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Chapter 3: Q. 85 (page 157)

Find the point on the line that is closest to the point 3,1.

[Hint: Express the distance dfrom the point to the line as a function of x,and then find the minimum value of dx2.

Short Answer

Expert verified

The point the line closest to 3,1is 2,2.

Step by step solution

01

Step 1. Given information

We need to determine the point on the line y=xthat is closest to the point 3,1.We need to express the distance dfrom the point to the line as a function of xand then find the minimum value of dx2.

02

Step 2. Simplify

We are searching for the line y=mx+cperpendicular to the line y=x.The orthogonal lines have an inversely proportional slope.

m=-11.

m=-1.

So, the line having the slope 1and passing through the point 3,1.

y-1=-1x-3.

y=-x+4.

Now to get the coordinate of the point x,xis the line y=x,solve.

x=-x+4.

2x=4.

x=2.

The point on the line closest to the3,1is2,2.

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Most popular questions from this chapter

In Problems 6鈥� 8, graph each quadratic function using transformations (shifting, compressing, stretching, and/or reflecting).

fx=x-22+2

Graph the function fby starting with the graph of y=x2and using transformations (shifting, compressing, stretching, and/or reflection). Verify your results using a graphing utility.

Hint: If necessary, write fin the form f(x)=a(x-h)2+k.

f(x)=3x2+6x

If h(x)=-3x+4,then

a) Determine the slope and y-intercept of each function.

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(c) Determine the average rate of change of each function.

(d) Determine whether the linear function is increasing, decreasing, or constant.

On one set of coordinate axes, graph the family of parabolas f(x)=x2+2x+cfor c = -3, c = 0, and c = 1. Describe the characteristics of a member of this family.

Match each graph to one the following functions.

f(x)=x2-1
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