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Chapter 1: Problem 18

Use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. $$f(x)=3 x^{2}-1.75$$

Short Answer

Expert verified
The graph of the function \(f(x)=3x^{2}-1.75\) is a parabola that opens upwards with a vertex at the point (0, -1.75). It would be best displayed in a viewing window where \(-10 \leq x \leq 10\) and \(-10 \leq y \leq 30\).

Step by step solution

01

Understanding the Function

The given function, \(f(x)=3x^{2}-1.75\), is a quadratic function. It should be noted that the graph of a quadratic function is a parabola. Given the coefficients in the function, the parabola will open upwards because the coefficient of \(x^2\) is positive.
02

Determining the Vertex

The vertex of a parabola given by the function \(f(x)=ax^2+bx+c\) is at the point \((-\frac{b}{2a}, f(-\frac{b}{2a})\). For this function, however, there is no \(x\) term, meaning the vertex will lie on the y-axis. Hence, the vertex is \((0, f(0)) = (0, -1.75)\).
03

Choosing the Viewing Window

The vertex and direction of opening of the parabola give clues about how to choose the viewing window for the graph. In this case, the vertex is at (0,-1.75) and the parabola opens upwards, it would be ideal to choose a viewing window so the vertex is roughly in the center. A good choice of viewing window might be \(-10 \leq x \leq 10\) and \(-10 \leq y \leq 30\).
04

Graphing the Function

Using the chosen viewing window, plot the function \(f(x) = 3x^2 - 1.75\) using the graphing utility. The resulting plot would be a parabola, opening upwards, with the vertex at the point (0, -1.75).

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