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Chapter 1: Problem 47

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

Short Answer

Expert verified
The x-intercepts are at \(x = 0\) and \(x = 2\), the y-intercept is at \(y = 0\). The graph is not symmetric with respect to the y-axis. It is a upward-opening parabola crossing through points (0, 0) and (2, 0).

Step by step solution

01

Identify the intercepts

To find the x-intercepts, set \(y = 0\) and solve for \(x\). With \(y = 0\), the equation becomes \(0 = x^{2} - 2x\), or \(x (x-2) = 0\), hence \(x = 0\) or \(x = 2\) are the x-intercepts. To find the y-intercept, set \(x=0\) in the equation, the equation becomes \(y = (0)^{2} - 2*(0)\), hence \(y=0\) is the y-intercept.
02

Test for symmetry

A function is symmetric with respect to the y-axis if \(f(-x) = f(x)\). We replace \(x\) with \(-x\) in the equation and simplify: if we end up with the original equation, it's symmetric. Thus \(y=(-x)^{2}-2(-x) = x^2 + 2x\). This is not the original equation, hence it's not symmetric with respect to the y-axis.
03

Sketch the graph

Plot the x-intercepts and y-intercept obtained from Step 1 on the graph. As it's a quadratic equation, the graph will be a parabola. Given it opens upward and is not symmetric to y-axis, it has a minimum point. In this case the graph is a parabola that opens upwards, crossing through points (0, 0) and (2, 0).

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