H Algebra 2/Trig · Unit 3
Use the factored equation to name the degree, the intercepts, the end behavior, and the turns you can see without a calculator.
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Build the sketch from the factors
Degree: add the exponents.
Each factor x − r gives an intercept. The exponent says whether it crosses, bounces, or flattens.
Then use the end behavior to aim the arms.
Turning points
A bounce on the x-axis is a max or a min. If the curve is below the axis there, the bounce is a maximum. If it is above, the bounce is a minimum.
There is also one turn in a gap between intercepts when the curve has to change direction to reach the next intercept.
Three intercepts, and the curve crosses at each one.
a. Add the exponents.
Degree: 1 + 1 + 1 = 3
b. Set each factor equal to 0.
x = −2, multiplicity 1. The graph crosses the x-axis.
x = 1, multiplicity 1. The graph crosses the x-axis.
x = 4, multiplicity 1. The graph crosses the x-axis.
c. Odd degree and a positive leading coefficient: the left end goes down and the right end goes up.
As x → −∞, y → −∞
As x → ∞, y → ∞
d. Even multiplicities put a max or a min on the x-axis.
There is a local maximum between x = −2 and x = 1.
There is a local minimum between x = 1 and x = 4.
A sketch of that shape:
Answer: Degree 3
x = −2, multiplicity 1; x = 1, multiplicity 1; x = 4, multiplicity 1
As x → −∞, y → −∞
As x → ∞, y → ∞
There is a local maximum between x = −2 and x = 1
There is a local minimum between x = 1 and x = 4
a. Add the exponents.
Degree: 1 + 1 + 1 = 3
b. Set each factor equal to 0.
x = −4, multiplicity 1. The graph crosses the x-axis.
x = −1, multiplicity 1. The graph crosses the x-axis.
x = 2, multiplicity 1. The graph crosses the x-axis.
c. Odd degree and a positive leading coefficient: the left end goes down and the right end goes up.
As x → −∞, y → −∞
As x → ∞, y → ∞
d. Even multiplicities put a max or a min on the x-axis.
There is a local maximum between x = −4 and x = −1.
There is a local minimum between x = −1 and x = 2.
A sketch of that shape:
Answer: Degree 3
x = −4, multiplicity 1; x = −1, multiplicity 1; x = 2, multiplicity 1
As x → −∞, y → −∞
As x → ∞, y → ∞
There is a local maximum between x = −4 and x = −1
There is a local minimum between x = −1 and x = 2
a. Add the exponents.
Degree: 1 + 1 + 1 = 3
b. Set each factor equal to 0.
x = −3, multiplicity 1. The graph crosses the x-axis.
x = 2, multiplicity 1. The graph crosses the x-axis.
x = 5, multiplicity 1. The graph crosses the x-axis.
c. Odd degree and a positive leading coefficient: the left end goes down and the right end goes up.
As x → −∞, y → −∞
As x → ∞, y → ∞
d. Even multiplicities put a max or a min on the x-axis.
There is a local maximum between x = −3 and x = 2.
There is a local minimum between x = 2 and x = 5.
A sketch of that shape:
Answer: Degree 3
x = −3, multiplicity 1; x = 2, multiplicity 1; x = 5, multiplicity 1
As x → −∞, y → −∞
As x → ∞, y → ∞
There is a local maximum between x = −3 and x = 2
There is a local minimum between x = 2 and x = 5
a. Add the exponents.
Degree: 1 + 1 + 1 = 3
b. Set each factor equal to 0.
x = −5, multiplicity 1. The graph crosses the x-axis.
x = −2, multiplicity 1. The graph crosses the x-axis.
x = 1, multiplicity 1. The graph crosses the x-axis.
c. Odd degree and a negative leading coefficient: the left end goes up and the right end goes down.
As x → −∞, y → ∞
As x → ∞, y → −∞
d. Even multiplicities put a max or a min on the x-axis.
There is a local minimum between x = −5 and x = −2.
There is a local maximum between x = −2 and x = 1.
A sketch of that shape:
Answer: Degree 3
x = −5, multiplicity 1; x = −2, multiplicity 1; x = 1, multiplicity 1
As x → −∞, y → ∞
As x → ∞, y → −∞
There is a local minimum between x = −5 and x = −2
There is a local maximum between x = −2 and x = 1
a. Add the exponents.
Degree: 1 + 1 + 1 = 3
b. Set each factor equal to 0.
x = −1, multiplicity 1. The graph crosses the x-axis.
x = 2, multiplicity 1. The graph crosses the x-axis.
x = 6, multiplicity 1. The graph crosses the x-axis.
c. Odd degree and a positive leading coefficient: the left end goes down and the right end goes up.
As x → −∞, y → −∞
As x → ∞, y → ∞
d. Even multiplicities put a max or a min on the x-axis.
There is a local maximum between x = −1 and x = 2.
There is a local minimum between x = 2 and x = 6.
A sketch of that shape:
Answer: Degree 3
x = −1, multiplicity 1; x = 2, multiplicity 1; x = 6, multiplicity 1
As x → −∞, y → −∞
As x → ∞, y → ∞
There is a local maximum between x = −1 and x = 2
There is a local minimum between x = 2 and x = 6
a. Add the exponents.
Degree: 1 + 1 + 1 = 3
b. Set each factor equal to 0.
x = −3, multiplicity 1. The graph crosses the x-axis.
x = 1, multiplicity 1. The graph crosses the x-axis.
x = 4, multiplicity 1. The graph crosses the x-axis.
c. Odd degree and a negative leading coefficient: the left end goes up and the right end goes down.
As x → −∞, y → ∞
As x → ∞, y → −∞
d. Even multiplicities put a max or a min on the x-axis.
There is a local minimum between x = −3 and x = 1.
There is a local maximum between x = 1 and x = 4.
A sketch of that shape:
Answer: Degree 3
x = −3, multiplicity 1; x = 1, multiplicity 1; x = 4, multiplicity 1
As x → −∞, y → ∞
As x → ∞, y → −∞
There is a local minimum between x = −3 and x = 1
There is a local maximum between x = 1 and x = 4
One intercept bounces.
a. Add the exponents.
Degree: 2 + 1 + 1 = 4
b. Set each factor equal to 0.
x = −4, multiplicity 2. The graph bounces off the x-axis.
x = 1, multiplicity 1. The graph crosses the x-axis.
x = 3, multiplicity 1. The graph crosses the x-axis.
c. Even degree and a positive leading coefficient: both ends go up.
As x → −∞, y → ∞
As x → ∞, y → ∞
d. Even multiplicities put a max or a min on the x-axis.
At x = −4 the graph is above the axis on both sides, so that bounce is a local minimum.
There is a local maximum between x = −4 and x = 1.
There is a local minimum between x = 1 and x = 3.
A sketch of that shape:
Answer: Degree 4
x = −4, multiplicity 2; x = 1, multiplicity 1; x = 3, multiplicity 1
As x → −∞, y → ∞
As x → ∞, y → ∞
Local minimum at (−4, 0)
There is a local maximum between x = −4 and x = 1
There is a local minimum between x = 1 and x = 3
a. Add the exponents.
Degree: 1 + 2 + 1 = 4
b. Set each factor equal to 0.
x = −2, multiplicity 1. The graph crosses the x-axis.
x = 3, multiplicity 2. The graph bounces off the x-axis.
x = 6, multiplicity 1. The graph crosses the x-axis.
c. Even degree and a positive leading coefficient: both ends go up.
As x → −∞, y → ∞
As x → ∞, y → ∞
d. Even multiplicities put a max or a min on the x-axis.
At x = 3 the graph is below the axis on both sides, so that bounce is a local maximum.
There is a local minimum between x = −2 and x = 3.
There is a local minimum between x = 3 and x = 6.
A sketch of that shape:
Answer: Degree 4
x = −2, multiplicity 1; x = 3, multiplicity 2; x = 6, multiplicity 1
As x → −∞, y → ∞
As x → ∞, y → ∞
Local maximum at (3, 0)
There is a local minimum between x = −2 and x = 3
There is a local minimum between x = 3 and x = 6
a. Add the exponents.
Degree: 2 + 1 + 1 = 4
b. Set each factor equal to 0.
x = −5, multiplicity 2. The graph bounces off the x-axis.
x = −1, multiplicity 1. The graph crosses the x-axis.
x = 2, multiplicity 1. The graph crosses the x-axis.
c. Even degree and a negative leading coefficient: both ends go down.
As x → −∞, y → −∞
As x → ∞, y → −∞
d. Even multiplicities put a max or a min on the x-axis.
At x = −5 the graph is below the axis on both sides, so that bounce is a local maximum.
There is a local minimum between x = −5 and x = −1.
There is a local maximum between x = −1 and x = 2.
A sketch of that shape:
Answer: Degree 4
x = −5, multiplicity 2; x = −1, multiplicity 1; x = 2, multiplicity 1
As x → −∞, y → −∞
As x → ∞, y → −∞
Local maximum at (−5, 0)
There is a local minimum between x = −5 and x = −1
There is a local maximum between x = −1 and x = 2
a. Add the exponents.
Degree: 2 + 1 + 1 = 4
b. Set each factor equal to 0.
x = −3, multiplicity 2. The graph bounces off the x-axis.
x = 2, multiplicity 1. The graph crosses the x-axis.
x = 4, multiplicity 1. The graph crosses the x-axis.
c. Even degree and a positive leading coefficient: both ends go up.
As x → −∞, y → ∞
As x → ∞, y → ∞
d. Even multiplicities put a max or a min on the x-axis.
At x = −3 the graph is above the axis on both sides, so that bounce is a local minimum.
There is a local maximum between x = −3 and x = 2.
There is a local minimum between x = 2 and x = 4.
A sketch of that shape:
Answer: Degree 4
x = −3, multiplicity 2; x = 2, multiplicity 1; x = 4, multiplicity 1
As x → −∞, y → ∞
As x → ∞, y → ∞
Local minimum at (−3, 0)
There is a local maximum between x = −3 and x = 2
There is a local minimum between x = 2 and x = 4
a. Add the exponents.
Degree: 1 + 2 + 1 = 4
b. Set each factor equal to 0.
x = −1, multiplicity 1. The graph crosses the x-axis.
x = 2, multiplicity 2. The graph bounces off the x-axis.
x = 5, multiplicity 1. The graph crosses the x-axis.
c. Even degree and a negative leading coefficient: both ends go down.
As x → −∞, y → −∞
As x → ∞, y → −∞
d. Even multiplicities put a max or a min on the x-axis.
At x = 2 the graph is above the axis on both sides, so that bounce is a local minimum.
There is a local maximum between x = −1 and x = 2.
There is a local maximum between x = 2 and x = 5.
A sketch of that shape:
Answer: Degree 4
x = −1, multiplicity 1; x = 2, multiplicity 2; x = 5, multiplicity 1
As x → −∞, y → −∞
As x → ∞, y → −∞
Local minimum at (2, 0)
There is a local maximum between x = −1 and x = 2
There is a local maximum between x = 2 and x = 5
a. Add the exponents.
Degree: 2 + 1 + 1 = 4
b. Set each factor equal to 0.
x = −6, multiplicity 2. The graph bounces off the x-axis.
x = −2, multiplicity 1. The graph crosses the x-axis.
x = 1, multiplicity 1. The graph crosses the x-axis.
c. Even degree and a positive leading coefficient: both ends go up.
As x → −∞, y → ∞
As x → ∞, y → ∞
d. Even multiplicities put a max or a min on the x-axis.
At x = −6 the graph is above the axis on both sides, so that bounce is a local minimum.
There is a local maximum between x = −6 and x = −2.
There is a local minimum between x = −2 and x = 1.
A sketch of that shape:
Answer: Degree 4
x = −6, multiplicity 2; x = −2, multiplicity 1; x = 1, multiplicity 1
As x → −∞, y → ∞
As x → ∞, y → ∞
Local minimum at (−6, 0)
There is a local maximum between x = −6 and x = −2
There is a local minimum between x = −2 and x = 1
Two bounces, or a fraction in front.
a. Add the exponents.
Degree: 2 + 2 + 1 = 5
b. Set each factor equal to 0.
x = −4, multiplicity 2. The graph bounces off the x-axis.
x = 1, multiplicity 2. The graph bounces off the x-axis.
x = 6, multiplicity 1. The graph crosses the x-axis.
c. Odd degree and a positive leading coefficient: the left end goes down and the right end goes up.
As x → −∞, y → −∞
As x → ∞, y → ∞
d. Even multiplicities put a max or a min on the x-axis.
At x = −4 the graph is below the axis on both sides, so that bounce is a local maximum.
At x = 1 the graph is below the axis on both sides, so that bounce is a local maximum.
There is a local minimum between x = −4 and x = 1.
There is a local minimum between x = 1 and x = 6.
A sketch of that shape:
Answer: Degree 5
x = −4, multiplicity 2; x = 1, multiplicity 2; x = 6, multiplicity 1
As x → −∞, y → −∞
As x → ∞, y → ∞
Local maximum at (−4, 0)
Local maximum at (1, 0)
There is a local minimum between x = −4 and x = 1
There is a local minimum between x = 1 and x = 6
a. Add the exponents.
Degree: 2 + 2 + 1 = 5
b. Set each factor equal to 0.
x = −5, multiplicity 2. The graph bounces off the x-axis.
x = 2, multiplicity 2. The graph bounces off the x-axis.
x = 4, multiplicity 1. The graph crosses the x-axis.
c. Odd degree and a positive leading coefficient: the left end goes down and the right end goes up.
As x → −∞, y → −∞
As x → ∞, y → ∞
d. Even multiplicities put a max or a min on the x-axis.
At x = −5 the graph is below the axis on both sides, so that bounce is a local maximum.
At x = 2 the graph is below the axis on both sides, so that bounce is a local maximum.
There is a local minimum between x = −5 and x = 2.
There is a local minimum between x = 2 and x = 4.
A sketch of that shape:
Answer: Degree 5
x = −5, multiplicity 2; x = 2, multiplicity 2; x = 4, multiplicity 1
As x → −∞, y → −∞
As x → ∞, y → ∞
Local maximum at (−5, 0)
Local maximum at (2, 0)
There is a local minimum between x = −5 and x = 2
There is a local minimum between x = 2 and x = 4
a. Add the exponents.
Degree: 2 + 1 + 1 = 4
b. Set each factor equal to 0.
x = −3, multiplicity 2. The graph bounces off the x-axis.
x = 1, multiplicity 1. The graph crosses the x-axis.
x = 5, multiplicity 1. The graph crosses the x-axis.
c. Even degree and a negative leading coefficient: both ends go down.
As x → −∞, y → −∞
As x → ∞, y → −∞
d. Even multiplicities put a max or a min on the x-axis.
At x = −3 the graph is below the axis on both sides, so that bounce is a local maximum.
There is a local minimum between x = −3 and x = 1.
There is a local maximum between x = 1 and x = 5.
A sketch of that shape:
Answer: Degree 4
x = −3, multiplicity 2; x = 1, multiplicity 1; x = 5, multiplicity 1
As x → −∞, y → −∞
As x → ∞, y → −∞
Local maximum at (−3, 0)
There is a local minimum between x = −3 and x = 1
There is a local maximum between x = 1 and x = 5
a. Add the exponents.
Degree: 2 + 2 = 4
b. Set each factor equal to 0.
x = −2, multiplicity 2. The graph bounces off the x-axis.
x = 3, multiplicity 2. The graph bounces off the x-axis.
c. Even degree and a positive leading coefficient: both ends go up.
As x → −∞, y → ∞
As x → ∞, y → ∞
d. Even multiplicities put a max or a min on the x-axis.
At x = −2 the graph is above the axis on both sides, so that bounce is a local minimum.
At x = 3 the graph is above the axis on both sides, so that bounce is a local minimum.
There is a local maximum between x = −2 and x = 3.
A sketch of that shape:
Answer: Degree 4
x = −2, multiplicity 2; x = 3, multiplicity 2
As x → −∞, y → ∞
As x → ∞, y → ∞
Local minimum at (−2, 0)
Local minimum at (3, 0)
There is a local maximum between x = −2 and x = 3
a. Add the exponents.
Degree: 2 + 2 + 1 = 5
b. Set each factor equal to 0.
x = −6, multiplicity 2. The graph bounces off the x-axis.
x = −1, multiplicity 2. The graph bounces off the x-axis.
x = 2, multiplicity 1. The graph crosses the x-axis.
c. Odd degree and a negative leading coefficient: the left end goes up and the right end goes down.
As x → −∞, y → ∞
As x → ∞, y → −∞
d. Even multiplicities put a max or a min on the x-axis.
At x = −6 the graph is above the axis on both sides, so that bounce is a local minimum.
At x = −1 the graph is above the axis on both sides, so that bounce is a local minimum.
There is a local maximum between x = −6 and x = −1.
There is a local maximum between x = −1 and x = 2.
A sketch of that shape:
Answer: Degree 5
x = −6, multiplicity 2; x = −1, multiplicity 2; x = 2, multiplicity 1
As x → −∞, y → ∞
As x → ∞, y → −∞
Local minimum at (−6, 0)
Local minimum at (−1, 0)
There is a local maximum between x = −6 and x = −1
There is a local maximum between x = −1 and x = 2
a. Add the exponents.
Degree: 3 + 1 = 4
b. Set each factor equal to 0.
x = −3, multiplicity 3. The graph flattens and still crosses the x-axis.
x = 2, multiplicity 1. The graph crosses the x-axis.
c. Even degree and a positive leading coefficient: both ends go up.
As x → −∞, y → ∞
As x → ∞, y → ∞
d. Even multiplicities put a max or a min on the x-axis.
There is a local minimum between x = −3 and x = 2.
A sketch of that shape:
Answer: Degree 4
x = −3, multiplicity 3; x = 2, multiplicity 1
As x → −∞, y → ∞
As x → ∞, y → ∞
There is a local minimum between x = −3 and x = 2
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