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Chapter 3: Problem 30

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. $$f(x)=2 x^{2}-7 x-4$$

Short Answer

Expert verified
The vertex is at (7/4, -33/8). The intercepts are at (4, 0), (-1/2, 0), and (0, -4). The axis of symmetry is \(x = 7/4\). The domain of the function is all real numbers, and the range is [-33/8, \infty).

Step by step solution

01

- Find the vertex.

The vertex of a parabola given in standard form, \(ax^{2} + bx + c\), is determined by the formula \((-b/2a, f(-b/2a))\). For this function, \(a = 2\) and \(b = -7\), so the vertex is at \(((7/2*2), f(7/2*2)) = (7/4, f(7/4))\). Substituting \(x = 7/4\) into \(f(x) = 2x^{2}-7x-4\) we get \(f(7/4) = -33/8\). So, the vertex is \( (7/4, -33/8) \).
02

- Determine the intercepts.

The x-intercepts are the solutions of the equation \(f(x) = 2x^{2}-7x-4 = 0\). Solving this using the quadratic formula, we find that \(x = 4, -1/2\). The y-intercept is found by setting \(x = 0\) in the equation, which gives \(f(0) = -4\). So the intercepts are at \((4, 0)\), \((-1/2, 0)\), and \((0, -4)\).
03

- Plot the Graph.

Plot the vertex at \((7/4, -33/8)\) and the intercepts at \((4, 0)\), \((-1/2, 0)\), and \((0, -4)\). The graph will be a parabola that opens upwards because the coefficient of \(x^{2}\) is positive.
04

- Find the Equation of the Axis of Symmetry.

The axis of symmetry of a parabola given in the standard form, \(ax^{2} + bx + c\), is given by \(x = -b/2a\). Therefore in this case it will be \(x = -(-7)/(2*2) = 7/4\).
05

- Determine the Domain and Range.

The domain of a quadratic function is all real numbers, because a quadratic function is defined for all real values of x. Considering that the parabola opens upwards and the y-coordinate of the vertex is \(-33/8\), the range of the function is \([-33/8, \infty)\). In interval notation, this is \(\[-33/8, \infty)\), indicating that it includes \(-33/8\) and all greater numbers.

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