The table below shows some values of a linear function f and an exponential function g. Fill in exact values for each of the missing entries.
X 0 1 2 3 4
f(x) 25 375
g(x) 25 375
To find the missing values, we need to determine the formulas for the linear function f(x) and the exponential function g(x).
For the linear function f(x), we can use the formula y = mx + b, where m is the slope and b is the y-intercept.
Using the given values, we can calculate the slope:
m = (f(x2) - f(x1)) / (x2 - x1)
m = (375 - 25) / (4 - 0)
m = 350 / 4
m = 87.5
Now, we can plug in the slope and any point (x, f(x)) to solve for b. Let's use the point (0, f(0)):
25 = (87.5)(0) + b
25 = b
Therefore, the formula for the linear function f(x) is:
f(x) = 87.5x + 25
Now, let's find the formula for the exponential function g(x). Exponential functions have the form y = a * b^x, where a is the initial value and b is the base.
Using the given values, we can plug in a and b:
25 = a * b^0
25 = a * 1
25 = a
Now, we can write the formula for the exponential function g(x):
g(x) = 25 * b^x
To find the value of b, we can use the point (1, g(1)):
375 = 25 * b^1
375 = 25b
b = 375 / 25
b = 15
Therefore, the formula for the exponential function g(x) is:
g(x) = 25 * (15)^x
Now, let's fill in the missing values in the table:
X 0 1 2 3 4
f(x) 25 112.5 200 287.5 375
g(x) 25 375 5625 84375 1265625
To fill in the missing values for the linear function f(x), we need to determine the equation of the line. We can use the formula for the equation of a line: y = mx + b, where m is the slope and b is the y-intercept.
Since we have points (0, 25) and (4, 375), we can calculate the slope using the formula m = (y2 - y1) / (x2 - x1).
Substituting the values, m = (375 - 25) / (4 - 0) = 350 / 4 = 87.5.
Now that we know the slope, we can substitute it into the equation y = mx + b and use one of the known points to solve for the y-intercept b.
Using the point (0, 25), we can substitute x = 0 and y = 25 into the equation:
25 = 87.5(0) + b
25 = b
Therefore, the equation for the linear function f(x) is:
f(x) = 87.5x + 25.
Now, to fill in the missing values for the exponential function g(x), we need to determine the equation of the exponential curve. The general form of an exponential function is y = a * b^x, where a is the initial value and b is the base of the exponent.
Using the points (0, 25) and (4, 375), we can set up two equations:
25 = a * b^0 ---> 25 = a * 1 ---> a = 25
375 = a * b^4
Now, substitute the value of a (which we found to be 25) into the second equation:
375 = 25 * b^4
Divide both sides by 25 to isolate b^4:
375 / 25 = b^4
15 = b^4
To find the value of b, take the fourth root of both sides:
∛(15) = b
Using a calculator, we find that the fourth root of 15 is approximately 1.985.
Therefore, the equation for the exponential function g(x) is:
g(x) = 25 * (1.985)^x.
Now, to fill in the missing values in the table, substitute each value of x into the corresponding equation for f(x) and g(x) to find the values of f(x) and g(x) at that x-coordinate.
For example:
f(1) = 87.5(1) + 25 = 112.5
g(1) = 25 * (1.985)^1 = 49.625
By substituting x = 1 into the equations, you can calculate the missing values in the table.
I think you are missing come values.
For linear, figure the constant slope.
For exponential, divide two known values to get a power of the growth factor.
See your previous post for an example.