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Chapter 3: Problem 28

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. $$f(x)=x^{2}-2 x-15$$

Short Answer

Expert verified
The x-intercepts of the graph are \(x=5\) and \(x=-3\). The vertex is \((1, -16)\). The equation of the axis of symmetry is \(x=1\). The domain is all real numbers and the range is \([-16, +\infty)\).

Step by step solution

01

Factoring the quadratic function

First, we need to factor the quadratic function. The factored form of the quadratic function \(f(x)=x^{2}-2x-15\) is \(f(x)=(x-5)(x+3)\). The roots of the equation, or the x-intercepts of the graph, are found by setting each factor equal to zero and solving for x. This yields \(x=5\) and \(x=-3\).
02

Finding the Vertex

The vertex of a parabola is located at the point \((h,k)\), where \(h\) and \(k\) are respectively given by the formulae \(h=-b/2a\) and \(k=f(h)\). In this case, we have \(a = 1\), \(b = -2\), and we get \(h=1\). Substituting \(h=1\) into the function, we find \(k = -16\). Thus, the vertex of the parabola is at \( (1, -16) \).
03

Axis of Symmetry

The axis of symmetry is a vertical line passing through the vertex. In the standard form of the quadratic function, the axis of symmetry is given by the equation \(x = h\). Therefore, for our function, the equation of the axis of symmetry is \(x=1\).
04

Determining the Domain and range

The domain of a function is the set of all possible input values (x-values). For all quadratic functions, the domain is all real numbers, characterized by \((-\infty, + \infty)\). The range (y-values) is determined by the direction the parabola opens and its vertex. Since this parabola opens upwards and has a minimum vertex at \(y= -16\), the range is \([-16, +\infty)\).

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