In Exercises 85–90, find the x-intercepts of the graph of each equation. Then use the x-intercepts to match the equation with its graph. [The graphs are labeled (a) through (f).]
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$y = x^{-2} - x^{-1} - 6$

Verified step by step guidance

1. To find the x-intercepts of the graph, set the equation equal to zero: \(y = x^{-2} - x^{-1} - 6 = 0\).
2. Rewrite the negative exponents as fractions to make the equation easier to work with: \(\frac{1}{x^2} - \frac{1}{x} - 6 = 0\).
3. Multiply both sides of the equation by \(x^2\) (assuming \(x \neq 0\)) to eliminate the denominators: \$1 - x - 6x^2 = 0$.
4. Rearrange the equation into standard quadratic form: \(-6x^2 - x + 1 = 0\). You can multiply both sides by \(-1\) to get \$6x^2 + x - 1 = 0$ for easier solving.
5. Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) with \(a=6\), \(b=1\), and \(c=-1\) to find the values of \(x\) where the graph crosses the x-axis.

Key Concepts

X-Intercepts of a Graph

X-intercepts are the points where the graph crosses the x-axis, meaning the y-value is zero. To find them, set y = 0 and solve the resulting equation for x. These points help identify key features of the graph and are essential for matching equations to their graphs.

Solving Equations with Negative Exponents

Negative exponents indicate reciprocals, such as x^(-1) = 1/x. When solving equations with negative exponents, rewrite terms as fractions to simplify and solve. Understanding this helps in manipulating the equation to find x-intercepts accurately.

Graph Behavior of Rational Functions

Functions with negative exponents often represent rational functions with variables in denominators. Their graphs can have vertical asymptotes and discontinuities where the denominator is zero. Recognizing these behaviors aids in matching equations to their correct graphs.