Use long division to rewrite the equation for g in the form quotient + remainder/divisor. Then use this form of the function's equation and transformations of f(x) = 1/x to graph g. g(x) = (2x+7)/(x+3)

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Identify the dividend and divisor for the long division. Here, the dividend is the numerator polynomial $2x + 7$ and the divisor is the denominator polynomial $x + 3$.

Set up the long division by dividing the leading term of the dividend (\$2x$) by the leading term of the divisor ($x$). This gives the first term of the quotient: $\frac{2x}{x} = 2$.

Multiply the entire divisor $x + 3$ by the quotient term $2$, resulting in $2(x + 3) = 2x + 6$.

Subtract this product from the original dividend: $(2x + 7) - (2x + 6)$. Simplify the subtraction to find the remainder.

Express the original function $g(x)$ as the quotient plus the remainder over the divisor: $g(x) = 2 + \frac{\text{remainder}}{x + 3}$. This form helps to analyze the graph by relating it to the parent function $f(x) = \frac{1}{x}$ with transformations.

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주요 개념

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

Polynomial Long Division

Polynomial long division is a method used to divide one polynomial by another, similar to numerical long division. It helps rewrite a rational function as a quotient plus a remainder over the divisor, simplifying the expression for analysis or graphing.

Rational Functions and Their Graphs

A rational function is a ratio of two polynomials. Understanding its graph involves identifying asymptotes, intercepts, and behavior at infinity, which can be made clearer by rewriting the function using long division.

Transformations of the Parent Function f(x) = 1/x

The function f(x) = 1/x is a basic rational function with a hyperbola shape. Graphing related functions involves applying transformations like shifts, stretches, and reflections to this parent graph based on the rewritten form of the function.