Find the linear-to-linear function whose graph passes through the points (1,1), (5,2) and (20,3). What is its horizontal asymptote?
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To find the linear-to-linear function whose graph passes through the given points, we need to determine the equation of a line. We can use the point-slope form of a linear equation, which is given by:
y - y1 = m(x - x1),
where (x1, y1) is a point on the line and m is the slope of the line.
Let's use the first two points (1,1) and (5,2) to calculate the slope of the line:
m = (y2 - y1) / (x2 - x1)
= (2 - 1) / (5 - 1)
= 1 / 4.
Using the point-slope form with the point (1,1):
y - 1 = (1/4)(x - 1).
Expanding this equation:
y - 1 = (1/4)(x) - (1/4)(1)
y - 1 = (1/4)x - 1/4.
Simplifying further:
y = (1/4)x - 1/4 + 1
y = (1/4)x - 1/4 + 4/4
y = (1/4)x + 3/4.
So, the equation of the line that passes through the points (1,1) and (5,2) is y = (1/4)x + 3/4.
Now, let's check if the third point (20,3) lies on this line:
3 = (1/4)(20) + 3/4
3 = 5 + 3/4
3 = 20/4 + 3/4
3 = 23/4.
Since 3 is not equal to 23/4, the point (20,3) does not lie on this line. Therefore, the function represented by this line does not pass through the given points.
As a result, we cannot determine the horizontal asymptote of this function because it is not the correct function for the given set of points.
To find the linear-to-linear function, we need to determine the equation that represents the line passing through the given points.
Let's start by finding the slope of the line using the formula:
slope = (change in y) / (change in x)
Using the first two points (1,1) and (5,2):
slope = (2 - 1) / (5 - 1)
slope = 1 / 4
So the slope of the line is 1/4.
Next, we need to find the y-intercept of the line. We can use any of the three given points. Let's use the first point (1,1).
Using the slope-intercept form of a linear equation:
y = mx + b
where m is the slope and b is the y-intercept, we can substitute the values we know:
1 = (1/4)(1) + b
1 = 1/4 + b
b = 1 - 1/4
b = 3/4
Therefore, the equation of the linear function passing through the given points is:
y = (1/4)x + 3/4
To find the horizontal asymptote of this function, we need to examine the behavior of the function as x approaches positive or negative infinity.
In a linear-to-linear function, the horizontal asymptote is determined by the ratio of the coefficients of the leading term in the numerator and denominator.
The leading term in the numerator is 1/4x, and the leading term in the denominator is 1.
Therefore, the ratio of the coefficients is 1/4.
Thus, the horizontal asymptote of the given function is y = 1/4.
This means that as x approaches positive or negative infinity, the function approaches a horizontal line at y = 1/4.