Question

Find the rate of change and the initial value of the linear function given as these values: (2.6,-6) (3.2,3)

Answer 1

To find the rate of change and initial value of a linear function, we can use the slope-intercept form of a linear equation: y = mx + b.

Let's use the first given point (2.6, -6) to find the initial value (y-intercept), b.

Substituting the x and y values of the point into the equation, we get:
-6 = 2.6m + b

Now, let's use the second given point (3.2, 3) to find the rate of change (slope), m.

Substituting the x and y values of the point into the equation, we get:
3 = 3.2m + b

We now have a system of two equations with two unknowns:

-6 = 2.6m + b (Equation 1)
3 = 3.2m + b (Equation 2)

We can solve this system of equations to find the values of m and b.

Subtracting Equation 1 from Equation 2, we eliminate b:
3 - (-6) = 3.2m - 2.6m + b - b
9 = 0.6m

Solving for m:
m = 9/0.6
m = 15

We now have the value of m, which represents the rate of change.

Substituting m = 15 into Equation 1, we can solve for b:

-6 = 2.6(15) + b
-6 = 39 + b

Solving for b:
b = -6 - 39
b = -45

The initial value (y-intercept) of the linear function is -45, and the rate of change (slope) is 15.