Question

Included in the figure is the equation of the parabola y = -x ^ 2 + 8x + 9. The point N is the vertex of the parabola. Find the coordinates if M point.

Answer 1

M: (-9/4, 0)

Explanation:

y = -x² + 8x + 9

y = -(x² - 8x - 9)

y = -((x - 4)² - 16 - 9)

y = -(x - 4)² + 25

N: (4, 25)

The other point which is on the line and the curve is the y-intercept: (0, 9)

So the difference between the y intercept and N is 4 along the x-axis (i.e. 4 - 0) and 16 on the y-axis (i.e. 25 - 9);

We can therefore find the gradient of the line:

m = change in y/change in x

m = 16/4

m = 4

Now, find the equation of the line using the coordinate (0, 9) or N and m like so:

y - y¹ = m(x - x¹)

y - 9 = 4(x - 0)

y = 4x + 9

Point M has y-coordinate 0;

We can plu this into the equation to get the x-coordinate:

0 = 4x + 9

4x = -9

x = -9/4

M: (-9/4, 0)