Questions: Use the vertex and intercepts to sketch the graph of the quadratic function. Give the equation for the parabola's axis of symmetry. Use the parabola to identify the function's domain and range. f(x)=(x-3)^2+1 The axis of symmetry is x=3. Identify the function's domain. The domain is (-∞, ∞). Identify the function's range. The range is [1, ∞).

Solution Steps

The given quadratic function is \( f(x) = (x - 3)^2 + 1 \). The standard form of a quadratic function is \( f(x) = a(x - h)^2 + k \), where \( (h, k) \) is the vertex of the parabola. Here, \( h = 3 \) and \( k = 1 \). The axis of symmetry is the vertical line that passes through the vertex, so the axis of symmetry is \( x = 3 \).

The domain of a quadratic function is all real numbers because a parabola extends infinitely in both the positive and negative directions along the x-axis. Therefore, the domain is \( (-\infty, \infty) \).

The vertex form of the quadratic function \( f(x) = (x - 3)^2 + 1 \) indicates that the parabola opens upwards (since the coefficient of the squared term is positive). The minimum value of the function occurs at the vertex, which is \( k = 1 \). Therefore, the range of the function is all real numbers greater than or equal to 1. In interval notation, the range is \( [1, \infty) \).

Final Answer