Final answer:
The equation of the parabola with x-intercepts (-5,0) and (-9,0) and a vertical stretch of 1/2 is f(x) = 1/2(x + 5)(x + 9). The vertex lies exactly halfway between the x-intercepts, which is at x = -7.
Step-by-step explanation:
To find the equation of a parabola given the x-intercepts and a vertical stretch, we first use the intercept form of a quadratic equation, which is f(x) = a(x - h)(x - k), where h and k are the x-intercepts and a is the stretching factor. Since the x-intercepts are given as (-5,0) and (-9,0), our equation initially looks like f(x) = a(x + 5)(x + 9). Knowing that the vertex is a minimum and there is a vertical stretch of 1/2, we can determine a to be positive and equal to 1/2.
The vertex of a parabola given by f(x) = a(x - h)2 + k is the point (h, k), and it lies exactly halfway between the x-intercepts for a symmetric parabola. For our parabola, this would be at the midpoint between -5 and -9, which is -7. Since the parabola opens upwards and has a minimum vertex, the y-coordinate of the vertex will be the lowest point on the parabola, but without additional information, we can only provide the x-coordinate of the vertex.
Hence, the final equation of the parabola becomes f(x) = 1/2(x + 5)(x + 9). Note that we've incorporated the vertical stretch into the coefficient a.