Find the vertex of each of the following parabolas by averaging the x intercepts. Then write each equation in graphing form.
Y=x^2-14x+40
y = (x-10)(x-4)
I expect you can take it from there, no?
To find the vertex of the parabola and write the equation in graphing form, follow these steps:
Step 1: Find the x-intercepts by setting y = 0 and solving the quadratic equation.
Setting y = 0 in the equation y = x^2 - 14x + 40, we get:
x^2 - 14x + 40 = 0
Using factoring or the quadratic formula, we find the x-intercepts:
(x - 10)(x - 4) = 0
Setting each factor equal to zero:
x - 10 = 0 or x - 4 = 0
Solving for x:
x = 10 or x = 4
So, the x-intercepts are x = 10 and x = 4.
Step 2: Find the average of the x-intercepts to get the x-coordinate of the vertex.
To find the x-coordinate of the vertex, we average the x-intercepts:
(10 + 4) / 2 = 14 / 2 = 7.
So, the x-coordinate of the vertex is 7.
Step 3: Substitute the x-coordinate of the vertex (7) into the original equation to find the y-coordinate of the vertex.
Substituting x = 7 into the equation y = x^2 - 14x + 40, we get:
y = (7)^2 - 14(7) + 40
y = 49 - 98 + 40
y = -9
So, the y-coordinate of the vertex is -9.
Step 4: Write the equation in graphing form.
The graphing form of a quadratic equation is given by:
y = a(x - h)^2 + k, where (h, k) represents the coordinates of the vertex.
From Step 3, we have (h, k) = (7, -9).
Substituting these values into the graphing form equation, we get:
y = a(x - 7)^2 - 9.
Therefore, the equation in graphing form is:
y = a(x - 7)^2 - 9.
To find the vertex of a parabola, you can use the formula x = -b/(2a) where a, b, and c are the coefficients of the quadratic equation in the form ax^2 + bx + c = 0. In this case, the quadratic equation is y = x^2 - 14x + 40.
First, let's find the x-intercepts of the equation by solving the equation for y = 0:
0 = x^2 - 14x + 40
To factorize the equation, we need to find two numbers that multiply to give 40 and add up to -14. In this case, the numbers are -4 and -10:
0 = (x - 4)(x - 10)
Setting each factor to zero and solving for x, we get:
x - 4 = 0 --> x = 4
x - 10 = 0 --> x = 10
The x-intercepts are x = 4 and x = 10.
To find the vertex, we take the average of the x-intercepts, which gives (4 + 10)/2 = 14/2 = 7. So, the x-coordinate of the vertex is 7.
To find the y-coordinate of the vertex, we substitute this x-coordinate into the equation:
y = (7)^2 - 14(7) + 40
y = 49 - 98 + 40
y = -9
Thus, the vertex is (7, -9).
Now, let's write the equation in graphing form. The form y = a(x - h)^2 + k is called the vertex form, where h and k are the coordinates of the vertex.
In this case:
h = 7 (x-coordinate of the vertex)
k = -9 (y-coordinate of the vertex)
Substituting the values into the vertex form, we have:
y = a(x - 7)^2 - 9
To find the coefficient a, we can use one of the points on the parabola, such as (4, 0):
0 = a(4 - 7)^2 - 9
0 = a(-3)^2 - 9
0 = 9a - 9
9 = 9a
a = 1
Therefore, the equation of the parabola in graphing form is y = (x - 7)^2 - 9.