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).]
<Image>
$y = x^{-2} - x^{-1} - 6$

검증된 단계별 안내

To find the x-intercepts of the graph, set the equation equal to zero: $y = x^{-2} - x^{-1} - 6 = 0$.

Rewrite the negative exponents as fractions to make the equation easier to work with: $\frac{1}{x^2} - \frac{1}{x} - 6 = 0$.

Multiply both sides of the equation by $x^2$ (assuming $x \neq 0$) to eliminate the denominators: $1 - x - 6x^2 = 0$.

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.

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.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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).] <Image>y=x−2−x−1−6y = x^{-2} - x^{-1} - 6 y=x−2−x−1−6