Coordinates A and B: (1, 0) and (3, 0)
Coordinates of P: (0, 3)
Coordinates of Q: (2, -1)
Step-by-step explanation:
You're pretty much given the x-intercepts from the original equation.
y = (x - 1)(x - 3). So to get the x-intercepts (points A and B), y has to equal 0. So,
0 = (x - 1)(x - 3).
and y = 0 when x equals 1 and 3. Because anything multiplied by 0 equals 0. So substituting 1 for the value of x, for example, we get:
0 = (1 - 1)(1 - 3)
0 = 0 * (-2)
So there are your x-intercepts, points A and B.
Now,
Expand the brackets of the equation.
y = (x - 1)(x - 3) becomes
y = x^2  -  4x  + 3
You get the y-intercept (point P) by substituting 0 for the value of x. Doing so gives:
y = 0^2 - 4*0 + 3
y = 0 - 0 + 3
y = 3
So, we have our y-intercept (0, 3).
Point Q, the x-coordinate of the vertex is calculated with the equation -b/2a
Taking our expanded equation: x^2 - 4x + 3
which is of the form ax^2 + bx + c
So we find the values of a, b and c and input the values into the equation
a = 1
b = -4
c = 3
So, inputting these into the vertex equation, gives:
-(-4) / 2 * 1
4 / 2
2
So we have the x-coordinate of the vertex. Now that we know that, we can just substitute it into the expanded equation to give the y-coordinate:
y = 2^2 - 4*2 + 3
y = 4 - 8 + 3
y = -1
So there's our y-coordinate, and now we have both the x and y coordinate for the vertex, point Q. (2, -1).
