Chapter 7.6, Problem 40WE

(a)

To determine

Find the number of times the graph of the parabola $y=x^2-6x+10$ intersects x -axis.

Answer to Problem 40WE

Zero

Explanation of Solution

Formula Used:

The parabola $y-k=a{(x-h)}^2$ has

Vertex = ( h , k )

If a > 0 , the parabola opens upward and the vertex is the minimum.

If a < 0 , the parabola opens downward and the vertex is the maximum.

Calculation:

Given equation :

  $\begin{array}{l}y=x^2-6x+10\\ \Rightarrow y=x^2-6x+9+1\\ \Rightarrow y-1=x^2-2(x)(3)+3^2\\ \Rightarrow y-1={(}^x\mathrm{............}[{(}^a=a^2+2ab+b^2]\end{array}$

It has vertex at (3,1) and since a = 1 > 0 , so the minimum is y = 1 , hence the parabola intersects x -axis zero times .

(b)

To determine

Find the number of times the graph of the parabola $y=x^2-6x+10$ intersects the line y = 1.

Answer to Problem 40WE

One time.

Explanation of Solution

Formula Used:

The parabola $y-k=a{(x-h)}^2$ has

Vertex = ( h , k )

axis of symmetry : x = h

Calculation:

Given equation :

  $\begin{array}{l}y=x^2-6x+10\\ \Rightarrow y=x^2-6x+9+1\\ \Rightarrow y-1=x^2-2(x)(3)+3^2\\ \Rightarrow y-1={(}^x\mathrm{............}[{(}^a=a^2+2ab+b^2]\end{array}$

It has vertex at (3,1) and since a = 1 > 0 , so the minimum is y = 1 , hence the parabola intersects y = 1 only one time , at the vertex.

(b)

To determine

Find the number of times the graph of the parabola $y=x^2-6x+10$ intersects the line y = 2.

Answer to Problem 40WE

Two times.

Explanation of Solution

Formula Used:

The parabola $y-k=a{(x-h)}^2$ has

Vertex = ( h , k )

axis of symmetry : x = h

Calculation:

Given equation :

  $\begin{array}{l}y=x^2-6x+10\\ \Rightarrow y=x^2-6x+9+1\\ \Rightarrow y-1=x^2-2(x)(3)+3^2\\ \Rightarrow y-1={(}^x\mathrm{............}[{(}^a=a^2+2ab+b^2]\end{array}$

It has vertex at (3,1) and since a = 1 > 0 , so the minimum is y = 1 and the axis of symmetry is x = 3.

So, for any value greater than y = 1 , the graph of the parabola is symmetric along x = 3.

Hence the line y = 2 intersects the parabola two times.