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Chapter 3: Problem 22

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. $$y-3=(x-1)^{2}$$

Short Answer

Expert verified
Vertex: (1,3), Axis of symmetry: x=1, x-intercepts: ((1-鈭�3,0) and (1+鈭�3,0)), y-intercept: (0,4), Domain: R, Range: [3, 鈭�)

Step by step solution

01

Identify the Vertex

The vertex is found in the form of \((h,k)\) where h and k are the coefficients in the equation. Therefore the vertex of the given equation \(y-3=(x-1)^{2}\) is \((1,3)\).
02

Determine Axis of Symmetry

The line of symmetry is given by the equation \(x=h\). Thus, for our equation, the line of symmetry is \(x=1\).
03

Determine the X and Y intercepts

Intercepts are found by setting y and x to zero respectively and solving the equation. For x-intercepts, we set y to 0 in the equation: \n0 = (x-1)^{2} + 3. Thus, x-intercepts: \((1-\sqrt{3}, 0)\) and \((1+\sqrt{3}, 0)\) \nFor y-intercept, substitute x = 0 in the equation: \ny-3 = (-1)^2 = 1. Thus, y-intercept: \(0, 4\)
04

Graph the Function

Now that we have all the necessary information, plot the vertex at (1,3), the x-intercepts at (1-鈭�3,0) and (1+鈭�3,0), and the y-intercept at (0,4). The axis of symmetry is x=1. Sketch the parabola
05

Determine the Domain and Range

The domain of a quadratic equation is all real numbers, as the x-values can be any real number. \nThe range can be found from the vertex, which is the highest or lowest point on the graph. Looking at the graph, the parabola opens upwards, so the vertex is the minimum point. Hence, the range of the function is [3, 鈭�).

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