AP Precalculus Practice Test Unit 1 FRQ 3 Rational Function Zeros Holes Asymptotes Intercepts

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Please subscribe! / nickperich • My AP Precalculus Practice Tests are carefully designed to help students build confidence for in-class assessments, support their work on AP Classroom assignments, and thoroughly prepare them for the AP Precalculus exam in May. • For AP Precalculus Unit 1, FRQ #3, you’ll be analyzing a rational function by examining its **zeros**, **holes**, **asymptotes**, and **intercepts**. Here’s a breakdown of each feature and how to approach them: • 1. *Zeros of the Rational Function* • *Zeros* of a rational function are the \\( x \\)-values where the function equals zero, which occur where the *numerator* of the function is zero, provided the *denominator* is not also zero at those points. • To find the zeros, set the numerator equal to zero and solve for \\( x \\). These values represent the points where the graph crosses or touches the x-axis. • 2. *Holes in the Graph* • *Holes* occur in the graph of a rational function at any \\( x \\)-value that causes both the numerator and denominator to equal zero simultaneously. • To identify a hole, factor the numerator and denominator. If there’s a common factor, set it equal to zero to find the x-value of the hole. • Substitute this x-value back into the simplified function (after canceling out the common factor) to determine the y-coordinate of the hole, which is an undefined point on the graph. • 3. *Asymptotes of the Rational Function* • *Vertical Asymptotes* occur at \\( x \\)-values that make the denominator zero, provided they don’t cancel with a factor in the numerator (those that cancel create holes instead). • To find vertical asymptotes, set the simplified denominator equal to zero and solve for \\( x \\). The graph will approach these vertical lines but never cross them. • *Horizontal Asymptotes* describe the end behavior of the function as \\( x \\) becomes very large in the positive or negative direction. • To determine the horizontal asymptote, compare the degrees of the numerator and denominator: • If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is the x-axis. • If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients of the numerator and denominator. • If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote (though there might be an **oblique asymptote**). • 4. *Intercepts of the Rational Function* • *x-Intercepts* are found by setting the numerator of the function equal to zero (as long as the denominator is not zero at those points), which gives the points where the graph crosses or touches the x-axis. • *y-Intercepts* are found by substituting \\( x = 0 \\) into the function (if defined at \\( x = 0 \\)), which gives the point where the graph crosses or touches the y-axis. • Putting It All Together • For a Free Response Question (FRQ) on this topic, your response would include: • 1. *Identifying the zeros* by setting the numerator equal to zero and checking that the denominator isn’t zero at those points. • 2. *Finding the location of any holes* by identifying and canceling common factors in the numerator and denominator and evaluating the function at those points. • 3. *Describing vertical and horizontal asymptotes* by analyzing where the denominator equals zero for vertical asymptotes and comparing degrees of the numerator and denominator for horizontal asymptotes. • 4. *Calculating intercepts* by setting \\( x \\) and \\( y \\) to zero in the function to determine the points where the graph crosses the axes. • This approach allows you to fully analyze the rational function’s key features, providing a clear picture of its graph and behavior. • I have many informative videos for Pre-Algebra, Algebra 1, Algebra 2, Geometry, Pre-Calculus, and Calculus. Please check it out: • / nickperich • Nick Perich • Norristown Area High School • Norristown Area School District • Norristown, Pa • #math #algebra #algebra2 #maths #math #shorts #funny #help #onlineclasses #onlinelearning #online #study

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