Find the linear function with the following properties.

Question

f(2)=10

f(10)=−2

Answer 1

Given two points, let's use two-point form

(y - 10) / (x - 2) = (-2 - 10) / (10 - 2)

Can you simplify and answer?

Answer 2

To find the linear function with the given properties, we can use the point-slope form of the equation for a line:

y - y1 = m(x - x1)

where (x1, y1) is a point on the line, and m is the slope of the line. In this case, we have two points: (2, 10) and (10, -2).

First, let's find the slope (m) using the two points:

m = (y2 - y1) / (x2 - x1)

m = (-2 - 10) / (10 - 2)

m = (-12) / (8)

m = -3/2

Now that we have the slope, we can use one of the points to find the equation of the line. Let's use (2, 10):

y - 10 = (-3/2)(x - 2)

Now, let's simplify and rewrite the equation in slope-intercept form (y = mx + b):

y - 10 = (-3/2)(x - 2)

Distribute the (-3/2) on the right side:

y - 10 = (-3/2)x + 3

Add 10 to both sides to isolate y:

y = (-3/2)x + 3 + 10

y = (-3/2)x + 13

So, the linear function that satisfies the given properties is:

f(x) = (-3/2)x + 13