To find the x-intercepts of a quadratic function, we need to determine the x values for which the function equals zero.
In this case, we have a parabola that opens downward, passes through the points (-2, 0) and (3, 0), and has a minimum point.
To find the x-intercepts, we can set the quadratic function equal to zero and solve for x. Let's denote the quadratic function as f(x).
Since the parabola passes through the points (-2, 0) and (3, 0), we know that these points are on the function graph. Therefore, we can set up the following equations:
1. When x = -2, f(x) = 0
f(-2) = a(-2)^2 + b(-2) + c = 0
2. When x = 3, f(x) = 0:
f(3) = a(3)^2 + b(3) + c = 0
We also know that the parabola has a minimum point, which means that its vertex lies on the symmetry axis. The axis of symmetry is the line that passes through the vertex and divides the parabola into two symmetric parts. The vertex's x-coordinate is given by the formula x = -b / (2a). In our case, since the parabola passes through the point (0, -6), we can find the symmetry axis as follows:
x = -b / (2a)
0 = -b / (2a)
Simplifying the equation, we find b = 0.
Substituting b = 0 in the equations we set up earlier, we get:
1. When x = -2:
a(-2)^2 + c = 0
2. When x = 3:
a(3)^2 + c = 0
Simplifying these equations, we have:
1. 4a + c = 0
2. 9a + c = 0
We can solve these two equations simultaneously to find the values of a and c.
Subtracting equation 1 from equation 2, we get:
9a + c - (4a + c) = 0 - 0
5a = 0
a = 0
Substituting a = 0 into equation 1, we find:
4(0) + c = 0
c = 0
Therefore, the quadratic function is f(x) = 0x^2 + 0x + 0, which simplifies to f(x) = 0.
Since the coefficient of x^2 is zero, the quadratic function reduces to a linear function with a slope of 0. This means that the graph is a horizontal line passing through the y-axis at y = 0.
In summary, the given information does not define a quadratic function with x-intercepts. The graph is a horizontal line passing through the Y-axis. Thus, the answer is none of the given options (a, b, c, d).