Find the vertex of each of the following parabolas by averaging the x intercept. Then write each equation in graphing form.
Y=(x-3)(x-11)
halfway between 3 and 11
(3+11)/2 = 7
so that is the x location of the vertex
y = (7-3)(7-11)
4*-4 = -16
so I get
(7, -16)
Well, let's find the x-intercepts first. To do that, we'll set y = 0.
0 = (x - 3)(x - 11)
To solve for x, we can set each factor equal to zero and solve for x individually:
x - 3 = 0 --> x = 3
x - 11 = 0 --> x = 11
Now, to find the vertex, we can average the x-intercepts:
Vertex x-coordinate = (3 + 11) / 2 = 14 / 2 = 7
To write the equation in graphing form, we can use the vertex form: y = a(x - h)^2 + k. In this equation, (h, k) represents the vertex.
Since the x-coordinate of the vertex is 7, we have:
Vertex = (7, k)
To find the value of k, we substitute the x-coordinate (7) into the equation and solve for y:
k = (7 - 3)(7 - 11)
k = 4(-4)
k = -16
So, the vertex is (7, -16), and the equation in graphing form is:
y = (x - 7)^2 - 16
To find the vertex of a parabola, we can first find the x-intercepts by setting the equation equal to zero and solving for x.
1. Setting y = 0:
0 = (x - 3)(x - 11)
Using the zero product property, we can set each factor equal to zero:
x - 3 = 0 --> x = 3
x - 11 = 0 --> x = 11
Next, we can find the average of the x-intercepts to determine the x-coordinate of the vertex:
x-coordinate of vertex = (x1 + x2) / 2
= (3 + 11) / 2
= 14 / 2
= 7
To find the y-coordinate of the vertex, we substitute the x-coordinate back into the original equation:
y = (7 - 3)(7 - 11)
= 4(-4)
= -16
Therefore, the vertex of the parabola Y = (x - 3)(x - 11) is (7, -16).
To write the equation in graphing form, we expand the equation:
Y = (x - 3)(x - 11)
= x^2 - 11x - 3x + 33
= x^2 - 14x + 33
So, the equation in graphing form is Y = x^2 - 14x + 33.
To find the vertex of a parabola, we don't need to average the x-intercepts. Instead, we can use the formula for the x-coordinate of the vertex, which is given by the formula:
x = -b / (2a)
Here, a, b, and c are the coefficients of the quadratic equation in the form ax^2 + bx + c.
In this case, the equation is in factored form:
Y = (x - 3)(x - 11)
To find the equation in graphing form, we need to multiply out the equation:
Y = x^2 - 14x + 33
Now that we have the equation in standard form, we can tell the vertex by using the formula for x:
x = -b / (2a)
= -(-14) / (2 * 1)
= 14 / 2
= 7
Now, substitute this value of x back into the equation to find the y-coordinate of the vertex:
Y = (7)^2 - 14(7) + 33
= 49 - 98 + 33
= -16
So, the vertex of the parabola is (7, -16).
The equation in graphing form is Y = x^2 - 14x + 33.