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Chapter 1: Problem 48

Identify any intercepts and test for symmetry. Then sketch the graph of the equation. $$y=-x^{2}-2 x$$

Short Answer

Expert verified
The function has intercepts at (0, 0) and (-2, 0) and symmetry about the origin. The graph of the function is a downward-opening parabola.

Step by step solution

01

Finding the Y-intercept

To find the y-intercept, set \(x = 0\) in the equation. This gives us \(y = -0^{2} - 2*0 = 0\), so our y-intercept is at (0, 0)
02

Finding the X-intercepts

To find the x-intercepts, set \(y = 0\) in the equation and solve for \(x\). That gives the equation \(0 = -x^{2} - 2x\). By factoring, we get \(0 = -x*(x + 2)\). Therefore, \(x = 0\) or \(x = -2\). So the x-intercepts are at (0, 0) and (-2, 0)
03

Testing for symmetry

Replace \(x\) with \(-x\) in \(y = -x^{2} - 2x\), which gives \(y = -(-x)^{2} - 2*(-x)\). This simplifies to \(y = -x^{2} + 2x\), which isn't the same as our original equation, so the function lacks symmetry about the y-axis. Next, verify if the function has symmetry about the origin by replacing \(y\) with \(-y\) and \(x\) with \(-x\). This results in \(-y = -x^{2} - 2x\), which is the same as the original equation once it's multiplied through by -1. Thus, the function does have symmetry about the origin.
04

Graphing the function

First plot our intercepts: (0, 0) and (-2, 0). Since there is no symmetry about the y-axis, we can't mirror one side over to the other. Instead, utilize symmetry about the origin to find additional points. For instance, since we know (2, -8) is on the graph, it must also include (-2, 8) due to origin symmetry. Plot additional points as needed, then sketch the parabola which opens downwards.

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Most popular questions from this chapter

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