Section 5.1 Solutions
1) 𝑓(𝑥) = (𝑥 − 3)2 (𝑥 + 1)
1a) List each x-intercept (zero) and its multiplicity (round to 2 decimal places when needed)
(x-3)2(x+1) = 0
(x-3)(x-3)(x+1) = 0
x–3=0
or x – 3 = 0
x+1=0
x=3
or x = 3
or x -1
Answer #1a: x-intercepts are (3, 0) multiplicity even (-1,0) multiplicity odd
1b) Determine whether the graph crosses or touches the x-axis at each x-intercept
A graph touches when the x-intercept has even multiplicity and crosses when the x-intercept has odd
multiplicity.
Answer #1b: touches the x-axis at x = 3, crosses the x-axis at x = -1
1c) Determine the maximum number of turning points on the graph
if you multiply the firsts in each parenthesis you get x*x*x = x3, hence this is a third degree polynomial
Answer #1c: has at most 2 turning points
1d) sketch a graph and approximate the turning points, also label the x-intercepts
1e) Describe the end behavior
Answer #1e: falls to the left, rises to the right
1f) state the intervals where the function is increasing and decreasing
The function increases from the start x =−∞ to the first turning point x = .33
it then decreases from the first turning point x = .33 to the second turning point x = 3
it then increases from the second turning point x = 3 to the end of the graph x = ∞
Answer #1f: increasing (−∞, . 𝟑𝟑) ∪ (𝟑, ∞) decreasing (. 𝟑𝟑, 𝟑)
3) f(x) = (x – 3)3(x + 4)
3a) List each x-intercept (zero) and its multiplicity (round to 2 decimal places when needed)
(x – 3)3(x + 4) = 0
Enough to solve x – 3 = 0 and x + 4 = 0
x–3=0
x +4=0
x=3
x = -4
Answer #3a: x-intercepts are (3,0) has odd multiplicity and (-4,0) odd multiplicity
3b) Determine whether the graph crosses or touches the x-axis at each x-intercept
A graph touches when the x-intercept has even multiplicity and crosses when the x-intercept has odd
multiplicity.
Answer #3b: graph crosses x-axis at both x-intercepts
3c) Determine the maximum number of turning points on the graph
Multiply the firsts in each parenthesis of the expanded polynomial
(x)(x)(x)(x) = x4, this is a 4th degree polynomial
Answer #3c: has at most 3 turning points
3d) sketch a graph and approximate the turning points, also label the x-intercepts
I rounded the y-coordinate of the min to an integer as the decimal was hard to read on my graph.
3e) Describe the end behavior
Answer #3e: rises to the left, rises to the right
3f) state the intervals where the function is increasing and decreasing
The graph decreases from the start x = -∞ to the turning point x = -2.25
it then increases from x = -2.25 to the end of the graph x = ∞
Answer #3f: increasing (−𝟐. 𝟐𝟓, ∞) decreasing (-∞, -2.25)
5) f(x) = (x+3)(x-3)(x+6)
5a) List each x-intercept (zero) and its multiplicity (round to 2 decimal places when needed)
(x+3)(x-3)(x+6) = 0
x+3=0
x–3=0
x+6=0
x= -3
x=3
x = -6
Answer #5a: (−𝟑, 𝟎) (3,0) (-6,0) all have odd multiplicity
5b) Determine whether the graph crosses or touches the x-axis at each x-intercept
A graph touches when the x-intercept has even multiplicity and crosses when the x-intercept has odd
multiplicity.
Answer #5b: graph crosses at each x-intercept (−𝟑, 𝟎) (3,0) (-6,0)
5c) Determine the maximum number of turning points on the graph
Multiply the firsts (x)(x)(x) = x3 this is a 3rd degree polynomial
Answer #5c: will have at most 2 turning points
5d) sketch a graph and approximate the turning points, also label the x-intercepts
The points were hard to read on the graph. I’ve just made dots on the graph to keep it clean.
The x-intercepts are (-6,0) (-3,0) and (3,0)
The maximum point is (-4.65,17.04)
The minimum point is (.65, -57.04)
5e) Describe the end behavior
Answer #5e: Falls to the left rises to the right
5f) state the intervals where the function is increasing and decreasing
The function increases from the start x =−∞ to the first turning point x = -4.65
it then decreases from the first turning point x = -4.65 to the second turning point x = .65
it then increases from the second turning point x = .65 to the end of the graph x = ∞
Answer #5f: increasing (−∞, −𝟒. 𝟔𝟓) ∪ (. 𝟔𝟓, ∞) decreasing (−𝟒. 𝟔𝟓, . 𝟔𝟓)
7) f(x) = (x+3)2(x – 2)
7a) List each x-intercept (zero) and its multiplicity (round to 2 decimal places when needed)
(x +3)(x+3)(x-2) = 0
x+3=0
x = -3
x+3=0
x = -3
x–2=0
x=2
Answer #7a: (-3,0) has even multiplicity, (2,0) has odd multiplicity
7b) Determine whether the graph crosses or touches the x-axis at each x-intercept
A graph touches when the x-intercept has even multiplicity and crosses when the x-intercept has odd
multiplicity.
Answer #7b: Graph touches at (-3,0) and crosses at (2,0)
7c) Determine the maximum number of turning points on the graph
Multiply the firsts in the expanded polynomial (x)(x)(x)= x3, this is a 3rd degree polynomial
Answer #7c: has at most 2 turning points
7d) sketch a graph and approximate the turning points, also label the x-intercepts
7e) Describe the end behavior
Answer #7e: rises to the left and rises to the right
7f) state the intervals where the function is increasing and decreasing
The function increases from the start x =−∞ to the first turning point x = -3
it then decreases from the first turning point x = -3 to the second turning point x = .33
it then increases from the second turning point x = .3 to the end of the graph x = ∞
Answer #7f: increasing (−∞, −𝟑) ∪ (. 𝟑𝟑, ∞) decreasing (−𝟑, . 𝟑𝟑)
9) 𝑓(𝑥) = 𝑥 2 + 6𝑥 − 7
9a) List each x-intercept (zero) and its multiplicity (round to 2 decimal places when needed)
x2 + 6x – 7 = 0
(x + 7)(x – 1) = 0
x+7=0
x = -7
x–1=0
x=1
Answer #9a: (-7,0) odd multiplicity, (1,0) odd multiplicity
9b) Determine whether the graph crosses or touches the x-axis at each x-intercept
A graph touches when the x-intercept has even multiplicity and crosses when the x-intercept has odd
multiplicity.
Answer #9b: graph crosses at both (-7,0) and (1,0)
9c) Determine the maximum number of turning points on the graph
This polynomial is multiplied out, the maximum number of turning points is one less than the highest
exponent of 2.
Answer #9c: has at most 1 turning points
9d) sketch a graph and approximate the turning points, also label the x-intercepts
9e) Describe the end behavior
Answer #9e: rises to the left and rises to the right
9f) state the intervals where the function is increasing and decreasing
The function increases from the start x =−∞ to the turning point x = -3
it then decreases from turning point x = -3 to the end of the graph x = ∞
Answer #9f: increasing (−𝟑, ∞) decreasing (−∞, −𝟑)
11) f(x) = x2 - 4
11a) List each x-intercept (zero) and its multiplicity (round to 2 decimal places when needed)
x2 – 4 = 0
(x + 2)(x – 2) = 0
x +2 = 0
x = -2
x–2=0
x=2
Answer #11a: (-2,0) odd multiplicity, (2,0) odd multiplicity
b) Determine whether the graph crosses or touches the x-axis at each x-intercept
A graph touches when the x-intercept has even multiplicity and crosses when the x-intercept has odd
multiplicity.
Answer #11b: Graph crosses the x-axis at (-2,0) and (2,0)
c) Determine the maximum number of turning points on the graph
This polynomial is multiplied out, the maximum number of turning points is one less than the highest
exponent of 2.
Answer #11c: has at most 1 turning point
11d) sketch a graph and approximate the turning points, also label the x-intercepts
11e) Describe the end behavior
Answer #11e: rises to the left, rises to the right
11f) state the intervals where the function is increasing and decreasing
Answer #11f: increasing (−∞, 𝟎) decreasing (𝟎, ∞)
13) f(x) = x3 – 6x2 + 5x
13a) List each x-intercept (zero) and its multiplicity (round to 2 decimal places when needed)
x3 – 6x2 + 5x = 0
x(x2 – 6x + 5) = 0
x(x-1)(x-5) = 0
x=0
x–1=0
x–5=0
x=0
x=1
x=5
Answer #13a: (0,0), odd multiplicity
(1,0) odd multiplicity
x = (5,0) odd multiplicity
13b) Determine whether the graph crosses or touches the x-axis at each x-intercept
A graph touches when the x-intercept has even multiplicity and crosses when the x-intercept has odd
multiplicity.
Answer #13b: graph crosses at each x-intercept
13c) Determine the maximum number of turning points on the graph
this is a 3nd degree polynomial
Answer has at most 2 turning points.
13f) it got hard to read the points when I showed them on the graph.
The maximum point is (.47, 1.13)
The minimum point is (3.53, -13.13)
13e) Describe the end behavior
Answer #13e: rise to left and rise to the right
13f) state the intervals where the function is increasing and decreasing
The function increases from the start x =−∞ to the maximum point x = .47
it then decreases from the first turning point x = .47 to the minimum point x = 3.53
it then increases from the minimum point x = 3.53 to the end of the graph x = ∞
Answer #13f: increases (−∞, . 𝟒𝟕) ∪ (𝟑. 𝟓𝟑, ∞) decreases (.47, 3.53)
15) 𝑓(𝑥) = −3𝑥 4 + 12𝑥 2
15a) List each x-intercept and (zero) its multiplicity (round to 2 decimal places when needed)
-3x4 + 12x2 = 0
-3x2(x2 – 4) = 0
-3x2 = 0 x2 – 4 = 0
(divide by -3)
x2 = 0
xx = 0
(x+2)(x-2) = 0
x+2=0
x- 2 = 0
Answer #15a: (0,0) has even multiplicity, (2,0) has odd multiplicity (-2,0) has odd multiplicity
15b) Determine whether the graph crosses or touches the x-axis at each x-intercept
Answer: graph touches at (0,0) and crosses at (-2,0) and crosses at (2,0)
15c) Determine the maximum number of turning points on the graph
This is a 4th degree polynomial
Answer #15c: has at most 3 turning points
15d) sketch a graph and approximate the turning points, also label the x-intercepts
15e) Describe the end behavior
Answer #15e: falls to the left, falls to the right
15f) state the intervals where the function is increasing and decreasing
Answer #15f: increasing (−∞, −𝟏. 𝟒𝟏) ∪ (𝟎, 𝟏. 𝟒𝟏) decreasing (-1.41, 0) (𝟏. 𝟒𝟏, ∞)
#17 – 26: Form a polynomial with whose x-intercepts are given. (multiply out your polynomial)
17) x-intercepts: (3,0), (-4,0); degree 2
First: Set x-coordinates of x-intercepts equal to 0 and then create factors.
x = 3 and x = -4
x- 3 = 0
x + 4 =0
Second: Create polynomial by multiplying the left side of each equation.
f(x) = (x-3)(x+4)
f(x) = x2 + 4x – 3x – 12
Answer #17: f(x) = x2 + x - 12
1
19) x-intercepts: (3,0), (2 , 0); degree 2
First: Set x-coordinates of x-intercepts equal to 0 and then create factors.
x=
1
2
x=3
1
2𝑥 = 2 ∗ 2
x–3=0
2x = 1
2x – 1 = 0
Second: Create polynomial by multiplying the left side of each equation.
f(x) = (2x-1)(x-3)
f(x) = 2x2 – 6x – 1x + 3
Answer #19: f(x) = 2x2 – 7x + 3
3
4
21) x-intercepts: (−3,0), ( , 0), (3,0) ; degree 3
First: Set x-coordinates of x-intercepts equal to 0 and then create factors.
3
x=4
3
4𝑥 = 4 ∗ 4
x=3
x=-3
x–3=0
x+3=0
4x = 3
4x – 3 = 0
Second: Create polynomial by multiplying the left side of each equation.
f(x) = (x – 3)(x+3)(4x-3)
f(x) = (x2 + 3x – 3x – 9)(4x-3)
f(x) = (x2 – 9)(4x – 3)
Answer #21: f(x) = x3 – 3x2 – 36x + 27
23) x-intercepts (√2, 0) (−√2, 0); degree 2
First: Set x-coordinates of x-intercepts equal to 0 and then create factors.
x = √2
x = - √2
x - √2 = 0
x + √2 = 0
Second: Create polynomial by multiplying the left side of each equation.
f(x) = (x - √2)( x + √2)
f(x) = x2 +√2𝑥-√2𝑥 - √4
Answer #23: f(x) = x2 - 2
25) x-intercepts (√2, 0) (−√2, 0) (6,0); degree 3
First: Set x-coordinates of x-intercepts equal to 0 and then create factors.
x = √2
x = - √2
x=6
x - √2 = 0
x + √2 = 0
x–6=0
Second: Create polynomial by multiplying the left side of each equation.
f(x) = (x - √2)( x + √2)(x-6)
f(x) = (x2 +√2𝑥-√2𝑥 - √4)(x-6)
f(x) = (x2 – 2)(x – 6)
Answer #25: f(x) = x3 – 6x2 – 2x + 12