determine the equation of a quadratic equation that passes through the point (1,4) and has a vertex of (3,8).
what is the y-intercept?
To determine the equation of a quadratic function, let's start by using the vertex form of a quadratic equation:
y = a(x - h)^2 + k
where (h, k) represents the vertex of the parabola.
Since the vertex is (3,8), we can substitute h = 3 and k = 8 into the equation:
y = a(x - 3)^2 + 8
Now, let's use the given point (1,4) to solve for the value of a:
4 = a(1 - 3)^2 + 8
4 = a(-2)^2 + 8
4 = 4a + 8
4a = -4
a = -1
Substituting the value of a back into the equation, we get:
y = -1(x - 3)^2 + 8
Expanding and simplifying:
y = -(x^2 - 6x + 9) + 8
y = -x^2 + 6x - 1
Therefore, the equation of the quadratic function that passes through the point (1,4) and has a vertex of (3,8) is y = -x^2 + 6x - 1.
To find the y-intercept, we set x = 0 and solve for y:
y = -0^2 + 6(0) - 1
y = -1
Therefore, the y-intercept is -1.