1.
3 Rates of Change in Linear and Quadratic Functions
AP Precalculus
Name: _____________________________ CA #1
What is the average rate of change for each function on the given intervals?
1. 𝑦 3𝑥 6 on 3 𝑥 2 2. 𝑦 8 2𝑥 on 5 𝑥 7
3. 𝑦 𝑥 3𝑥 5 on 2 𝑥 4 4. 𝑦 2𝑥 6𝑥 4 on 3 𝑥 2
What is the rate of change of the average rates of change for each function over consecutive equal-length
intervals?
5. 𝑦 21𝑥 1 6. 𝑦 8 𝑥
7. 𝑓 𝑥 𝑥 3𝑥 7. 8. 𝑓 𝑥 3𝑥 𝑥 1.
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The values of a function are given at selected 𝒙-values in the table below. The function’s concavity does
not change. Determine if the function is concave up or concave down. Justify your answer.
9.
𝒙 4 7 10 13 16
𝒈 𝒙 35 20 13 10 9
10.
𝒙 5 3 1 1 3
𝒉 𝒙 100 95 80 50 10
Answers to 1.3 CA #1
1. 3 2. 2 3. 5 4. 16 5. 0 6. 0 7. 2
9. Concave up because the rate of change is 10. Concave down because the rate of change
8. 6 increasing over equal-length input-value is decreasing over equal-length input-
intervals. value intervals.
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1.4 Polynomial Functions and Rates of Change
AP Precalculus Name: _____________________________ CA #1
Find the leading coefficient and the degree of each polynomial.
1. 𝑓 𝑥 8𝑥 6𝑥 5 2. 𝑓 𝑥 10𝑥 7
L.C. _____ Degree: _____ L.C. _____ Degree: _____
3. 𝑓 𝑥 8𝑥 3𝑥 2𝑥 4. 𝑓 𝑥 2𝑥 8𝑥
L.C. _____ Degree: _____ L.C. _____ Degree: _____
Let 𝒇 𝒙 be a polynomial function with the given values. Are there any guaranteed extrema? If
so, state where they occur.
5. 𝑓 0 2, 𝑓 1 0, and 𝑓 11 0. 6. 𝑓 8 0, 𝑓 0 1, and 𝑓 5 3.
Is there a global maximum or minimum for each function?
7. 𝑓 𝑥 2𝑥 4𝑥 𝑥 8. 𝑓 𝑥 3𝑥 2 9. 𝑓 𝑥 7𝑥 2𝑥 1
Find the following extrema. If there are none, cross it off and write NONE.
10. Absolute min of _____ when 𝑥
11. Absolute max of _____ when 𝑥
12. Relative min(s) at 𝑥
13. Relative max(es) at 𝑥
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14. Absolute min of _____ when 𝑥
15. Absolute max of _____ when 𝑥
16. Relative min(s) at 𝑥
17. Relative max(es) at 𝑥
Answers to 1.4 CA #1
1a. 8 2a. 10 3a. 3 4a. 8
5. Yes, on 1 𝑥 11.
1b. 2 2b. 3 3b. 5 4b. 4
6. No guarantee. 7. none 8. min 9. max 10. min of 4 when 𝑥 3
11. none 12. 2 and 3 13. 1 14. none 15. max of 2 when 𝑥 1
16. 1 17. 4 and 1
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1.5A Polynomial Functions and Complex Zeros
AP Precalculus Name: _____________________________ CA #1
For each polynomial function, find the intervals for each condition.
1. ℎ 𝑥 𝑥 8𝑥 7. When is ℎ 𝑥 0? 2. 𝑝 𝑥 𝑥 𝑥 2 𝑥 4 𝑥 8 . When is
𝑝 𝑥 0?
3. 𝑓 𝑥 𝑥 𝑥 6 𝑥 1 𝑥 3 . When is 4. 𝑔 𝑥 𝑥 𝑥 12𝑥. When is 𝑔 𝑥 0?
𝑓 𝑥 0?
For each polynomial, the degree is listed along with all of its real zeros. Find the number of NON-REAL
zeros the polynomial has.
5. The degree is 4 with real zeros 6. The degree is 13 with real zeros 7. The degree is 5 with real zeros at
at 𝑥 4 and 𝑥 4. at 𝑥 9, 𝑥 0, and 𝑥 13. 𝑥 3 and 𝑥 8. 𝑥 3 has a
𝑥 0 has a multiplicity of 5. multiplicity of 4.
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Given one non-real zero of a polynomial, find another zero.
8. 9 2𝑖 9. 3 7𝑖
Find the degree of the polynomial from the given input and output values.
10.
Input 0 1 2 3 4 5 6 7
Output 7 3 5 31 99 227 433 735
11.
Input 0 1 2 3 4 5 6 7
Output 8 17 58 113 116 47 538 1567
Answers to 1.5A CA #1
1. ∞, 1 ∪ 7, ∞ 2. ∞, 8 ∪ 0, 4 3. ∞, 6 ∪ 3, ∞ 4. 3, 0 ∪ 4, ∞
5. 2 6. 6 7. 0 8. 9 2𝑖 9. 3 7𝑖 10. 3 11. 4
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1.5B Even and Odd Polynomials
AP Precalculus Name: _____________________________ CA #1
State whether the following graphs represent functions that are even, odd, or neither.
1. 2. 3.
y y y
x x
State if the following functions are even, odd, or neither.
4. 𝑓 𝑥 𝑥 3𝑥 5. 𝑓 𝑥 3𝑥 5𝑥 2𝑥 6. 𝑓 𝑥 2𝑥 𝑥 𝑥
7. 𝑦 𝑥 3𝑥 5 8. 𝑓 𝑥 𝑥 5
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1.6 Polynomial Functions and End Behavior
AP Precalculus Name: _____________________________ CA #1
Describe the end behavior of each function using limit notation.
1. 𝑔 𝑥 5𝑥 6𝑥 2. 𝑝 𝑥 7𝑥 1
3. 𝑓 𝑥 3𝑥 𝑥 2𝑥 4. 𝑓 𝑥 6𝑥 3𝑥 7
5. y 6. y
x
x
Sketch the graph of a polynomial function that could match each statement.
7. lim 𝑝 𝑥 ∞ and lim 𝑝 𝑥 ∞ 8. lim 𝑝 𝑥 ∞ and lim 𝑝 𝑥 ∞
→ → → →
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1.7A Rational Functions and End Behavior
AP Precalculus Name: _____________________________ CA #1
State the domain of the following rational functions. Use interval notation.
1. 𝑓 𝑥 2. ℎ 𝑥 3. 𝑑 𝑡
Domain: Domain: Domain:
Use the graph of the rational function 𝒇 to find the following.
4. Domain:
End Behavior:
Is there horizontal aysmptote?
If so, write the equation of the horizontal aysmptote.
5. Domain:
End Behavior:
Is there horizontal aysmptote?
If so, write the equation of the horizontal aysmptote.
Continued on the back.
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CALCULATOR ACTIVE Complete the table to answer the following.
6. 𝑓 𝑥
𝒙 10,000 1,000 100 100 1,000 10,000
𝒇 𝒙
End Behavior: Is there horizontal aysmptote?
If so, write the equation of the horizontal aysmptote.
7. 𝑑 𝑡
𝒕 5,000 500 50 50 500 5,000
𝒅 𝒕
End Behavior: Is there horizontal aysmptote?
If so, write the equation of the horizontal aysmptote.
Answers to 1.7A CA #1
1. ∞, 4 ∪ 4, 4 ∪ 4, ∞ 6. 7.
2. ∞, 7 ∪ 7, 2 ∪ 2, ∞ 𝑥 𝑓 𝑥 𝑡 𝑑 𝑡
3. ∞, ∪ ,∞
-10,000 0.9996 -5,000 1.2 10
-1,000 0.99602 -500 125124
4. Domain: ∞, 3 ∪ 3, ∞ -100 0.9615 -50 1261.1
End Behavior: 100 1.0417 50 1236.1
lim 𝑓 𝑥 ∞ and lim 𝑓 𝑥 ∞ 1,000 1.004 500 124874
𝑥→ ∞ 𝑥→∞
10,000 1.0004 5,000 1.2 10
Horizontal Asymptote: none
lim 𝑓 𝑥 1 lim 𝑓 𝑥 ∞
5. Domain: ∞, 0 ∪ 0, 2 ∪ 2, ∞ 𝑥→ ∞ 𝑡→ ∞
End Behavior: lim 𝑓 𝑥 1 lim 𝑓 𝑥 ∞
𝑥→∞ 𝑡→∞
lim 𝑓 𝑥 2 and lim 𝑓 𝑥 2 HA: 𝑦 1 HA: none
𝑥→ ∞ 𝑥→∞
Horizontal Asymptote: 𝑦 2
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