Question

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. f(x)=(x−1)2+2

Accepted Answer

Identify the given quadratic function: $$f(x) = (x - 1)^2$ + 2. $$This is in vertex form, $$f(x) = a(x - h)^2$ + k$$, where $$(h, k) is $$the vertex. Find the vertex by comparing: here, $$h = 1$$ and $$k = 2$$, so the vertex is at the point $$(1, 2). $$Determine the axis of symmetry, which is the vertical line passing through the vertex. The equation is $$x = h$$, so here it is $$x = 1. $$Find the y-intercept by evaluating $$f(0)$$: substitute $$x = 0$$ into the function to get $$f(0) = (0 - 1)^2$ + 2. $$This gives the point where the graph crosses the y-axis. Find the x-intercepts by setting $$f(x) = 0$$ and solving for $$x$$: solve $$(x - 1)^2$ + 2 = 0. $$Since the square term plus 2 cannot be zero for real $$x$$, check if there are real x-intercepts or not. Determine the domain and range: the domain of any quadratic function is all real numbers, $$(-\infty, \infty). $$The range depends on the vertex and the direction of the parabola (opens upward since the coefficient of the squared term is positive). The minimum value is $$k = 2$$, so the range is $$[2, \infty).$$

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. f(x)=(x−1)²+2

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Key Concepts

Vertex of a Quadratic Function

Axis of Symmetry

Domain and Range of Quadratic Functions