Final answer:
The resulting quadratic function is f(x) = x² + 30x - 400.
The value of b is 30 and c is -400.
Step-by-step explanation:
To find the values of b and c for the quadratic function, we need to consider the transformations it undergoes. First, when the function is reflected in the y-axis, the coefficient of x becomes -b.
Then, when the function is stretched horizontally with a scale factor of a, the coefficient of x becomes (1/a)b. Lastly, when the function is translated horizontally through t, the constant term c becomes c - (b/a)t + (bt^2)/a.
Since the function f(x) = x² + bx + c has the same x-intercepts as the resulting quadratic function, the solutions of the original function will remain the same after the transformations.
Hence, we can equate the quadratic formula for the original function to the quadratic formula for the resulting function. By comparing the coefficients, we can determine the values of b and c.
Let's substitute the values of a, b, and c into the quadratic formula and solve for b and c.
The quadratic formula for the original function f(x) = x² + bx + c is:
x = (-b ± sqrt(b² - 4ac)) / (2a)
Let's substitute the values of a, b, and c into the quadratic formula:
x = (-b ± sqrt(b² - 4ac)) / (2a)
= (-b ± sqrt((10.0)² - 4(1.00)(-200))) / (2(1.00))
= (-b ± sqrt(100 + 800)) / 2
= (-b ± sqrt(900)) / 2
= (-b ± 30) / 2
The solutions of the quadratic equation are x = (-b + 30) / 2 and x = (-b - 30) / 2. Since we want the same x-intercepts, these solutions must be equal to the original x-intercepts of f(x). Therefore, we can set the quadratic formula equal to zero and solve for b:
(-b + 30) / 2 = 0
-b + 30 = 0
-b = -30
b = 30
Now that we know b is 30, we can substitute it into the quadratic formula and solve for c:
x = (-30 ± sqrt((30)² - 4(1.00)(-200))) / (2(1.00))
= (-30 ± sqrt(900 + 800)) / 2
= (-30 ± sqrt(1700)) / 2
= (-30 ± sqrt(25 * 68)) / 2
= (-30 ± 5sqrt(68)) / 2
Since we know that x = 10 is one x-intercept of f(x), we can substitute this value into the quadratic formula and solve for c:
(10)² + (30)(10) + c = 0
100 + 300 + c = 0
c = -400
Therefore, the quadratic function resulting from reflecting f(x) in the y-axis, stretching it horizontally, and translating it has the equation f(x) = x² + 30x - 400.