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Chapter 1: Problem 88

Use a graphing utility. Graph: \(f(x)=x^{2}-|2 x-3|\)

Short Answer

Expert verified
The function \(f(x) = x^2 - |2x - 3|\) results in a distorted 'W' shape.

Step by step solution

01

Understanding the Quadratic Component

In order to plot the graph, first, the quadratic component \(x^2\) should be examined. This is a upward opening parabola with vertex at the origin (0, 0).
02

Understanding the Absolute Value Component

Next, plot the absolute value component |2x - 3|. This is a v-shaped graph with vertex at \((1.5, 0)\). The expression inside the absolute value function is a linear function which crosses the x-axis at 1.5. Hence its absolute function dips down to 0 at \(x = 1.5\) and raises back upwards in both directions.
03

Graphing the Final Function

Subtract the graph of the absolute value function from the quadratic function. Wherever the absolute value graph is above the quadratic graph, the resulting graph will dip below the X-axis. This creates a downward-parabolic function around the vertex of the absolute value graph (\(x = 1.5\)) and maintains an upward-parabolic shape elsewhere. The final graph hence looks like a distorted 'W' shape.

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Most popular questions from this chapter

Sketch the graph of the piecewisedefined function. $$v(x)=\left\\{\begin{array}{ll} 2 x-2 & \text { if } x \neq 3 \\ 1 & \text { if } x=3 \end{array}\right.$$

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