AP Precalc 1.8-1.9 Zeros & AsymptotesAP Precalc 1.8-1.9 Zeros & Asymptotes

A P P R E C A L C U L U S · UNIT1

Zeros and Vertical Asymptotes

Topics 1.8 and 1.9 Study Guide

In one sentence: zeros come from the numerator, vertical asymptotes come from the denominator, and a factor that cancels leaves a hole instead of either one.

Why it works, in plain language. A fraction equals 0 only when its top is 0 (and the bottom is not). So zeros come from the numerator. A fraction is undefined when its bottom is 0. If the top is not 0 there, dividing a nonzero number by something tiny gives a huge number, so the graph shoots off to ±∞ and you get a vertical asymptote. If a factor is 0 on both top and bottom, the huge and tiny cancel and the graph just has one missing point, a hole. The sign of a fraction can only flip at a zero or at a spot where it is undefined, which is why sign tables only need those few critical numbers. Step by step: classify every special x-value.

(1) Factor top and bottom completely.

(2) Circle factors that appear in both: those x-values are holes.

(3) Remaining factors of the top: zeros.

(4) Remaining factors of the bottom: vertical asymptotes.

(5) For limits near a vertical asymptote, check the multiplicity: odd means the sides go opposite ways, even means both sides match.

Step by step: solve f(x) > 0 or < 0. (1) List zeros and undefined x-values. (2) Draw a number line and mark them. (3) Pick one test value in each region and find the sign. (4) Choose the regions you need. Include zeros only if the inequality has "or equal to", and never include undefined values. 1.8 Zeros of a rational function

1. Factor numerator and denominator.

2. Set the numerator = 0 and solve.

3. Throw out any solution that also makes the denominator 0. That x-value is a hole, not a zero. Examples. f(x) = (x − 1)/(x² − 9): zero x = 1.

d(t) = (t + 3)(t − 1)/(4t + 12) = (t + 3)(t − 1)/(4(t + 3)): zero t = 1. (t = −3 is a hole, not a zero.) h(x) = (x² − 3x − 10)/(x² + 6x) = (x − 5)(x + 2)/(x(x + 6)): zeros x = 5 and x = −2. Calculator active: graph it and read the x-intercepts (for example x ≈ −5.372 and 0.372 for (x² + 5x − 2)/(x² − 9)).

AP Precalculus 1 of 4

Sign tables

The sign of f can only change at a zero or where f is undefined (asymptote or hole). Those are your critical numbers.

1. List the critical numbers in order and mark zeros "0" and undefined points "DNE". 2. Test one x in each interval (or read the graph). Write positive or negative.

3. For "f(x) ≥ 0" include zeros (closed bracket) but never DNE points (always parenthesis). Example. r(x) = (x² − 4)/(x + 3) ≥ 0. Critical numbers −3 (DNE), −2 (zero), 2 (zero). Signs on (−∞, −3), (−3,−2), (−2,2), (2,∞): negative, positive, negative, positive. Answer (−3, −2] ∪ [2, ∞). r(x) = (x² − 4x)/(x² + 8x + 16) < 0: zeros 0 and 4, DNE at −4. Negative only on (0, 4) because (x + 4)² is never negative.

Full analysis. f(x) = (x² − 2x − 24)/(x − 6) = (x − 6)(x + 4)/(x − 6). Domain (−∞, 6) ∪ (6, ∞). Zero x = −4 (x = 6 is a hole). HA: none. y-intercept (0, 4). f > 0 on (−4, 6) ∪ (6, ∞).

g(x) = 4(x + 5)(x − 2)/(x² − 4): zero x = −5 (x = 2 is a hole), HA y = 4, y-intercept (0, 10), g < 0 on (−5, −2).

Test-prep. f(x) = (x² + 2x − 24)/(4 − x) = (x + 6)(x − 4)/(−(x − 4)): x-intercept at x = −6 and a hole at x = 4. If a sign table has zeros at −3 and 2 and f is undefined at 0, then f has exactly two distinct real zeros, and the graph has a hole or vertical asymptote at x = 0.

1.9 Vertical asymptotes

A vertical asymptote sits at x = a when the denominator is 0 at a and the factor does not cancel. Describe it with one-sided limits. Near a vertical asymptote the outputs go to +∞ or −∞: limx→a⁻ f(x) = ±∞ and limx→a⁺ f(x) = ±∞

Odd multiplicity of the denominator factor: the two sides go in opposite directions (−∞ on one side, +∞ on the other).

Even multiplicity: both sides go the same direction (both +∞ or both −∞).

Examples. f(x) = x(x + 2)/(x² − 4) = x(x + 2)/((x + 2)(x − 2)): VA x = 2 only. Domain (−∞, −2) ∪ (−2, 2) ∪ (2, ∞).

d(t) = (t² + 4t − 12)/(t − 2)² = (t + 6)(t − 2)/(t − 2)²: VA t = 2 (the factor (t − 2) cancels once, leaving one in the denominator).

h(x) = (x³ − 3x²)/(x² + 8x + 15) = x²(x − 3)/((x + 3)(x + 5)): VA x = −3 and x = −5. AP Precalculus 2 of 4

Tables near an asymptote

f(x) = (x² − 1)/(x − 4). x = 3.9, 3.99, 3.999 give −142.1, −1492, −14992. x = 4.001, 4.01, 4.1 give 15008, 1508, 158.1. So VA x = 4, limx→4⁻ = −∞ and limx→4⁺ = ∞.

f(x) = (x² − 2x)/(x + 2): near x = −2 the outputs are −86.1, −806, −8006 on the left and 7994, 794, 74.1 on the right: VA x = −2, left −∞, right ∞.

If a table shows d(t) huge and positive on both sides of t = 0 (5,589 and 128,341 for example), then limt→0 d(t) = ∞, d(0) is undefined, and there is no y-intercept.

Sketching from limits

Draw a dashed vertical line for each asymptote and a dashed horizontal line for each HA. Plot the given points. For "limx→−3⁻ = −∞ and limx→−3⁺ = ∞" the curve plunges down on the left of −3 and shoots up on the right.

Hole or VA? Free-response pattern: the numerator has one zero at x = 3. The denominator has zeros at both 3 and 6, with the same multiplicity at 3.

Domain: (−∞, 3) ∪ (3, 6) ∪ (6, ∞). x = 3 is a hole because (x − 3) is a common factor. x = 6 is a vertical asymptote because only the denominator is 0 there.

If the numerator's multiplicity at 3 were greater, there would still be a hole. If the denominator's multiplicity were greater, x = 3 would be a vertical asymptote.

Traps. (1) Cancel first, then find the asymptotes, but remember the cancelled x-value is still excluded from the domain. (2) A zero of the numerator never makes a VA. (3) "f has an x-intercept at −6 and a hole at 4" is different from "VA at 4". (4) Always give both one-sided limits, they may differ. Quick check

1. Zeros of (x − 2)(x + 3)/(x − 2)? x = −3 only (x = 2 is a hole)

2. VA of (x + 1)/((x − 5)(x + 1))? x = 5

3. (x − 4)²/(x + 1)² near x = −1: both sides go to? +∞

4. Can f(x) ≥ 0 ever include a point where f is undefined? No

AP Precalculus 3 of 4

Practice set

Try every question before you look at the key. Show your work.