How do you find the vertex and intercepts for #y=x^2 – 12x + 36#?

Question

Replaced equation (with 2 solutions for #x#) into a corresponding function with vertex and intercepts.

Eva Adamson · Accepted Answer

Vertex at #(6,0)#
y-intercept: #36#
x-intercept: 6#

#y=x^2-12x+36# #rArr y = (x-6)^2+0# this is the vertex form of a parabola with vertex at #(6,0)# The y-intercept is the value of #y# when #x=0# #color(white)("XXX")y=(0)^2-12(0)+36 = 36# The x-intercept is the value of #x# when #y=0# #color(white)("XXX")0=x^2-12x+36 = (x-6)(x-6)# #color(white)("XXX")rArr x=6# (single intercept point) graph{x^2-12x+36 [-25.27, 54.73, -1.6, 38.4]}

Isaiah Anthony · Answer

undefined To find the vertex and intercepts for the quadratic function $ y = x^2 - 12x + 36 $, you can follow these steps:

1. To find the x-coordinate of the vertex, use the formula $ x = -\frac{b}{2a} $, where $ a = 1 $ and $ b = -12 $.
2. Once you have the x-coordinate of the vertex, substitute it into the equation to find the y-coordinate.
3. The vertex of the parabola is the point (x, y) obtained from step 2.
4. To find the x-intercepts, set $ y = 0 $ and solve the quadratic equation $ x^2 - 12x + 36 = 0 $ using factoring, completing the square, or the quadratic formula.
5. The solutions to this equation will give you the x-coordinates of the x-intercepts.
6. To find the y-intercept, substitute $ x = 0 $ into the equation and solve for y.

The vertex is (6, 0), and since the quadratic function is a perfect square trinomial, it has only one x-intercept (6, 0). The y-intercept is (0, 36).

Ethan Adkins · Answer

undefined To find the vertex of the quadratic function y = x^2 - 12x + 36, you can use the formula for the x-coordinate of the vertex, which is given by x = -b/2a, where a and b are the coefficients of the quadratic equation.

First, identify the coefficients a and b from the quadratic equation y = x^2 - 12x + 36. In this equation, a = 1 and b = -12.

Substitute these values into the formula x = -b/2a to find the x-coordinate of the vertex:

x = -(-12) / (2*1)
x = 12 / 2
x = 6

Now that you have the x-coordinate of the vertex, you can find the y-coordinate by substituting the value of x into the original equation:

y = (6)^2 - 12(6) + 36
y = 36 - 72 + 36
y = 0

So, the vertex of the quadratic function y = x^2 - 12x + 36 is (6, 0).

To find the x-intercepts, set y = 0 in the equation and solve for x:

0 = x^2 - 12x + 36

This equation can be factored as (x - 6)(x - 6) = 0. Since both factors are the same, the x-intercept occurs at x = 6.

To find the y-intercept, set x = 0 in the equation and solve for y:

y = (0)^2 - 12(0) + 36
y = 36

So, the y-intercept of the quadratic function y = x^2 - 12x + 36 is 36.