ch 3 review quarter test 1 and ch 3 test. graphs of quadratic functions where a, b, and c are real...
TRANSCRIPT
![Page 1: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/1.jpg)
Ch 3 reviewQuarter test 1
And Ch 3 TEST
![Page 2: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/2.jpg)
Graphs of Quadratic Functions
cbxaxxf 2)(Where a, b, and c are real numbers and a 0
Standard Form
•Domain: all real numbers
•Range: depends on the minimum and maximum
•The graph is a parabola.
![Page 3: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/3.jpg)
if 0a
The graph of x2 is shifted “h” units horizontally and “k” units vertically.
opens:
axis of symmetry:
vertex:
k is the
range:
V(h, k) / minimum
x = h
up
x = h
(h, k)
minimum
ky
positive
![Page 4: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/4.jpg)
if
The graph of x2 is shifted “h” units horizontally, “k” units vertically, and reflected over x-axis.
opens:
axis of symmetry:
vertex:
k is the
range:
V(h, k) / maximum
x = h
down
x = h
(h, k)
maximum
negative0a
ky
![Page 5: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/5.jpg)
cbxaxxf 2)(Standard form: Vertex form: khxaxf 2)()(
can “see” the transformations…
The vertex form is easier to graph…to change from standard form to vertex form, either complete the square (YUCK!) or memorize this formula:
a
bh
2
and
a
bfk
2 h
Therefore, the vertex is at
a
bf
a
b
2,
2
and the axis of symmetry is . a
bx
2
k = f(h)
![Page 6: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/6.jpg)
Example: Let . Find the vertex, axis of symmetry, the minimum or maximum value, and the intercepts. Use these to graph f(x). State the domain and range and give the intervals of increase and decrease. Then write the equation in vertex form and list the transformations that were made to the parent function, f(x) = x2.
163)( 2 xxxf
1st identify a, b, and c: a = 3 b = 6 c = 1
Next find h and k: 16
6
)3(2
6
2
a
bh
2)1()( fhfk
So, the vertex is (-1, -2) and the axis of symmetry
is x = -1. Since a > 0, then the graph opens up
and has a minimum value at -2.
To find y-intercepts evaluate f(0): 1)0( f
To find x-intercepts (roots/zeros) use the quadratic formula:
a
acbbx
2
42
)3(2
)1)(3(4)6()6( 2
6
246
-0.184and
-1.816
The intercepts are at (0, 1), (-0.184, 0), and
(-1.816, 0).
![Page 7: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/7.jpg)
To graph, plot the vertex, intercepts, utilize the axis of symmetry.
V(-1, -2) y-int: (0,1) axis of sym: x = -1
So, to be symmetrical, another point will be at (-2, 1).
Check using your graphing calculator!
Domain: all real numbers
Range:
2y
Decreasing:
Increasing: ),1( )1,(
Vertex form: f(x) = a(x - h)2 + k
a = 3 h = -1 k = -2
2)1(3)( 2 xxf
It is the graph of x2 shifted left 1, vertically stretched by
3 and shifted down 2.
![Page 8: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/8.jpg)
Example: Find the standard form equation of the quadratic function whose vertex is (1, -5) and whose y-intercept is -3.
h k (0, -3)
Vertex form: f(x) = a(x - h)2 + k
Fill in the information that was given:
)5()10(3 2 a Solve for a…
5)1(3 2 a53 a
2a
Write the equation in vertex form then simplify to standard form:
5)1(2)( 2 xxf
5)12(2)( 2 xxxf
5242)( 2 xxxf
342)( 2 xxxf
![Page 9: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/9.jpg)
Power Functions
naxxf )(The polynomial that the graph resembles (the end behavior model)…
EX: The power function of the polynomial is…
33xy 153)( 23 xxxxf
22 )5(2)( xxxg 42xy
)35(2)( 2 xxxf 36xy
![Page 10: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/10.jpg)
Properties of Power Functions with Even Degrees
1. f is an even function
a. The graph is symmetric with respect to the y-axis
b. f(-x) = f(x)
2. Domain: all real numbers
3. The graph always contains the points (0, 0), (1, 1) and (-1, 1)
4. As the exponent increases in magnitude, the graph becomes more vertical when x < -1 or x > 1; but for x near the origin, the graph tends to flatten out and be closer to the x-axis.
The graph always contains the points (0, 0), (1, 1) and (-1, 1)*
*Points used to make transformations
![Page 11: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/11.jpg)
EX: Graph y = x4, y = x8 and y = x12 all on the same screen.
Let and be your viewing window.
What do you notice?
11 x 10 y
![Page 12: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/12.jpg)
Properties of Power Functions with Odd Degrees
1. f is an odd function
a. The graph is symmetric with respect to the origin
b. f(-x) = -f(x)
2. Domain: all real numbers
Range: all real numbers
3. The graph always contains the points (0, 0), (1, 1) and (-1, -1)
4. As the exponent increases in magnitude, the graph becomes more vertical when x < -1 or x > 1; but for x near the origin, the graph tends to flatten out and be closer to the x-axis.
The graph always contains the points (0, 0), (1, 1) and (-1, -1)*
*Points used to make transformations
![Page 13: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/13.jpg)
EX: Graph y = x3, y = x7 and y = x11 all on the same screen.
Let and be your viewing window.
What do you notice?
11 x 11 y
![Page 14: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/14.jpg)
Graphs of polynomial functions are smooth (no sharp corners or cusps) and continuous (no gaps or holes…it can be
drawn without lifting your pencil)…
Is a polynomial graph Is not a polynomial graph
![Page 15: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/15.jpg)
We can apply what we learned about transformations in Chapter 2
and what we just learned about power functions to graph
polynomials…
![Page 16: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/16.jpg)
EXAMPLE: Graph f(x) = 1 – (x – 2)4 using transformations.
Step 1: y = x4 Step 2: y = (x – 2)4
Step 3: y = - (x – 2)4 Step 4: y = 1 – (x – 2)4
Start with (0, 0), (1, 1) & (-1, 1)
Shift right 2 units
Reflect over x-axis
Shift up 1 unit
![Page 17: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/17.jpg)
EXAMPLE: Graph f(x) = 2(x + 1)5 using transformations. Check your work with your graphing calculator.
x5…(0, 0), (1, 1) & (-1, -1)
(x + 1)5…shift left 1 unit
2(x + 1)5…vertical stretch by factor 2
multiply the y-values by 2
![Page 18: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/18.jpg)
Zeros and the Equation of a Polynomial Function
If f is a polynomial function and r is a real number for which f(r) = 0, then r is called a real zero of f, or a root of f. If r is a real zero/root of f then:
a. r is an x-intercept of the graph of f, and
b. (x – r) is a factor of f
In other words…if you know a zero/root, then you know a factor…if you know a factor, then you know a zero/root
EX: If (x – 4) is a factor, then 4 is a zero/root…
If -3 is a zero/root, then (x + 3) is a factor…
![Page 19: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/19.jpg)
EXAMPLE: Find a polynomial of degree 3 with zeros -4, 1, and 3. (Let a = 1)
If x = -4, then the factor that solves to that is…
)4())4(( xx
If x = 1, then the factor that solves to that is… )1( x
If x = 3, then the factor that solves to that is… )3( x
)3)(1)(4()( xxxxf
![Page 20: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/20.jpg)
)3)(1)(4()( xxxxf
)3)(43()( 2 xxxxf
)(xf 3x 23x 23x x9 x4 12
1213)( 3 xxxf
![Page 21: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/21.jpg)
Rational Function
A function of the form , where p and q are
polynomial functions and q is not the zero
polynomial.
The domain is the set of all real numbers EXCEPT those for which the denominator q is zero.
)(
)(
xq
xp
2
1)(
x
xxfEX:
* Enter in calculator as (x + 1)/(x – 2)...MUST put parentheses!
*
![Page 22: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/22.jpg)
Domain and Vertical Asymptotes
To find the domain of a rational function, find the zeros of the denominator…this is where the denominator would be zero…this is where x cannot exist.
The vertical asymptote(s) of a rational function are where x cannot exist…it is the virtual boundary line on the graph. Vertical asymptotes are defined by the equation ‘x =‘
![Page 23: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/23.jpg)
How the graph reacts on either side of a vertical asymptote:
Goes in opposite directions as it approaches the asymptote:
Goes in the same direction as it approaches the asymptote:
THE GRAPH WILL NEVER CROSS THE VERTICAL ASYMPTOTE!!!
![Page 24: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/24.jpg)
EXAMPLE: Find the domain and vertical asymptotes of the rational functions.
a.5
42)(
2
x
xxf
The graph will not exist where the denominator equals zero!
x + 5 = 0
x = -5 When x = -5, the graph will not exist!
5xThe domain is and the VA is 5x5xThe domain is and the VA is 5x5xThe domain is and the VA is 5x
![Page 25: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/25.jpg)
EXAMPLE: Find the domain and vertical asymptotes of the rational functions.
2x
b.4
1)(
2
xxf c.
1)(
2
3
x
xxf
x2 – 4 = 0
(x + 2)(x – 2) = 0
x = -2 x = 2
Domain:
VA: 2,2 xx
x2 + 1 = 0
x2 = -1
x = not real
Domain: all real #’s
VA: none
![Page 26: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/26.jpg)
Intercepts on the x and y axes
To find the y-intercepts of a rational function, that is where x = 0, evaluate f(0).
To find the x-intercepts of a rational function, first make sure the function is in lowest terms, that is the numerator and denominator have no common factors. Then, find the zeros of the numerator.
factor top & bottom first!!
The zeros of the numerator are the x-intercepts (zeros) of the rational function.
![Page 27: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/27.jpg)
EXAMPLE: Find the x and y intercepts of the rational functions.
a.5
42)(
2
x
xxf
5
)2(2 2
x
xNo common factors…in
lowest terms.
y-intercept: x-intercept:
5
4
50
4)0(2)0(
2
f042 2 x
42 2 x22 x2x
The y-intercept is at and the x-intercepts
are at and 5
42 2
![Page 28: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/28.jpg)
EXAMPLE: Find the x and y intercepts of the rational functions.
b. No common factors…in lowest terms.
y-intercept: x-intercept:
The y-intercept is at and there are no x-intercepts 41
4
1)(
2
xxf
)2)(2(
1
xx
4
1
4)0(
1)0(
2
f none…the numerator has
no x in it to solve for!
![Page 29: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/29.jpg)
EXAMPLE: Find the x and y intercepts of the rational functions.
c. Cannot be factored…in lowest terms.
y-intercept: x-intercept:
01
0
10
)0()0(
2
3
f03 x0x
The y-intercept is at 0 and the x-intercept is at 0.
1)(
2
3
x
xxf
![Page 30: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/30.jpg)
Graphing Rational Functions Using Transformations
a. Analyze the graph of 21)(
xxR
1st find the domain and any vertical asymptotes:
x2 = 0
x = 0
Domain: 0xVA: x = 0
Next, find the x & y-intercepts:
x-intercepts: none
y-intercepts: none
undefinedf 0
1
)0(
1)0(
2
![Page 31: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/31.jpg)
Graphing Rational Functions Using Transformations
a. Analyze the graph of 21)(
xxR
Is it even?
To graph without using a calculator, identify a few points on the graph by plugging in x-values:
)(1
)(
1)(
22xR
xxxR
It is an even function, so it is symmetric to the y-axis.
f(1) = 1 f(-1) = 1
f(2) = ¼ f(-2) = ¼
![Page 32: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/32.jpg)
Graphing Rational Functions Using Transformations
b. Use transformations to graph
Check the domain and any vertical asymptotes:
(x – 2)2 = 0
x – 2 =0
x = 2
Domain: 2xVA: x = 2
Next, check the y-intercept:
y-intercepts: 1.25 25.114
11
)20(
1)0(
2
f
1)2(
1)(
2
xxR
It is the graph of shifted right 2 and up 1.21
x
Shift the VA and points right 2 and up 1...
![Page 33: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/33.jpg)
Graphing Rational Functions Using Transformations
c. Analyze the graph of and use it to graph
Find the domain and any vertical asymptotes:
x - 2 = 0
x = 2
Domain: 2x
VA: x = 2
Next, find the x & y-intercepts:
x-intercepts: 3
)2(
11)(
xxfxxg 1)(
It is the graph of shifted right 2, reflected over the x-axis, and shifted up 1.
x1
2
1
2
2
2
11
xx
x
x 2
12
x
x2
3
x
x x – 3 = 0
x = 3
![Page 34: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/34.jpg)
Graphing Rational Functions Using Transformations
c. Analyze the graph of and use it to graph
Find the domain and any vertical asymptotes:
x - 2 = 0
x = 2
Domain: 2x
VA: x = 2
Next, find the x & y-intercepts:
x-intercepts: 3
y-intercepts: 1.52
3
2
11
)20(
11)0(
f
)2(
11)(
xxfxxg 1)(
It is the graph of shifted right 2, reflected over the x-axis, and shifted up 1.
x1
![Page 35: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/35.jpg)
Properties of Rational Functions
![Page 36: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/36.jpg)
Holes (Points of Discontinuity)
x-values for a rational function that cannot exist, BUT are not asymptotes. These
occur whenever the numerator and denominator have a common factor.
Must factor both top and bottom first!!
![Page 37: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/37.jpg)
EXAMPLE: Find the domain and vertical asymptote(s).
34
1)(
2
xx
xxf
4
2)(
2
x
xxf
)3)(1(
1
xx
x a hole occurs at x + 1 = 0
Domain:
VA: x = -3
A hole occurs at x = -1
3,1 x
![Page 38: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/38.jpg)
34
1)(
2
xx
xxf
![Page 39: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/39.jpg)
EXAMPLE: Find the domain and vertical asymptote(s).
34
1)(
2
xx
xxf
4
2)(
2
x
xxf
)3)(1(
1
xx
x)2)(2(
2
xx
x
Domain:
VA: x = -3
A hole occurs at x = -1
a hole occurs at x - 2 = 0
A hole occurs at x = 2
Domain:
VA: x = -2
3,1 x 2x
![Page 40: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/40.jpg)
Horizontal Asymptotesdescribe a certain behavior of the graph as
or as , that is its end behavior.
How the graph behaves on the far ends of the x-axis.
x x
The graph of a function may intersect a horizontal asymptote.
The graph of a function will never intersect a vertical asymptote.
Always written as y =
![Page 41: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/41.jpg)
![Page 42: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/42.jpg)
Three Types of Rational Functions
Balanced…the degree of the numerator and denominator are equal
4
132)(
2
2
x
xxxf Horizontal Asymptote: b
ay
2
22
x
xH.A.
The horizontal asymptote is where y = 2.
2
![Page 43: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/43.jpg)
Three Types of Rational Functions
Bottom Heavy…the degree of the denominator is larger than the degree of the numerator.
Horizontal Asymptote: 0y
The horizontal asymptote is where y = 0.
4
3)(
2
x
xxf
![Page 44: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/44.jpg)
Three Types of Rational Functions
Top Heavy…the degree of the numerator is larger than the degree of the denominator
Has NO HORIZONTAL ASYMPTOTE
There is no horizontal asymptote.
3
2)(
2
x
xxf
has an oblique asymptote instead…
![Page 45: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/45.jpg)
Oblique (Slant) Asymptotean asymptote that is neither vertical nor
horizontal, but also describes the end behavior of a graph. Has the equation “y =“ and has an x in it. It is found by dividing the polynomial: top bottom (quotient only)
Top Heavy rational functions have oblique or
slant asymptotes instead of a horizontal asymptote.
![Page 46: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/46.jpg)
EXAMPLE: Find the oblique asymptote of
1)(
2
4
x
xxxf
Note: The textbook considers only linear equations oblique asymptotes…
divide the polynomials using long division…
00010 2342 xxxxxx
2x
4x 30x 2x2x
1
x 02x x0 1
x 1 ignore remainder
The oblique asymptote is y = x2 + 1.
![Page 47: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/47.jpg)
EXAMPLE: For each function, find the domain, x and y-intercepts, vertical asymptotes, horizontal asymptotes, and slant asymptotes.
a.4
132)(
2
2
x
xxxf
)2)(2(
)1)(12(
xx
xx
D: ___________________
x-int: _________________
y-int: _________________
VA: _________________
HA: _________________
Now find domain and vertical asymptotes
x + 2 = 0 x – 2 = 0
x = -2 x = 2
x = 2, x = -2
2x
Balanced
1st find the horizontal asymptote
22
2
2
x
xy
y = 2
![Page 48: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/48.jpg)
EXAMPLE: For each function, find the domain, x and y-intercepts, vertical asymptotes, horizontal asymptotes, and slant asymptotes.
a.4
132)(
2
2
x
xxxf
)2)(2(
)1)(12(
xx
xx
D: ___________________
x-int: _________________
y-int: _________________
VA: _________________
HA: _________________
x = 2, x = -2
2x
y = 2
Find the x-intercepts:
Find the y-intercepts:
4
1
4)0(
1)0(3)0(2)0(
2
2
f
41
2x + 1 = 0 x + 1 = 0
x = -1/2 x = -1 -1/2, -1
![Page 49: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/49.jpg)
EXAMPLE: For each function, find the domain, x and y-intercepts, vertical asymptotes, horizontal asymptotes, slant asymptotes.
b.
D: ___________________
x-int: _________________
y-int: _________________
VA: _________________
HA: _________________
Find domain and vertical asymptotes
x + 1 = 0 x – 1 = 0
x = -1 x = 1
x = 1, x = -1
1x
Bottom-heavy
y = 0
1
3)(
2
x
xxf
)1)(1(
3
xx
x
Find the x-intercepts:
Find the y-intercepts:
31
3
10
30)0(
2
f
x - 3 = 0
x = 3
3
3
![Page 50: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/50.jpg)
EXAMPLE: For each function, find the domain, x and y-intercepts, vertical asymptotes, horizontal asymptotes, slant asymptotes.
c.D: ___________________
x-int: _________________
y-int: _________________
VA: _________________
HA: _________________None
Find the oblique asymptote:
3
2)(
2
x
xxf Top-heavy
oblique asymptote: __________
0023 2 xxx
3 2 0 0
2
66
18
18y = 2x - 6
y = 2x - 6
![Page 51: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/51.jpg)
EXAMPLE: For each function, find the domain, x and y-intercepts, vertical asymptotes, horizontal asymptotes, slant asymptotes.
c.D: ___________________
x-int: _________________
y-int: _________________
VA: _________________
HA: _________________
x = -3
3x
None
Find the x-intercepts: Find the y-intercepts:
030
)0(2)0(
2
f2x2 = 0
x = 0
03
2)(
2
x
xxf
oblique asymptote: __________
Find domain and vertical asymptotes
x + 3 = 0
x = -3
0
y = 2x - 6
![Page 52: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/52.jpg)
Real and Non-real Zeros of Polynomial Functions
![Page 53: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/53.jpg)
The zeros of a polynomial function can be found by finding its factors.
The real zeros (roots) are the x-values where the graph crosses the x-axis. In this section, you will be finding both real and non-real
(imaginary) roots.
![Page 54: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/54.jpg)
Can “SEE” Real roots x-intercepts
Cannot “see” imaginary roots...must use algebraic
method, such as the quadratic formula, to find them
NON-REAL
REAL
![Page 55: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/55.jpg)
Remainder and Factor Theorems
Recall: Division Algorithm for Polynomials
)(
)()(
)(
)(
xg
xrxq
xg
xf
dividend
divisor
quotient
remainder
If the remainder is zero (0) then, g(x) divides evenly into f(x) and )()()( xqxgxf
EX: 12/4 = 3 with remainder 0, so 4 x 3 = 12
Remainder Theorem
Let f be a polynomial function. If f(x) is divided by x – c, then the remainder is f(c). f(c) = the remainder!
![Page 56: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/56.jpg)
EXAMPLE: Find the remainder if f(x) = x3 – 4x2 – 5 is divided by x – 3.
THIS CAN BE DONE IN ONE OF 3 WAYS!!
Using Synthetic Division: 3 1 4 0 5
1
3
1
3
39
14 Remainder
Using the Remainder Theorem: (3)3 – 4(3)2 – 5 = -14f(3) =
Using Graphing: Let y1 = x3 – 4x2 - 5
When x = 3, y = -14 Look at table
The remainder
is -14.
![Page 57: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/57.jpg)
Factor Theorem
Let f be a polynomial function. Then x – c is a factor of f(x) iff f(c) = 0.
1. If f(c) = 0, then x – c is a factor of f(x).
2. If x – c is a factor of f(x), then f(c) = 0.
Basically, if the remainder is zero, you have a factor and a zero/root…and if you have a factor or zero/root, the remainder will be zero…
![Page 58: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/58.jpg)
EXAMPLE: Determine whether x – 1 is a factor of f(x) = 2x3 – x2 + 2x – 3. If so, then factor f(x).
1st check for a remainder of 0… 2(1)3 – (1)2 + 2(1) – 3 = 0f(1) =
Since the remainder is 0, then (x – 1) is a factor of f(x).
Now factor f(x) using synthetic division…
1 2 1 2 3
2
2
1
1
3
3
02x2 + x + 3
cannot be factored any further...
)32)(1()( 2 xxxxf
![Page 59: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/59.jpg)
Complex Zeros (Roots) of a Polynomial Function
Fundamental Theorem of Algebra
Every polynomial function f(x) of degree n has exactly n numbers of real + imaginary zeros...that is, there are exactly n complex zeros.
Furthermore, a polynomial of odd degree has at least one real zero. WHY?? Goes in opposite directions, so it
MUST go through the x-axis!
Complex Roots (Conjugate Pairs) Theorem
Let f be a polynomial function. If a + bi is a complex zero of f, then a – bi is also a zero of the function.
Irrational AND Imaginary zeros must come in pairs!
![Page 60: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/60.jpg)
3 are listed, so there are 2 more...3 + 2 = degree 5
imaginary and irrational zeros must come in conjugate pairs...
![Page 61: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/61.jpg)
Steps for finding zeros (roots) of polynomial functions:
1. Determine the number of real and non-real roots the function will have by graphing.
2. Find the real zeros (x-intercepts) on your graph. If no real zeroes, then polynomial WILL BE FACTORABLE.
3. Factor the function using synthetic division. Continue to factor until you get a quadratic factor.
4. Solve each of the factors for the roots. Answer in exact form (not decimals).
where it crosses x-axis
real
TOTAL ZEROS = DEGREE
f(c) = 0
or get a polynomial that is factorable...
exact form
First check to see if the polynomial can be factored by “normal” means!!
![Page 62: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/62.jpg)
EXAMPLE: Factor and find the zeros of the polynomial function.
a. 67112)( 23 xxxxf
Step 1: Graph and find # of real & non-real zeros
crosses the x-axis 3 times, so there are 3 real and 0 non-real
![Page 63: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/63.jpg)
EXAMPLE: Factor and find the zeros of the polynomial function.
a. 67112)( 23 xxxxf
Step 2: Find the real zeros on your graph.
Is 1 a zero? Does f(1) = 0?
Is -6 a zero? Does f(-6) = 0? Yes
All three are real and can be found using your calculator!
Yes
Is -1/2 a zero? Does f(-1/2) = 0? Yes
So, f(x) factored is (x – 1)(2x + 1)(x + 6) and the zeros are x = -6, -1/2, 1
![Page 64: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/64.jpg)
EXAMPLE: Factor and find the zeros of the polynomial function.
b. 483284)( 2345 xxxxxxf
crosses the x-axis once and touches once...
What are the possibilities?
![Page 65: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/65.jpg)
Odd multiplicity Even multiplicity
3 real: multiplicity of 1 and multiplicity of 2 + 2 non-real
![Page 66: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/66.jpg)
EXAMPLE: Factor and find the zeros of the polynomial function.
According to the graph, the real zeros are at x = -3 and x = 2
We must use synthetic division since we cannot factor by grouping. Choose one of the real zeros to use for the division.
Let’s start with x = -3…
b. 483284)( 2345 xxxxxxf
![Page 67: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/67.jpg)
EXAMPLE: Factor and find the zeros of the polynomial function.
b. 483284)( 2345 xxxxxxf
If x = -3, then... 3 1 1 4 8
1
34
12
8
2416
161684 234 xxxx
32 4848
16
48
0
Not a quadratic and not factorable by grouping, so divide/factor again using synthetic division and another real zero…
![Page 68: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/68.jpg)
EXAMPLE: Factor and find the zeros of the polynomial function.
b. 483284)( 2345 xxxxxxf
2
1
2
2 4
14
4 8 16 842 23 xxx16816
8 0
Use x = 2 to factor x4 – 4x3 + 8x2 – 16x + 16 further…
)2(2 xx )2(4 x
)4)(2( 2 xx
So, f(x) factored is (x + 3)(x - 2)(x - 2)(x2 + 4) =
(x + 3)(x – 2)2(x2 + 4)
![Page 69: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/69.jpg)
EXAMPLE: Factor and find the zeros of the polynomial function.
b. 483284)( 2345 xxxxxxf
)4()2)(3()( 22 xxxxf
3x 2xmult. 2
042 x
42 x
ix 24
The zeros are -3, 2 (mult 2), 2i and -2i.
![Page 70: Ch 3 review Quarter test 1 And Ch 3 TEST. Graphs of Quadratic Functions Where a, b, and c are real numbers and a 0 Standard Form Domain: all real numbers](https://reader034.vdocuments.site/reader034/viewer/2022051517/5697bfd41a28abf838cac97f/html5/thumbnails/70.jpg)
EXAMPLE: Factor and find the zeros of the polynomial function.
c. 18452553)( 234 xxxxxf
)9)(13)(2()( 2 xxxxf
iix 3,3,31,2