Given the points (3,52488) and (-4,-393216), determine
(a) the linear function ,
(b) the power function ,
for these two points.
( N.B. If you use your calculator immediately without thinking, it will say that there are no solutions, but there are solutions. Find an equivalent equation for one of the equations so that there are no negative values.)
any two points can be joined by a single line. In this case it's
y-52488 = ((-393216-52488)/(-4-3))(x-3)
There are many power functions which can go through any two points.
To find the linear function and power function for the given points, let's start by determining the slope of the line passing through the two points.
(a) Linear function:
The slope, denoted by m, of the line passing through two points (x₁, y₁) and (x₂, y₂) is given by the formula:
m = (y₂ - y₁) / (x₂ - x₁)
Let's calculate the slope using the given points:
m = (-393216 - 52488) / (-4 - 3)
m = -445704 / -7
m = 63672
Now that we have the slope, let's use the point-slope form of a linear equation to derive the linear function:
y - y₁ = m(x - x₁)
We can choose either of the given points. Let's use (3, 52488):
y - 52488 = 63672(x - 3)
Expanding the equation:
y - 52488 = 63672x - 191016
Rearranging the equation to the slope-intercept form:
y = 63672x - 138528
Therefore, the linear function for the given points is y = 63672x - 138528.
(b) Power function:
To find the power function, we need to transform the original coordinates into a new set of values such that there are no negative numbers.
Let's divide each y-coordinate by 52488, which is the y-coordinate of the point (3, 52488):
New point (3, 52488):
New y = 52488 / 52488 = 1
New point (-4, -393216):
New y = -393216 / 52488 ≈ -7.481
Now that we have the transformed points, we can find the power function.
The general form of a power function is y = a * x^b, where a and b are constants.
Substituting the new point (3, 1):
1 = a * 3^b
Substituting the new point (-4, -7.481):
-7.481 = a * (-4)^b
Dividing the second equation by the first equation:
-7.481 / 1 = (a * (-4)^b) / (a * 3^b)
Simplifying:
-7.481 = (-4/3)^b
Now solve for b:
b = log(-7.481) / log(-4/3)
Note: The logarithm of a negative number is not defined, so we need to take the logarithm of the absolute value and account for the sign later.
Using a calculator or software, calculate:
b ≈ log(7.481) / log(4/3) ≈ -0.979
Now substitute the value of b into either of the original equations:
1 = a * 3^(-0.979)
Solve for a:
a ≈ 1 / 3^(-0.979) ≈ 1.369
Therefore, the power function for the given points is:
y = 1.369 * x^(-0.979)
Please note that due to the use of approximations, the values presented here may not be exact.