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Chapter 3: Problem 29

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. $$f(x)=x^{2}+3 x-10$$

Short Answer

Expert verified
The x-intercepts of the function are at \(x = 2\) and \(x = -5\). The vertex is located at point \((-1.5, -12.75)\) and the axis of symmetry is the line \(x = -1.5\). The domain of the function is all real numbers or \(x \in (-\infty, \infty)\) and the range of the function is \([-12.25, \infty)\).

Step by step solution

01

Determine the x-intercepts

The x-intercepts of the graph can be found by solving the equation \(f(x) = 0\), which gives us \(x^{2} + 3x - 10 = 0\). Factor this equation to find that \(x = 2\) and \(x = -5\) are the x-intercepts.
02

Determine the vertex and axis of symmetry

The vertex of a quadratic function in the form \(f(x) = ax^{2} + bx + c\) is at \((-b / 2a, f(-b / 2a))\). Using this formula with \(a = 1\) and \(b = 3\) gives the x-coordinate of the vertex as \(-3 / 2 = -1.5\). Plugging \(-1.5\) into the function gives the y-coordinate as \(-1.5^{2} + 3(-1.5) - 10 = -12.25\). Therefore, the vertex is \((-1.5, -12.25)\). The axis of symmetry of a parabola is the vertical line that passes through the vertex, so the axis of symmetry for this graph is \(x = -1.5\).
03

Determine the domain and range

The domain of a quadratic function is all real numbers, so the domain of \(f(x) = x^{2} + 3x - 10\) is \(x \in (-\infty, \infty)\). The range of a parabolic function opens upward (since \(a > 0\) in the quadratic function), so the range of this function is \([f(-1.5), \infty)\) or \([-12.25, \infty)\).

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