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Reflection of a line across the line y = x

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(a) Write the equation of a line that intersects the negative x-axis and the positive y-axis at points not equidistant from the origin (0, 0).

(b) Draw the line.

(c) Draw the line that is the reflection of your line across the line y=x.

(d) Find the equation of the line drawn in Part (c). Do not convert fractions, if any, to decimals.

(e) Identify and write the slope, the x-intercept, and the y-intercept of each of the two lines above.

(f) Make a general statement based on your comparison of the slopes, the x-intercepts, and the y- intercepts.

(g) Mathematically, using inverse functions and composition of functions, determine whether the two lines above are inverses of each other.

(h) Determine the point at which the two lines intersect.

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Solution Preview

(a) Consider the line that intersects the negative x-axis at the point with x-coordinate -4, and the positive y-axis at the point with y-coordinate 3. We can use the coordinates of those two points to determine the point-slope form of the equation of the line.

The coordinates of the point at ...

Solution Summary

The solution is obtained by a very systematic approach, which is presented and explained in detail. First, the equation of a line with the given characteristics is determined. Then that line is drawn and its reflection across the line y = x is obtained. The equation of the second line is determined, and key parameters for each line (namely, the slope, x-intercept, and y-intercept) are computed. This information is used to make a general conjecture as to how the key parameters for one of the two lines are related to those for the other line. All of that information is used to show that the two lines are inverses of each other. Finally, the point of intersection of the two lines is found.