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Chapter 1: Problem 60

Use a graphing utility to graph the equation. Use a standard setting. Approximate any intercepts. $$y=x^{2}+x-2$$

Short Answer

Expert verified
The graph of the equation \(y=x^{2}+x-2\) is a parabola that intersects the x-axis at points \((-2,0)\) and \((1,0)\), and intersects the y-axis at point \((0,-2)\).

Step by step solution

01

Understanding the Quadratic Equation

The given equation is a quadratic equation of the form \(y=ax^{2}+bx+c\), where \(a=1\), \(b=1\), and \(c=-2\). The graph of this equation will be a parabola.
02

Graphing the equation

Using a graphing utility, plot the equation \(y=x^{2}+x-2\). Make sure you set the calculator window to standard settings. Plot the points from \(x=-6\) to \(x=6\) and look for where the graph intersects the x-axis (these are the x-intercepts) and where it intersects the y-axis (these are the y-intercepts).
03

Finding the Intercepts

The x-intercepts are points where the graph crosses the x-axis. This happens when \(y=0\). To find these points, set \(y\) to zero and solve for \(x\). So, \(x^{2}+x-2=0\). By factoring, this breaks down into \((x+2)(x-1)=0\). For the equation to equal zero, either \(x+2=0\) or \(x-1=0\). Solving for \(x\) gives \(x=-2\) and \(x=1\). So, the x-intercepts are \((-2,0)\) and \((1,0)\).The y-intercept is the point where the graph crosses the y-axis. This happens when \(x=0\). To find this point, set \(x\) to zero and solve for \(y\). So, \(y=(0)^{2}+0-2\), which simplifies to \(y=-2\). So, the y-intercept is \((0,-2)\).
04

Summary

After graphing the equation \(y=x^{2}+x-2\) and finding the intercepts, we see that the graph intersects the x-axis at points \((-2,0)\) and \((1,0)\), and intersects the y-axis at point \((0,-2)\). This gives us a good idea of what the graph will look like and where it is positioned relative to the axes.

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