Find the maxima and minima coordinates of the vertext of the parabola of y= x^2-8x+1
This is a parabola, so it can either open upwards and have a min but not a max or open downwards and have a max but not a min. From the sign of the leading term (+1), we know this opens up.
The minimum position of this parabola would be the point of symmetry. To find this, we solve for the zeroes and average those.
The zeroes are found when y = 0. Some quadratic formula later, we get roots of (8 +/- sqrt(60))/2. The average is 4.
To find the coordinates of the vertex of the parabola, we need to first rewrite the equation in vertex form. The vertex form of a parabola is given by y = a(x - h)^2 + k, where (h, k) represents the coordinates of the vertex.
Considering the equation y = x^2 - 8x + 1, we can use the process of completing the square to rewrite it in vertex form. Here are the steps:
1. Begin by isolating the x terms on one side:
y = x^2 - 8x + 1
y + 7 = x^2 - 8x
2. Complete the square by adding the square of half the coefficient of x to both sides of the equation:
y + 7 + 4 = x^2 - 8x + 4
y + 11 = (x - 4)^2
Now, the equation is in vertex form: y = (x - h)^2 + k, where (h, k) is the vertex.
Comparing our equation with the vertex form, we can see that the vertex has coordinates (h, k) = (4, 11).
Therefore, the vertex of the parabola y = x^2 - 8x + 1 is located at (4, 11).
To find the maximum or minimum coordinates of a parabola, we need to rewrite the equation in vertex form, which is given by:
y = a(x-h)^2 + k
where (h, k) represents the coordinates of the vertex.
Given the equation y = x^2 - 8x + 1, we can rewrite it in vertex form by completing the square.
1. Start by isolating the x-terms to one side of the equation:
y - 1 = x^2 - 8x
2. To complete the square, take half of the coefficient of the x-term (-8) and square it:
(-8/2)^2 = 16
3. Add the result from step 2 to both sides of the equation:
y - 1 + 16 = x^2 - 8x + 16
4. Simplify:
y + 15 = (x - 4)^2
Now the equation is in vertex form. We can identify the coordinates of the vertex as (h, k), where h represents the x-coordinate of the vertex and k represents the y-coordinate.
Comparing the equation y + 15 = (x - 4)^2, we see that (h, k) = (4, -15).
Therefore, the vertex coordinates of the parabola y = x^2 - 8x + 1 are (4, -15).