How can I find the coordinates of the points at the curve y=x2-x-12 where it cuts the x axis and y axis?
Before I start giving the solution to the problem, I will rewrite the given equation as follows to avoid confusion due to different interpretation of 'x2' to mean 'x*2' or 'x^2'.
y = x^2 - x - 12 . . . (1)
The method to find to points where a curve having above equation cuts x axis is to substitute the value y = 0 in above equation, and solve the equation for value of x. Please note as this is a quadratic equation of x, there will be two possible values of x. That is the cure cuts y axis at two places. To fond values of x for these these two point we need to find the factors of x. These will give 'x' coordinates of these points. 'y' coordinates will always be '0'.
To find the point where the curve cuts y axis we substitute the value x = o in the above equation and solve the equation for value of y. Here there will be only one possible value of y, indicating that the curve cuts y axis at one point only. The x coordinate will always be equal to '0'.
To find where curve cuts x axis
Substituting value of y = 0 in equation (1) we get:
x^2 - x - 12 = 0
Therefore: x^2 - 4x + 3x - 12 = 0
Therefore: (x^2 - 4x) + (3x - 12) = 0
Therefore: (x - 4) *(x + 3) = 0
Therefore: x = 4 and x = -3
To find where curve cuts y axis
Substituting value of x = 0 in equation (1) we get:
y = 0 - 0 - 12 = -12
The curve cuts x axis at point (4, 0) and (-3, 0).
It cuts y axis at (0, -12)
To find the coordinates of the points of intersection of the curve y=x^2-x-12 with x and y axes.
The equation of x axis is y=0. Therefore,the points of intersection of y=x^2-x-12 and y=0 are got by eliminating y between these two equations or putting y=0 in y= x^2-x-12 and solving for x to get x intercepts:
0=(x+3)(x-4). or x+3=0 or x-4=0 ==>
x=-3 or x=4, are the points on x axis where the curve intersects x axis. So, the coordinates of the points of intersection of x axis are (-3 , 0) and (4 , 0)
Similarly the equation of y axis is x=0. So,Put x= 0 in y = x^2-x-12 and solve for y to get the point , where the given curve intersects y axis: So,
y=0^2-0-12 =-12, is the point where the curve intersects y axis. So, the coordinates of the point of intersection of y axis is: (0 , -12)