Chapter 9.6, Problem 59E

a.

To determine

To calculate the coordinates of a point

Answer to Problem 59E

Point A is $(0,4)$

Explanation of Solution

Given information:

Equation is,

  $y=-x^2+6x+4$

  

Calculation:

Concluding from graph, point A is $(0,y)$

Hence, for value of y, substituting the values,

  $y=-{(0)}^2+6(0)+4$

So,

  $y=0+0+4$

On solving,

  $y=4$

Hence, point A is,

  $(0,4)$

b.

To determine

To calculate the coordinates of a point

Answer to Problem 59E

Point B is $5,9$

Explanation of Solution

Given information: Equation is,

  $y=-x^2+6x+4$

Point A is,

  $(0,4)$

  

Formula used:

Equation of line is

  $y-y_1^{}=m(x-x_1^{})$

Slope is,

  $m=\frac{y_2^{}-y_1^{}}{x_2^{}-x_1^{}}$

Calculation:

For slope, substituting the values,

  $m=\frac{2-0}{6-4}$

So,

  $m=\frac{2}{2}$

On solving,

  $m=1$

For equation of line AB,

  $y-4=1(x-0)$

So,

  $y-4=x$

So,

  $y=x+4$

For solving $y=x+4$ and $y=-x^2+6x+4$ , subtracting them

  $y-y=x+4-(-x^2+6x+4)$

Hence,

  $0=x+4+x^2-6x-4$

On solving,

  $x^2-5x=0$

So,

  $x(x-5)=0$

So,

  $x-5=0$

On solving,

  $x=5$

Substituting the values, in equation of line,

  $y=5+4$

On solving,

  $y=9$

Hence, point B is,

  $5,9$