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Linear function and inversely proportional function example1

Question:

The graph of a linear function y1 = mx + b intersects with the graph of an inversely proportional function y2 = k/x at the point p (1, 4). The linear function y1 intersects with the x-axis at -2.

(1). What are the equations of y1 and y2?

(2). When x > 1, what is the relationship between y1 and y2?

the graph of a linear function y = m x + b and the graph of an inversely proportional function y = k/x.

Solution:

(1). The line equation y1 = m x + b is the slope intercept form of a line equation. Because we are given two points on the line l1, we use the two-points form of a line equation, which is y - yo = m (x - xo), in which m is the slope of the line and (xo, yo) is a known point. We name this point as point A, it has the x-coordinate -2 and y-coordinate 0, that is, the coordinate of point A is (-2, 0). Let the point A be the first point, the point P be the second point. The slope of line L1 is: m = (y coordinate of the second point subtract the y-coordinate of the first point) / (x coordinate of the second point subtract the x-coordinate of the first point) = (4 - 0) / [1 - (-2)] = 4 / (1 + 2) = 4/3. So, the slope of line y1 is 4/3. The equation of line L1 is:

y - yo = m (x - xo)
y - 0 = (4/3)[x - (-2)] = (4/3)(x + 2)
y = (4/3) x + 8/3
the line intersects with the y-axis at 8/3

So, the line equation is: y1 = (4/3) x + 8/3

Now, we will find the inversely proportional function y2 = k/x, k is unknow, but we know that the point P (1, 4) lies on y2. So, we substitute the coordinate of the point P into the inversely proportional function to get the value of k.

y2 = k/x and P (1, 4) lies on y2
4 = k/1, so, k = 4
the equation of y2 is: y2 = 4/x

So, the inversely proportional function is y2 = 4/x.

(2). when x = 1, y1 = y2 = 4.

when x > 1, the linear function increases as x increase. The inversely proportional function decreases as x increase. So, when x > 1, y1 > y2.