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CHARACTERISTICS OF GRAPH WORKSHEET
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i) Domain ii) Range iii) Maximum iv) Minimum |
v) Discrete or Continuous? vi) y – intercept: vii) x – intercept: viii) 7𝑓(5)= |
Problem 2 :
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i) Domain: ii) Range: iii) Maximum: iv) Minimum: |
v) Interval of Increase: vi) Interval of Decrease: vii) 𝑓(2) + 𝑓(9) = |
Problem 3 :
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i) Domain ii) Increasing iii) Range iv) Decreasing v) x-intercept (s) |
vi) Positive vii) y-intercept viii) Negative ix) Maximum x) Minimum xi) End behaviour |
Problem 4 :
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i) Domain ii) Range iii) x-intercept(s) iv) y-intercept v) Maximum |
vi) Increasing vii) Decreasing viii) Positive ix) Negative x) Minimum xi) End behaviors |
Problem 5 :
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i) Domain ii) Increasing iii) Range iv) Decreasing v) x-intercept(s) |
vi) Positive vii) y-intercept viii) Maximum ix) Minimum x) End behavior |
Problem 6 :
For each polynomial function, identify the following characteristics.
a) The type of function whether it is even or odd degree.
b) The end behavior of the graph of the function
c) the number of possible x-intercepts.
d) where the graph has a maximum or minimum value
e) the y-intercept.
Then, match each function to its corresponding graph.
i) f(x) = 2x3 - 4x2 + x + 2
ii) f(x) = -x4 + 10x2 + 5x - 6
iii) f(x) = -2x5 + 5x3 - x + 1
iv) f(x) = x4 - 5x3 + 16
Answer Key
1) 1)
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i) Domain: (-5, 5] ii) Range : [-4, 4] iii) Maximum: y = 4 iv) Minimum: y = -4 |
v) Discrete or Continuous? Continuous vi) y – intercept: y = 0 vii) x – intercept: x = 0 and x = 5 viii) 7 𝑓(5) = 0 |
2)
i) Domain : [0, 12]
ii) Range: [0, 8]
iii) Maximum: y = 8
iv) Minimum: y = 0
v) Interval of Increase: (0, 3)
vi) Interval of Decrease: (9, 12)
vii) 𝑓(2) = 4 and 𝑓(9) = 8
f(2) + f(9) = 4 + 8 ==> 12
3)
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Domain Increasing Range Decreasing x-intercept(s) Positive y-intercept Negative Maximum Minimum |
[-3, 14] (-3, 9) ꓴ (11, 14) [-5, 10] (9,14) (4.5,0) (4.5,14] (0,-3) [-3,4.5) (9,10) (-3,-5) |
4)
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Domain Increasing Range Decreasing x-intercept(s) Positive y-intercept Negative Maximum Minimum End Behavior |
(-∞, ∞) (-2,-0.5) ꓴ (1, ∞) (0, ∞) (-∞,-2) ꓴ (-0.5,1) (-2,0) & (1,0) (-∞,-2) ꓴ (-2,1) ꓴ(1, ∞) (0,4) none (-0.5,5.063) (-2,0) & (1,0) x → −∞, y → +∞; 𝑎𝑠 x → +∞, y → +∞ |
5)
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i) Domain ii) Increasing iii) Range iv) Decreasing v) x-intercept(s) vi) Positive vii) y-intercept viii) Maximum ix) Minimum x) End behavior |
(-∞, ∞) (2, ∞) (1, ∞) (-∞,2) none (-∞, ∞) (0,5) none (2,1) x → −∞, y → +∞; 𝑎𝑠 x → +∞, y → +∞ |
6)
i) f(x) = 2x3 - 4x2 + x + 2
a) Degree of the polynomial = Odd degree polynomial
b) Sign of leading coefficient = positive
when,
lim x--> ∞, f(X) --> ∞
lim x--> -∞, f(X) --> -∞
c) The number of possible x-intercepts are 3.
d) Using rational root theorem,
Factors of constant p = ±1, ±2
Factors of Leading coefficient q = ±1, ±2
p/q = ±1, ±1/2, ±2
e) y-intercept :
Put x = 0
f(0) = 2(0)3 - 4(0)2 + 0 + 2
= 2
By analyzing the end behavior, rational roots and y-intercept. It is clear that option D is correct.
ii) f(x) = -x4 + 10x2 + 5x - 6
a) Degree of the polynomial = even degree polynomial
b) Sign of leading coefficient = negative
when,
lim x--> ∞, f(X) --> -∞
lim x--> -∞, f(X) --> -∞
c) The number of possible x-intercepts are 4.
d) Using rational root theorem,
Factors of constant p = ±1, ±2, ±3, ±6
Factors of Leading coefficient q = ±1
p/q = ±1, ±2, ±3, ±6
e) y-intercept :
Put x = 0
f(0) = -(0)4 + 10(0)2 + 5(0) - 6
= -6
By analyzing the end behavior, rational roots and y-intercept. It is clear that option B is correct.
iii) f(x) = -2x5 + 5x3 - x + 1
a) Degree of the polynomial = odd degree polynomial
b) Sign of leading coefficient = negative
when,
lim x--> ∞, f(X) --> -∞
lim x--> -∞, f(X) --> ∞
c) The number of possible x-intercepts are 5.
d) Using rational root theorem,
Factors of constant p = ±1
Factors of Leading coefficient q = ±1, ±2
p/q = ±1, ±1/2
e) y-intercept :
Put x = 0
f(0) = -2(0)5 + 5(0)3 - 0 + 1
= 1
By analyzing the end behavior, rational roots and y-intercept. It is clear that option A is correct.
iv) f(x) = x4 - 5x3 + 16
a) Degree of the polynomial = even degree polynomial
b) Sign of leading coefficient = positive
when,
lim x--> ∞, f(X) --> ∞
lim x--> -∞, f(X) --> ∞
c) The number of possible x-intercepts are 4.
d) Using rational root theorem,
Factors of constant p = ±1, ±2, ±4, ±8, ±16
Factors of Leading coefficient q = ±1
p/q = ±1, ±2, ±4, ±8, ±16
e) y-intercept :
Put x = 0
f(0) = 04 - 5(0)3 + 16
= 16
By analyzing the end behavior, rational roots and y-intercept. It is clear that option C is correct.
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