Final answer:
To find the quadratic function f(x) with given roots x = -3 ± √22 and y-intercept (0,4), we use the factored form of a quadratic equation to determine 'a', and thereby derive the full equation f(x) in standard form.
Step-by-step explanation:
If the roots of a quadratic function f(x) are x = -3 ± √22 and the y-intercept is 4, we can determine f(x) by using the factored form of the quadratic equation, which is f(x) = a(x - r_{1})(x - r_{2}), where r_{1} and r_{2} are the roots of the equation. Given the roots, we rewrite them in the form applicable for the factored equation: x = -3 + √22 and x = -3 - √22. The y-intercept gives us the point (0, 4), which means f(0) = 4. From the y-intercept, we can find the value of a, by substituting x = 0 and f(x) = 4 into the factored form and solving for a. Eventually, we obtain f(x) in the standard quadratic equation form ax² + bx + c.
To find a, we would set up the equation 4 = a(0 + 3 - √22)(0 + 3 + √22), which simplifies to 4 = a * ((3 - √22)(3 + √22)). This then simplifies to 4 = a * (9 - 22), and thus a = -4/13. The final quadratic function that fits all the given conditions is: f(x) = -4/13 (x + 3 - √22)(x + 3 + √22), which expands to the standard form.