Answer:
Q(-2,-7)
See attachment
Step-by-step explanation:
We need to form a simultaneous equation and solve.
The point P has coordinates (1,8). Let the other point Q also have coordinate (x,y).
Then the average rate of change is the slope of the secant line connecting P(1,8) and Q(x,y) and this has a value of 5.
[tex]\implies \frac{8-y}{1-x}=5[/tex]
[tex]\implies 8-y=5(1-x)[/tex]
[tex]\implies y=5x-3...(1)[/tex]
This point Q also lies on the given parabola whose equation is [tex]y=-(x-2)^2+9...(2)[/tex]
Put equation (1) into (2) to get:
[tex]5x+3=-(x-2)^2+9[/tex]
[tex]5x+3=-(x^2-4x+4)+9[/tex]
[tex]5x+3=-x^2+4x-4+9[/tex]
[tex]5x+3=-x^2+4x+5[/tex]
[tex]x^2+x-2=0[/tex]
[tex](x-1)(x+2)=0[/tex]
[tex]x=1,x=-2[/tex]
When x=-2, y=5(-2)-3=-7
Therefore the required point is Q(-2,-7)
