Final answer:
To find the equation for a parabola that opens down and has the x-intercepts (1,0) and (7,0), we can first find the vertex, which is the highest point on the parabola. Then, we can use the vertex form of the equation to find the final equation.
Step-by-step explanation:
To find the equation for a parabola that opens down and has the x-intercepts (1,0) and (7,0), we can start by finding the vertex of the parabola. Since the parabola opens down, the vertex will be the highest point on the parabola. The x-coordinate of the vertex is the midpoint between the x-intercepts, which is (1+7)/2 = 4. The y-coordinate of the vertex is 0, since the vertex lies on the x-axis. So the vertex is (4,0). To find the equation for a parabola that opens down and has the x-intercepts (1,0) and (7,0), we can start by finding the vertex of the parabola. Since the parabola opens down, the vertex will be the highest point on the parabola. The x-coordinate of the vertex is the midpoint between the x-intercepts, which is (1+7)/2 = 4. The y-coordinate of the vertex is 0, since the vertex lies on the x-axis. So the vertex is (4,0).
Now that we have the vertex, we can use the vertex form of the equation: y = a(x-h)^2 + k, where (h,k) is the vertex. Plugging in the values from the vertex, we get: y = a(x-4)^2 + 0. Since the parabola opens down, a must be negative. Let's say a = -1. The equation of the parabola is then y = -(x-4)^2.
So the equation for a parabola that opens down and has the x-intercepts (1,0) and (7,0) is y = -(x-4)^2.