Sketch a complete graph of h(x)=(x^2-9)/(x^2-4) Clearly label all intercepts and asymptotes.

Name: Problem 41, Chapter 9: Precalculus
Uploaded: 2020-12-19T11:27:54-08:00
Duration: 1 min 28 s
Channel: Harmender Singh Yadav
Description: Sketch a complete graph of h(x)=(x^2-9)/(x^2-4) Clearly label all intercepts and asymptotes.

step 1 The given equation in this question is H of x equals to x squared minus 9 divided by x squared m

Sketch a complete graph of h(x)=(x^2-9)/(x^2-4) Clearly...

Key Concepts

Rational Functions

Rational functions are functions expressed as the ratio of two polynomials. Their graphs are analyzed by examining factors in the numerator and denominator, determining where the function is undefined, and investigating the end behavior, which may reveal asymptotes or other features. This careful study allows one to understand discontinuities and long?term behavior of the function.

Intercepts

Intercepts are the points where the graph of the function crosses the axes. The x-intercepts are found by setting the numerator equal to zero (provided these zeros do not also make the denominator zero), while the y-intercept is found by evaluating the function at x = 0, if 0 is within the domain. These intercepts provide key anchor points for sketching the graph.

Vertical Asymptotes

Vertical asymptotes occur at the values of x where the denominator is zero and the numerator is nonzero, indicating that the function grows without bound near these points. These asymptotes represent discontinuities in the graph where the function is undefined, and they guide the sketch by marking regions of unbounded behavior.

Horizontal Asymptotes

Horizontal asymptotes describe the end behavior of a rational function as x approaches positive or negative infinity. By comparing the degrees of the numerator and denominator, one can determine if a horizontal asymptote exists and compute its value (for example, if the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients). This helps in understanding how the function behaves far from the origin.

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