Final answer:
The quadratic function f(x) = 4x² - 4x + 3 can be rewritten as 4(x - 0.5)² + 2, with integers a = 4, b = -1/2, and c = 2. Its domain is all real numbers, and its range is [2, +∞) since the parabola opens upwards with its vertex at (0.5, 2).
Step-by-step explanation:
The student is asking to rewrite the quadratic function f(x) = 4x² - 4x + 3 into the form a(x + b)² + c, where a, b, and c are integers, and to find its domain and range.
To rewrite the function, we need to complete the square. The function given is already in the form ax² + bx + c, so we can start by writing down only the x-terms and completing the square for them:
Factor out the coefficient of x² from the first two terms: 4(x² - x) + 3.
Add and subtract the square of half the coefficient of x inside the parentheses: 4[(x - 0.5)² - 0.25] + 3.
Distribute and simplify: 4(x - 0.5)² - 4²(0.25) + 3 = 4(x - 0.5)² - 1 + 3 = 4(x - 0.5)² + 2.
So, f(x) can be written as 4(x - 0.5)² + 2, where a = 4, b = -0.5 (which can be expressed as b = -1/2 when looking for integers), and c = 2. Since we're asked for integers, we express b as -1/2 since 0.5 fraction form is expressed as 1/2 which are both integers.
The domain of f(x) is all real numbers, which is (-∞, +∞), because there are no restrictions on the values that x can take in a quadratic function. The range of f(x) can be determined by finding the vertex of the parabola. Since the coefficient of the x² term is positive, the parabola opens upwards, and the vertex represents the minimum point. The y-coordinate of the vertex (which is c) will be the lowest value of the range, thus the range is [c, +∞), or in this case, [2, +∞).