Final answer:
The x-intercepts of the parabola with vertex (3, -2) and y-intercept (0, 7) can be found using the vertex form of a quadratic equation. Once the equation is established, we can set y to zero and solve for x to get the approximate x-intercepts (1.59, 0) and (4.41, 0).
Step-by-step explanation:
To find the x-intercepts of a parabola given its vertex and y-intercept, we can use the vertex form of a quadratic equation, which is y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola. Since the vertex given is (3, -2), we substitute h with 3 and k with -2 to get the equation y = a(x - 3)^2 - 2. Additionally, we know the y-intercept is (0, 7), allowing us to solve for a by substituting x with 0 and y with 7, leading to 7 = a(-3)^2 - 2, which simplifies to 9a = 9. Hence a is 1, and our equation becomes y = (x - 3)^2 - 2.
To find the x-intercepts, we set y to 0 and solve for x. This gives us the equation 0 = (x - 3)^2 - 2. Solving for x, we have x - 3 = ±√2, and thus x = 3 ± √2. Approximating the square root of 2 as 1.41, we get the x-intercepts as approximately x = 1.59 and x = 4.41, which corresponds to the x-intercepts ((1.59, 0)), ((4.41, 0)).
Note that these calculations are approximate and the exact x-intercepts can be calculated using a more precise value for the square root of 2 or by using a calculator.