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Chapter 0: Problem 50

Sketch the graph of the first function by plotting points if necessary. Then use transformation(s) to obtain the graph of the second function. \(y=x^{2}, \quad y=\left|x^{2}-1\right|\)

Short Answer

Expert verified
To sketch the graph of \(y = x^2\), plot points like \((-2, 4)\), \((-1, 1)\), \((0, 0)\), \((1, 1)\), \((2, 4)\). This gives us a parabola with its vertex at the origin. Now, apply the transformation to obtain the graph of \(y = |x^2 - 1|\). Shift the parabola one unit downwards, giving it a new vertex at \((0, -1)\), and reflect the negative portions with respect to the x-axis to make it non-negative. The resulting graph is a vertically translated and reflected parabola with its vertex at \((0, -1)\).

Step by step solution

01

Sketch the graph of y=x^2.

The graph of the function \(y=x^2\) is a parabola with its vertex at the origin, opening upwards. As this is a basic quadratic function, we can plot a few points to get the shape of the graph. Points to be plotted: \((-2, 4)\), \((-1, 1)\), \((0, 0)\), \((1, 1)\), \((2, 4)\) By connecting these points, we obtain the graph of the quadratic function \(y=x^2\).
02

Determine the transformation for y=|x^2 - 1|.

First, we need to understand the transformation that converts \(y=x^2\) into \(y=|x^2 - 1|\). The transformation here involves the subtraction of 1 from \(x^2\), followed by taking the absolute value. Let's consider the direct transformation (without the absolute value): \(y=x^2 - 1\). This is a vertical translation of the original function \(y=x^2\) one unit downwards. Now let's analyze the transformation involving the absolute value: \(y=|x^2 - 1|\). This operation reflects the negative part of the graph of the transformed function \(y=x^2 - 1\) with respect to the x-axis, making it always non-negative.
03

Sketch the graph of y=|x^2 - 1|.

To sketch the graph of \(y = |x^2 - 1|\), we will follow the determined transformation. 1. From the graph of the original function, \(y=x^2\), shift the parabola one unit downwards to obtain the graph of \(y=x^2 - 1\). The new vertex will be at \((0, -1)\). 2. Reflect the negative part of the obtained graph with respect to the x-axis to make them non-negative, as the absolute value makes the function always positive or zero. As a result, we obtain the graph of the function \(y = |x^2 - 1|\), which is a vertically translated and reflected parabola in the negative region with its vertex at \((0, -1)\).

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Most popular questions from this chapter

Let \(f(x)=\left(1+\frac{1}{x}\right)^{x}\), where \(x>0\). a. Plot the graph of \(f\) using the window \([0,10] \times[0,3]\), and then using the window \([0,100] \times[0,3] .\) Does \(f(x)\) appear to approach a unique number as \(x\) gets larger and larger? b. Use the evaluation function of your graphing utility to fill in the accompanying table. Use the table of values to estimate, accurate to five decimal places, the number that \(f(x)\) seems to approach as \(x\) increases without bound. Note: We will see in Section \(2.8\) that this number, written \(e\), is given by \(2.71828 \ldots\)

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