The vertex of the graph of a quadratic function is (3, 1) and the y-intercept is (0, 10). Which of the following is the equation of the function?
The choices are:
A.y=2x^2-12x+10
B. y=-1/2x^2+3x-11/2
C. y=x^2-6x+10
D. 2x^2-7x+4
y = a(x-3)^2 + 1
10 = a(0-3)^2+1, so a = 1
Looks like (C) to me
To determine the equation of the quadratic function, we need to use the vertex form of a quadratic equation. The vertex form is given by:
f(x) = a(x - h)^2 + k
where (h, k) represents the coordinates of the vertex.
Given that the vertex is (3, 1), we can substitute these values into the equation:
f(x) = a(x - 3)^2 + 1
Now, we need to find the value of 'a'. We can use the information about the y-intercept, which is given as (0, 10). The y-intercept occurs when x = 0, so we can substitute these values into the equation:
10 = a(0 - 3)^2 + 1
10 = 9a + 1
9a = 10 - 1
9a = 9
a = 1
Substituting the value of 'a' back into the equation, we get:
f(x) = 1(x - 3)^2 + 1
Simplifying this equation, we have:
f(x) = (x - 3)^2 + 1
Therefore, the correct equation of the quadratic function is f(x) = (x - 3)^2 + 1. Answer: A.
To find the equation of the quadratic function, we need to know the vertex and one additional point on the graph, such as the y-intercept.
The vertex form of a quadratic function is given by:
y = a(x - h)^2 + k
where (h, k) is the vertex of the parabola.
Given that the vertex is (3, 1), we can substitute these values into the equation:
y = a(x - 3)^2 + 1
Now, we also know the y-intercept is (0, 10), which means that when x = 0, y = 10. Substituting these values into the equation, we get:
10 = a(0 - 3)^2 + 1
10 = a(9) + 1
10 = 9a + 1
9a + 1 = 10
9a = 9
a = 1
Now that we have the value of a, we can substitute it back into the equation:
y = 1(x - 3)^2 + 1
Simplifying further:
y = (x - 3)^2 + 1
Therefore, the equation of the quadratic function is:
y = (x - 3)^2 + 1