, let's analyze each of the given function options, namely:
f(x) = -x - 6
f(x) = 2x + 10
f(x) = -0.5x + 4
f(x) = 3x - 14
We substitute x with -4 and 6 into each equation and check which one yields the desired values.
Option a:
f(-4) = -(-4) - 6 = 4 - 6 = -2 (not equal to 2)
f(6) = -(6) - 6 = -6 - 6 = -12 (not equal to -3)
Option b:
f(-4) = 2(-4) + 10 = -8 + 10 = 2 (matches)
f(6) = 2(6) + 10 = 12 + 10 = 22 (not equal to -3)
Option c:
f(-4) = -0.5(-4) + 4 = 2 + 4 = 6 (not equal to 2)
f(6) = -0.5(6) + 4 = -3 + 4 = 1 (not equal to -3)
Option d:
f(-4) = 3(-4) - 14 = -12 - 14 = -26 (not equal to 2)
f(6) = 3(6) - 14 = 18 - 14 = 4 (not equal to -3)
None of the options given directly match the conditions f(-4) = 2 and f(6) = -3. However, we can write the linear equation that satisfies these conditions.
The general form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept. To find m, we use the two points (-4, 2) and (6, -3).
Slope (m) = (y2 - y1) / (x2 - x1) = (-3 - 2) / (6 - (-4)) = -5 / 10 = -1/2
Now we have the slope, so we use one of the points to find b:
2 = (-1/2)(-4) + b
2 = 2 + b
b = 2 - 2
b = 0
The linear function is f(x) = -1/2x + 0, or simplified, f(x) = -0.5x.