5.4 - Quadratic Functionserinfry/math150/supporting_docs/notes/Math_15… · It is nice when a quadratic function is given in standard form _____ because it makes it easy to see that

5.4 - Quadratic Functions

Definition: A _____________________ function is one that can be written in the form

f (x) = ___________________________ where a, b, and c are real numbers and a≠ 0.

(What do we have if a=0? _______________________________________________)

This form is called the _________________________ of a _____________________

The quadratic form f (x) = a(x − h)2 + k is called ______________________________

The graph of a quadratic function is called a _________________________________

Definition: Extreme functional values are the maximum and/or minimum values of the

function.

The x-intercepts of a quadratic function are also called the _____________or the __________.

1. Let g(x) = −(x − 2)2 +1

Determine the

a) vertex ______________

b) axis of symmetry _______________

c) maximum value of the function __________

d) x-intercepts ________________________

©Fry TAMU Spring 2017 Math 150 Notes Section 5.4 Page! 92

It is nice when a quadratic function is given in standard form ___________________________

because it makes it easy to see that the vertex is at __________________________________

Also determining the x-intercepts is relatively easy.

When a quadratic function is given in general form ___________________________________

it is easy to determine that the y-intercept is _________________, but determining the vertex

is no longer easy. Sometimes you will want to convert a quadratic function from general form

into standard form by _______________________________________________

Observations for understanding completing the square:

Expand the following expressions, looking for patterns:

(x − 3)2 = x2 ____ x ____ (x + 4)2 = x2 ____ x ____

(x − 5)2 = x2 ____ x ____ (x − 9)2 = x2 ____ x ____

Now work backward...

x2 +16x ____ = (x _____)2 x2 + 20x ____ = (x _____)2

x2 −14x ____ = (x _____)2 x2 −12x ____ = (x _____)2


Converting quadratics from general form to standard form

2 Let f (x) = x2 − 2x + 4

Put the function in standard form to determine the

a) vertex ______________

b) axis of symmetry ______________

c) minimum value of the function____________

d) x-intercepts ________________________

3. Let g(x) = x2 − 8x +17


a) vertex ______________



d) x-intercepts ________________________


4. Let h(x) = x2 − 5x + 6


a) vertex ______________



d) x-intercepts ________________________

5. Let f (x) = 3x2 +12x + 4


a) vertex ______________



d) x-intercepts ________________________


6. Let g(x) = −4x2 +12x + 7


a) vertex ______________


c) maximum value of the function____________

d) x-intercepts ________________________

7. Consider the function f (x) = ax2 + bx + c

Determine

a) the vertex ______________! b) the axis of symmetry _________________________

c) the extreme value of the function_________________________


d) x-intercepts ___________________________

We have just derived ________________________________________

When ax2 + bx + c = 0 , x = ____________________________________

Notice that axis of symmetry is always half way between the zeros.

b2 − 4ac is called _________________________________

It is often denoted by an uppercase ______________ D = __________________

When a quadratic function starts in general form but is converted to standard form, and the x-intercepts are determined from standard form, it is called, “solving by completing the square.”

When a quadratic function starts in general form, but it is not necessary to determine the vertex, the quadratic formula can be used to determine the x-intercepts.

Now here is a third technique for determining the x-intercepts of a quadratic function.

Finding the Zeros of Quadratic Functions by Factoring

Another technique for solving quadratic equations is by factoring. This technique is based on

the Zero-Product Principle. Think about it this way:

If a, b∈! and if ab =1 , must either a =1 or b =1 ? _____________________________

If ab = 0 , must either a = 0 or b = 0 ? _____________________________


To find where the graph of f (x) = x2 + 7x + 6 intersects the x-axis, we solve x2 + 7x + 6 = 0 .

x2 + 7x + 6 = ______________________,

So if _____________________________ = 0

then either __________________ = 0 or ________________ = 0

so ___________________ or ___________________

are solutions to the equation x2 + 7x + 6 = 0

Consider the quadratic function f (x) = x2 + x − 6 .

f (−3) = ___________________________ and f (2) = ___________________________

and f (x) can be factored so that f (x) = _______________________________________

This is not a coincidence!

Consider the quadratic function f (x) = ax2 + bx + c .

f (m) = 0 and f (n) = 0 iff f (x) can be factored so that f (x) = a x −m( ) x − n( )

In other words if a parabola crosses the x -axis at say x = 2 and x = −4 , then the factors of the

parabola’s quadratic function are ___________ and ___________

If you know that a parabola intersects the x -axis at x = −3

and x = 2 , can you be sure that its function is

f (x) = (x + 3)(x − 2) ? _______________

It could be that f (x) = _________________________

or f (x) = _________________________


Factoring quadratics, lead coefficient 1:

To factor x2 +10x +16 , determine if there are 2 numbers whose product is _______ and

whose sum is ___________. Those two numbers are ______ and ______ so

x2 +10x +16 = ___________________________________________________________

To factor x2 − 2x − 48 , determine if there are 2 numbers whose product is _______ and

whose sum is ___________. Those two numbers are ______ and ______ so

x2 − 2x − 48 = __________________________________________________________

Factoring quadratics with lead coefficient is not 1

Use the Blankety-Blank Method!

To factor: 10x2 +11x + 3 determine if there are 2 numbers whose product is _______

and whose sum is ___________ Those two numbers are _________ and ________

Now rewrite 10x2 +11x + 3 =___________________________________________________

To factor: 3x2 −14x − 5 determine if there are 2 numbers whose product is _______

and whose sum is ___________ Those two numbers are _________ and ________

Now rewrite 3x2 −14x − 5 =____________________________________________________

Important Fact: a2 + b2 does not factor over the real numbers. So x2 + 4 cannot be factored

using real numbers.

Notice that the graph of f (x) = x2 + 4 does not intersect the x -axis.


Factor using special patterns:

Look what happens when you multiply and simplify:

(x − 3)(x + 3)= _________________________________

Difference of Squares a2 − b2 =

Square of a Binomial a2 + 2ab + b2 =

a2 − 2ab + b2 =

8. Find the x-intercepts of the following functions by factoring

a) f (x) = x2 −144 b) g(x) = 16x2 − 81

c) h(x) = x2 +10x + 25 d) p(x) = 16x2 + 8x +1

e) r(x) = 6x2 + 7x − 5 f) v(x) = 20x2 + 7x − 3

9. Find the x-intercepts of the following functions using the quadratic equation

a) f (x) = x2 −144 b) g(x) = 16x2 − 81


10. Given the parabola on the right, determine the quadratic function that yields this graph. Then find the determinant of that quadratic.

When the discriminant is ___________________ there are _____________ real roots.

In these cases, the parabola intersects the x -axis ____________________________.


11. Given the parabola on the left, determine the quadratic function that yields this graph. Then find the determinant of that quadratic.

When the discriminant is ___________________ there is ________________ real root.

It is sometimes called a __________________________________.

In these cases, the parabola intersects the x -axis ____________________________.


12. Given the parabola on the left, determine the quadratic function that yields this graph. Then find the determinant of that quadratic.

When the discriminant is ___________________ there are __________ real roots.

In these cases, the parabola ___________________ intersect the x -axis.


Applications

13. A toy rocket is launched from a 304 foot cliff. If the height, in feet, of the rocket is given by .h(t) = −16t 2 + 288t + 304 where t is the number of seconds after liftoff.

Notice h(0) = __________________________________________

This corresponds to the rocket starting ____________________________________

What kind of function is h(t) ? __________________________________________

So the graph of this function is a ________________________________________

The maximum value of this function will occur at the _________________

Rewrite h(t) in standard form:

a) find the maximum height of the rocket. ______________________________

b) How long does it take for the rocket to reach that maximum height? ___________

c) When does the rocket hit the ground? ______________________________________

d) What is the domain of h(t) ? _______________________________


14. Joe has 30 ft of fence to make a rectangular kennel for his dogs, but plans to use his garage as one side. What dimensions produce the greatest area?

Definition: Extreme functional values are the maximum and/or minimum values of the function on its domain.

Minimum value __________ Minimum value __________ Minimum value __________

Maximum value __________ Maximum value __________ Maximum value __________


Local ExtremaThere is a local maximum at x = a if there is an open interval I containing a , and

_____________________________ ∀ x ∈ I

It is also important to notice on the graph that at a local maximum, the graph changes

from _______________________________________________

There is a local minimum at x = b if there is an open interval I containing b , and

_____________________________ ∀ x ∈ I

It is also important to notice on the graph that at a local minimum, the graph changes

from ____________________________________________

15. Consider the graph of the function f (x) = x3 − x2 − 6x .

a) What is the approximate value for the

local maximum ? _________________

b) About where does this local maximum

occur? _______________________

c) Find an approximate value for the local

minimum._______________________

d) Indicate the approximate location of this minimum value. ________________

Note: Sometimes you will want to know the maximum (or minimum) value of the function.

! Sometimes you will want to know where (for what x value) the maximum value occurs.


Extra Problems for 5.4 -- Quadratic Functions

1. The maximum value of the function g(x) = (5 − x)(x + 3)6

is ________________

2. Here is a quadratic function in general form: f (x) = −3x2 + 24x + 28

Convert it to standard form: _________________________________________________

3. Determine the coordinates of the x-intercepts of the graphs of

a) f (x) = 5x2 − 7x + 2

b) g(x) = 4x2 + 28x + 49 c) h(x) = 4x2 + 25x + 36


4. As part of a holiday tradition physics students went to the roof of the building and fired an orange projectile straight up into the air. They were able to determine that the height in feet of the projectile could be described by the function f (t) = −16t 2 + 64t + 96 where t is time in seconds after the projectile was fired into the air.

a) How tall was the building? ___________________________ ft

b) How high did the projectile go? ________________________ft

c) How long until it smashed onto the ground? ______________________sec

5. If ax2 + bx + c = 0 ,

! ! ! then x = _______________________________________


2.2 - Factoring (Polynomials)Which of these are examples of polynomials?

Polynomial Not a polynomial

Polynomials should not have

x2 + 2x +1

x7 + π

2( ) x2 + 2x +14 x −1

1x+ 3x2

5x3 + 3x2 − 2( )14

5x3 + 3x2 − 2( )2

x2 + sin x

5

Polynomials are expressions of the form anxn + an−1x

n−1 +!+ a1x + a0 where ai ∈! and n∈!

n is called ___________________________________________

an is called ___________________________________________

a0 is called ___________________________________________

If n = 2, the polynomial is called a _____________________________ Example:

If n = 3, the polynomial is called a ______________________________ Example:

A polynomial with 2 terms is called a _____________________________ Example:

A polynomial with 3 terms is called a _____________________________ Example:

©Fry TAMU Spring 2017 Math 150 Notes Section 2.2 Page 109

Factoring polynomials helps us when we need to find the zeros of polynomials.

Techniques for Factoring Polynomials

Common Factors: 12x4 + 4x2 + 2x = ____________________________________

Factor by Grouping: 3x3 + x2 −12x − 4 (Factoring by grouping is your best bet if you have a long cubic!)

3x3 + x2 −12x − 4 = ___________________________________________

Look what happens when you multiply

(a + b)(a2 − ab + b2 )

Similarly look what happens when you multiply

(a − b)(a2 + ab + b2 )

Sum of Cubes a3 + b3

Difference of Cubes a3 − b3


1. Factor

a) 27 − x3 b) 64x3 +1

c) x6 + 8 d) x5 − 4x3 − x2 + 4

e) 3x3 + 6x2 −12x − 24 f) 16x4 − 25x2

Factoring polynomials in quadratic form

2. Factor

a) x4 − 5x2 − 6 b) x6 + 2x3 +1


c) 4x12 − 9x6 + 2

Factoring some non-polynomials with similar techniques3. Factor

a) x + 3 x + 2 b) 12x23

⎛⎝⎜

⎞⎠⎟ + 20x

13

⎛⎝⎜

⎞⎠⎟ + 3

c) x2 + 3x + 4x13

⎛⎝⎜

⎞⎠⎟


Extra Problems for 2.2 -- Factoring

1. Factor completely over the integers

a) 27x4 − 64x = _____________________________________________________________

b) 8ax − 6x −12a + 9 = = ______________________________________________________

c) 169a6 −121b4 = __________________________________________________________

d) 6x2 +13x − 5 = __________________________________________________________

2. Factor completely over the real numbers x2 − 2 = ___________________________

3. Factor completely x2 + 2 = _______________________________________________


Chapter 6.1 (Graphing) Polynomial FunctionsPolynomial Degree Leading Term Constant

Term

f (x) = 4x3 + 2x − 5

g(x) = −17x5 + 6x3

h(x) = πx6 − 17

c(x) =14

p(x) = pnxn + pn−1x

n−1 +!+ p2x2 + p1x + p0

where all of the exponents are non-negative integers, the pi's are real numbers and pn ≠ 0

r(x) = −2 + 3x − 5x4

When we graph polynomials, we’ll pay special attention to

! the x and y intercepts

! where the graph is above or below the x-axis

! the behavior of the function as x approaches −∞ .

! the behavior of the function as x approaches ∞ .

We will use notation like this: “As x→∞ , x3 → ∞ .” It reads, “As x goes to infinity, x cubed

goes to infinity.” In this case it means that when the values of x are very large and positive, x

cubed is very large and positive. Notice how the graph of the function goes up on the right

side.

“As x→−∞ , x3 → −∞ .” This reads, “As x goes to negative infinity, x cubed goes to

negative infinity.” It means that when the values of x are very large and negative, x cubed is

very large and negative. In this case, the graph of the function goes down on the left side.

Later we will study functions that level off for large values of x and other functions that oscillate

forever. In calculus you will learn a precise definition for “very large.”


Cubics:

f (x) = x3 ! ! ! ! ! ! ! ! g(x) = −(x − 2)3

Constant Term: Constant Term:

y-intercept: y-intercept:

Leading Term: Leading Term:

As x→∞ , x3 → As x→∞ , −(x − 2)3

As x→−∞ , x3 → As x→−∞ , −(x − 2)3

x-intercept: x-intercept:

Notice that the graph only intersects the x-

axis in one place -- at x = ________.

This is the only place that the function

changes sign.

When x < 0, x3 < 0

When x > 0 x3 > 0

Notice that the graph only intersects the x-

axis in one place -- at x = ________.

This is the only place that the function

changes sign.

When x < 2, −(x − 2)3 > 0

When x > 2 −(x − 2)3 < 0


h(x) = −2(x + 3)(x −1)2

Constant Term: _________________

So the y-intercept is _______________

Leading Term: __________________

As x→∞ , −2(x + 3)(x −1)2 __________

As x→−∞ , −2(x + 3)(x −1)2 __________

x-intercepts:______________________________

Now we would like to solve −2(x + 3)(x −1)2 > 0 (because when −2(x + 3)(x −1)2 > 0, the graph of h(x) will be ____________ the x-axis.)

Redraw the x-axis

and plot the zeros

list the factors

to create a sign table

determine the sign of each factor

use the signs of the factors

to determine the sign of the product

h(x) > 0 when ________________________________

so that is where the graph of h(x) is above the x-axis


p(x) = 14

⎛⎝⎜

⎞⎠⎟ (x + 2)(x − 2)(3x − 4)

Constant Term: _________________

So the y-intercept is _______________

Leading Term: __________________

As x→∞ , 14

⎛⎝⎜

⎞⎠⎟ (x + 2)(x − 2)(3x − 4) ____________

As x→−∞ , 14

⎛⎝⎜

⎞⎠⎟ (x + 2)(x − 2)(3x − 4) ____________

x-intercepts:_________________________

Solve p(x) > 0 .

Draw a number line

and plot the zeros

list the factors

to create a sign table

determine the sign of each factor

use the signs of the factors

to determine the sign of the product

p(x) > 0 when ________________________________

so that is where the graph of p(x) is above the x-axis


True or False:

Some cubics never intersect the x-axis. _____________________

Some cubics intersect the x-axis in exactly one place. _____________________

Some cubics intersect the x-axis in exactly two places. _____________________

Some cubics intersect the x-axis in three places. _____________________

Some cubics intersect the x-axis in four places. _____________________

Quarticsf (x) = x4 +1 ! ! ! ! ! ! ! g(x) = − 1

4⎛⎝⎜

⎞⎠⎟ (x − 2)

4


y-intercepts: y-intercepts:

x-intercepts: x-intercepts:


As x→∞ , x4 +1 As x→∞ , − 14

⎛⎝⎜

⎞⎠⎟(x − 2)4

As x→−∞ , x4 +1 As x→−∞ , − 14

⎛⎝⎜

⎞⎠⎟(x − 2)4


h(x) = −x3(2x + 7)

y-intercept _______________

Leading Term: __________________

As x→∞ , −x3(2x + 7) ______________

As x→−∞ , −x3(2x + 7)_______________

x-intercepts:_________________________

Solve h(x) = −x3(2x + 7) > 0

p(x) = (x2 −1)(x + 2)2

y-intercept _______________

Leading Term: __________________

As x→∞ , p(x) = (x2 −1)(x + 2)2______________

As x→−∞ , p(x) = (x2 −1)(x + 2)2______________

x-intercepts:_________________________

Solve p(x) = (x2 −1)(x + 2)2 > 0


Polynomials of Higher Degree

In general, to understand the behavior of a polynomial,

1. Plot the y-intercept

2. Determine the behavior of the polynomial for large positive values of x and for large negative values of x. The behavior of the polynomial at these extremes will be dominated by the leading term, the term with the highest power of x.

3. Find the x-intercepts

4. Determine where the polynomial is positive and negative because this will tell you where the graph is above and below the x-axis.

f (x) = x5 ! ! ! ! ! ! ! g(x) = −x5 ! ! !



As x→∞ , x5 As x→∞ , −x5

As x→−∞ , x5 As x→−∞ , −x5

x-intercepts: x-intercepts:

x5 > 0 −x5 > 0


h(x) = (3x −1)2(x − 2)(x +1)(x + 3) ! ! ! ! ! ! !

y-intercept is _______________

Leading Term: __________________

As x→∞ ,

h(x) = (3x −1)2(x − 2)(x +1)(x + 3) ___________

As x→−∞ ,

h(x) = (3x −1)2(x − 2)(x +1)(x + 3) ___________

x-intercepts:_________________________

Solve h(x) = (3x −1)2(x − 2)(x +1)(x + 3) > 0


f (x) = x(x − 3)2(2x + 5)2

Degree of the polynomial:

y-intercept:

Leading Term:

As x→∞ , x(x − 3)2 (2x + 5)2

As x→−∞ , x(x − 3)2 (2x + 5)2

x-intercepts:

Solve: x(x − 3)2 (2x + 5)2 > 0


True or False: All linear functions intersect the x-axis.

True or False: All polynomial functions of degree zero intersect the x-axis.

True or False: All polynomial functions of degree one intersect the x-axis.

True or False: All quadratic functions intersect the x-axis.

True or False: All cubic functions intersect the x-axis.

True or False: All quartic functions intersect the x-axis.

True or False: All quintic functions intersect the x-axis.

True or False: All polynomials functions of even degree intersect the x-axis.

True or False: All polynomials functions of an odd degree intersect the x-axis.


Extra Problems for Section 6.1

1. a) If f (x) = 7x5 + 6 , then as x→−∞ , f (x)→ _______________

b) If f (x) = −7x5 + 6 , then as x→−∞ , f (x)→ _______________

c) If f (x) = 7x4 + 6 , then as x→−∞ , f (x)→ _______________

d) If f (x) = −7x4 + 6 , then as , x→−∞ , f (x)→ _______________

2 a) f (x) = x x −1( )2

b) g(x) = −x x −1( )2

c) h(x) = x x +1( )2

d) p(x) = −x x +1( )2

e) r(x) = −x x +1( )(x −1)

3 a) f (x) = x x +1( ) x −1( )

b) g(x) = −x x +1( ) x −1( )

c) h(x) = x2 x +1( )2 x −1( )

d) p(x) = −x2 x +1( )2 x −1( )

e) r(x) = −x2 x +1( ) x −1( )2


4 a) f (x) = x x +1( )2 x −1( )

b) g(x) = −x x +1( )2 x −1( )

c) h(x) = x x +1( ) x −1( )2

d) p(x) = −x x +1( ) x −1( )2

e) r(x) = − x +1( )2 x −1( )2

5 a) f (x) = x +1( )2 x −1( )3

b) g(x) = − x +1( )2 x −1( )3

c) h(x) = x +1( )3 x −1( )2

d) p(x) = − x +1( )3 x −1( )2

e) r(x) = x x +1( ) x −1( )

6 a) f (x) = x +1( ) x −1( )3

b) g(x) = − x +1( ) x −1( )3

c) h(x) = x +1( )2 x −1( )3

d) p(x) = − x +1( )2 x −1( )3

e) r(x) = x +1( )2 x −1( )2


2.1 - (Dividing) Polynomials

Dividing Polynomials (This is a lot like long division of integers!)

1. 4x3 − x2 − 5x + 2

x −1 =

What is the dividend? ___________________________ What is the divisor? ____________

What is the quotient? _______________________________


2. 3x3 − 2x2 + 5x2 −1

= _________________________________________!

What is the dividend? ___________________________ What is the divisor? ____________

What is the quotient? _______________________________


3. Use polynomial long division to determine

16x4 + 8x3 + 2x +14x2 − 2x +1

= _________________________________


6.2 - Rational Functions

Definition: A rational function is ___________________________________________

The domain of a rational function is all real numbers, except those values where the

denominator is ____________

1. g(x) = 1x +1

a) domain _____________________

b) y -intercept __________ c) x -intercept(s) __________

A fraction is zero when its _________________________

is zero and its ____________________ is NOT zero.

That is why the graph of 1x +1

never ____________________

the ____________________. (The numerator is never zero.)

d) vertical asymptote _________________ Vertical asymptotes occur where the

________________________ is zero, but the ___________________________ is not zero.

e) critical value(s) for g(x) __________________ f) Solve g(x) > 0 _____________

g) for large x , g(x) acts like _________________________ , (This is the quotient of the

_____________________________________.)

so as x→∞ , g(x) --> _____________ and as x→−∞ g(x) --> _____________

When the value of a function approaches a constant value for large values of x, we say that

the graph of the function has a ________________________________________________

The graph of g(x) has a horizontal asymptote of __________________________

©Fry TAMU Spring 2017 Math 150 Notes Section 6.2 Page !129

2. h(x) = xx +1

a) domain _________________

b) y -intercept ______________!

c) x -intercept(s) ______________

d) vertical asymptote _________________

e) critical value(s) for h(x) ____________________ (Critical values are those that make either the

numerator or the denominator zero.)

f) Solve h(x) > 0 _________________

g) for large x , (think about estimating) h(x) acts like ____________________________ ,

(This is the quotient of the _______________________________)

so as x→∞ , h(x) --> _____________ and as x→−∞ h(x) --> _____________

Here we say that h(x) has a horizontal asymptote of ___________________


3. p(x) = x2 +1x +1

a) domain _________________

b) y -intercept ______________!

c) x -intercept(s) ______________

d) vertical asymptote _________________

e) critical value(s) for p(x) ________________ f) Solve p(x) > 0 _________________

g) For large x , (think about estimating) p(x) acts like ___________________________ ,

(This is the quotient of the ______________________________________)

so as x→∞ , p(x) --> _____________ and as x→−∞ p(x) --> _____________

Does the graph of p(x) have a horizontal asymptote? ____________________

Some people would say that the graph of p(x) has a ______________________ asymptote.


A hole in the graph indicates a place where the function is undefined, but the function’s behavior is not asymptotic.

Understanding Holes in Graphs:

4. f (x) = x2 −1x +1

! a) domain _________________

b) y -intercept ______________!

c) x -intercept(s) ______________

d) vertical asymptote(s) _________________

e) there is a hole at _____________________

f) the y - coordinate of hole _______________________

g) critical value(s) for f (x) _____________________ h) Solve f (x) > 0 ______________

i) For large x , f (x) acts like _________________________________ ,

so as x→∞ , f (x) --> _____________ and as x→−∞ f (x) --> _____________

j) horizontal asymptote ____________________


5. g(x) = x2 −1x2 +1

!

a) domain _________________

b) y -intercept ______________!

c) x -intercept(s) ______________




g) critical value(s) for g(x) _____________________ h) Solve g(x) > 0 ______________

i) For large x , g(x) acts like _________________________________ ,

so as x→∞ , g(x) --> _____________ and as x→−∞ g(x) --> _____________



6. h(x) = x3 − 273 x2 − 9( )(x −1)

!

a) domain ______________________________

b) y -intercept ______________!

c) x -intercept(s) ______________




g) critical value(s) for h(x) _____________________ h) Solve h(x) > 0 ______________

i) for large x , h(x) acts like _________________________________ ,

so as x→∞ , h(x) --> _____________ and as x→−∞ h(x) --> _____________



7. p(x) = 2x3 + 6x2

x3 + 3x2 − 4x −12

a) domain _______________________________

b) y -intercept ________________!

c) x -intercept(s) ______________


e) there is a hole at ________________ f) the y - coordinate of hole ________________

g) critical value(s) for p(x) _____________________ h) Solve p(x) > 0 ______________

i) For large x , p(x) acts like _________________________________ ,

so as x→∞ , p(x) --> _____________ and as x→−∞ p(x) --> _____________



To graph a rational function,

a) Evaluate the function at x = 0, this is the ______________________________

b) Find the values for x for which the numerator is zero, but the denominator is not zero.

This is where the graph ____________________________________________

c) Find the values for x for which the denominator is zero, but the numerator is not zero.

This is where the graph ___________________________________________

d) Find the values of x for which both the numerator and the denominator are zero.

This is where there is _______________________________________

e) To find the y - coordinate of the hole: If there is a hole at x = a , then x − a( ) is a factor of

both the numerator and the denominator. The rational function f (x) can be written in the

form f (x) =(x − a)x − a( )

⎡

⎣⎢

⎤

⎦⎥p(x)q(x)

⎡⎣⎢

⎤⎦⎥ . It could be that p(x) = 1and/or q(x) = 1 .

! Let f ∗(x) = p(x)q(x)

, then the y - coordinate of the hole is f ∗ a( ) .

f) Simplify the quotient of the leading terms of the numerator and the denominator. The end

behavior of this function is the same as the end behavior of the given function.

g) If the function tends to a constant c as x gets very large, then we say that the graph has a

horizontal asymptote of y = c .

h) Determine where the function is greater than 0.

! This is where the graph of the function is __________________________.



1. f (x) = x2 − x −122(x2 −16)

a) List the coordinates of all of the x -intercepts.___________________________! If there are none, write NONE in the blank provided.

b) List the equations of all of the vertical asymptotes.________________________! If there are none, write NONE in the blank provided.

c) List the equations of all of the horizontal asymptotes._______________________! If there are none, write NONE in the blank provided.

d) List the coordinates of all the holes.______________________________! If there are none, write NONE in the blank provided.

e) On what intervals is the graph of f (x) = x2 − x −122(x2 −16)

above the x-axis?

Write your answer using interval notation: _________________


2. g(x) = 8x5 − 3

7x4 −1 ! ! as x→∞ , g(x) -->

3. h(x) = 8x4 − 3

7x5 −1 ! ! as x→∞ , h(x) -->

4. f (x) = x2 + 5x + 6x2 + 2x

State the domain of f (x) ______________________________

b) List the coordinates of all of the x -intercepts. ___________________________

c) List the coordinates of all of the y -intercepts. ___________________________

d) List the equations of all of the vertical asymptotes.___________________________

e) List the coordinates of all the holes. __________________________

f) List the equations of all of the horizontal asymptotes ______________________________


Bonus Section: Rational Expressions (This is not in your text book.)

A rational expression is the quotient of two _________________________________

In this section we’ll add, subtract, multiply, and divide rational expressions. Our goal will be to

simplify the result so that it is expressed as the quotient of two factored polynomials.

We will also need to be aware of possible restrictions on the values of our independent

variable(s), usually x . Rational expressions are undefined when _________________

So the rational expression xx

= ___________ as long as _____________________

1. For the following rational expressions, list any restrictions that exist for x .

a) x −1x

! ! ! ! ! ! Restrictions on x : _____________________

b) x + 2x2 + 4

! ! ! ! ! ! Restrictions on x : _____________________

c) x7 − x4

x 5−x3! ! ! ! ! ! Restrictions on x : ______________________

What is the difference between a rational expression and a rational function? Very little.

x +1x + 2

is a rational expression whereas f (x) = x +1x + 2

is a rational function.

Simplifying rational expressions is like reducing rational numbers: 1518

= 33

⎛⎝⎜

⎞⎠⎟56

⎛⎝⎜

⎞⎠⎟ =

56

Notice that 1518

= 3+126 +12

, but we would not cancel the 12’s!

When simplifying rational expressions we look for common ______________________ in the numerator and denominator.

©Fry TAMU Spring 2017 Math 150 Notes Bonus Section I Rational Expressions Page!139

In the previous section, the rational functions were already factored. Sometimes they are found in a messier form. In this section we’ll practice the algebraic skills necessary to write a given rational function as the quotient of two factored polynomials.

2. f (x) = 2x + 2

+ 3x − 3

Domain ________________________


3. g(x) = x − 4x +1

− x2 − 8x +16x2 −1

Domain ________________________

4. h(x) = 3

x + 2⎛⎝⎜

⎞⎠⎟ i

x2 − x − 2x2 −1

⎛⎝⎜

⎞⎠⎟

Domain ________________________


5. f (x) = x2 − x − 6x + 4

÷ (x2 + 7x +12)

Domain ________________________

6. g(x) =x3 −1x +1x −1

x2 + 2x +1

! ! ! Domain ________________________


A compound or complex fraction is an expression containing fractions within the numerator and/or the denominator. To simplify a compound fraction, first simplify the numerator, then simplify the denominator, and then perform the necessary division.

7. h(x) =3x+ 12x − 75x+1

Begin by getting a common denominator in the numerator and a common denominator in the denominator.


This is a function of 2 variables. We won’t try to graph it, we’ll just simplify it.

8. f (x, h) =

3x + h( )2 −

3x2

h

Note: In the traditional presentation of Math 150, students learn to simplify rational expressions before the concepts of function and domain are defined. So in some WebAssign problems for 1D, you’ll be asked for the “restrictions on x.” You will want to report the values of x that make the function undefined even before it is simplified.


Extra Problems for Bonus Section I Rational Expressions:

1. Fully simplify: a) 3x−1 + 7x−2

3+ 4x−1 − 7x−2

b) 3x2

3x− 3x2

c) 9 − 1

x2

9 + 6x+ 1x2


9.2 - Trigonometric Functions (the second part)

Another way to define the trigonometric functions:

Given a circle of radius one, (often called the unit circle)

with a radial line drawn at an angle θ , measured

counterclockwise from the positive x-axis, the radial line

intersects the circle at a point (x, y). The trigonometric

functions can then be defined as

sinθ cosθ tanθ

cscθ secθ cotθ

Find cos 3π4

⎛⎝⎜

⎞⎠⎟


Let θ be an angle in standard position. The reference angle θR is the acute

angle formed by the terminal side of θ and the x -axis.

cos 5π6

⎛⎝⎜

⎞⎠⎟

sin 4π3

⎛⎝⎜

⎞⎠⎟


sin −π6

⎛⎝⎜

⎞⎠⎟

tan 2π3

⎛⎝⎜

⎞⎠⎟

Reference Angle Theorem: Let trig θ( ) be any one of the six trigonometric functions defined above (on page 146).

! ! ! ! Then trig θ( ) = ±trig θR( )! !! ! The correct sign is determined by the quadrant of θ .


tan 7π6

⎛⎝⎜

⎞⎠⎟

csc 5π3

⎛⎝⎜

⎞⎠⎟


1. Given that sinθ = 35

and θ is acute, determine the values of

a) cosθ = _____________ b) tanθ = _______________

2. Given that sinθ = 35

and θ is NOT acute, determine the values of

a) cosθ = _____________ b) tanθ = _______________


3. Given that tanθ = −2 and cosθ < 0 determine the value of sinθ .

4. Given that secθ = 75

and tanθ < 0 determine the value of tanθ .



If tanθ = − 65

and cosθ < 0 , then sinθ = ________

From http://www.stitz-zeager.com/szprecalculus07042013.pdf pages 737 and 738


http://www.stitz-zeager.com/szprecalculus07042013.pdf

http://www.stitz-zeager.com/szprecalculus07042013.pdf

3.1 Solving EquationsThe solution of an equation is a value (or a set of values) that yields a true statement (an identity) when substituted for the variable in an equation.

The word “solve” means determine ______________________________________

Solving Linear Equations

Linear equations are equations involving only polynomials of degree one.

Examples include 2x +1= −4 and 4x + 2 = −x − 3

The algebraic techniques to find the solutions to these equations are simple, but I want you to keep in mind that there is a geometric interpretation associated with the equation. We are looking for the intersection of the graphs of two linear functions. The solutions we find are the x-coordinates of the intersections of the graphs.

Here are the graphs of ! ! ! ! ! ! Here are the graphs of

______________ and! ! ! ! ! ! ______________ and

______________ ! ! ! ! ! ! ! ______________

See how the lines intersect at ! ! ! ! ! See how the lines intersect at

x =______________! ! ! ! ! ! ! x =______________

See how substituting x =________! ! ! ! See how substituting x =_______

into 2x +1= −4 ! ! ! ! ! ! ! ! into 4x + 2 = −x − 3

makes the statement true.! ! ! ! ! ! makes the statement true.


Solving Quadratic Equations

Quadratic equations are equations involving only polynomials of degree two.

Examples include x2 − x − 6 = 2x − 2 ! and! 5x2 − 5x = 2x2 − 3x +1

Geometrically, such equations could represent the intersection of a parabola and a line or the intersection of two parabolas. The solutions we find are the x-coordinates of the intersections of the graphs.

x2 − x − 6 = 2x − 2 5x2 − 5x = 2x2 − 3x +112

⎛⎝⎜

⎞⎠⎟ x

2 = x −1


Solve Quadratic Equations by Factoring

1. Solve a) 1− 2x2 = −x

Put in general form

Factor

Set each factor equal to zero

b) 6x2 = x +15

Put in general form

Factor

Set each factor equal to zero


Solve by completing the square.

2. Solve the following quadratic equation by completing the square: 2x2 + x − 8 = 0

If ax2 + bx + c = 0 , then x = _________________________________

This is called ________________________________________________

3. Solve using the quadratic formula: 2x2 = 6x − 3


Solving equations in quadratic form

4. Solve

a) x4 − 5x2 − 6 = 0 b) x10 + 2x5 +1= 0

c) 4x12 − 9x6 + 2 = 0


d) x43

⎛⎝⎜

⎞⎠⎟ − 5x

23

⎛⎝⎜

⎞⎠⎟ + 4 = 0


Solving Rational EquationsRational equations are equations involving rational functions. Solving rational equations is

equivalent to finding the intersection(s) between the graphs of 2 rational functions.

Algebraically what we do is to “set one side of the equation equal to zero” and remember that a

fraction is zero when _________________________________________________________

The mistake: Consider solving 2x= 3x

. A lot of people want to start by cross multiplying or

multiplying both sides of the equation by x. This would yield ________________________.

But neither 2

nor 3

are defined. This equation does not have a solution. The risk occurs

when multiplying both sides of an equation by something that could be zero.

5. Solve

a) 12x

= 2 + 15x

The algebra will give us the x-coordinate of the intersection.


b) 4x − 4

− 3x −1

= 1


Radical Equations

To Solve Radical equations:I. Isolate one radical.II. Raise both sides of the equation to the appropriate power to remove the radical.! (Usually this means square both sides of the equation.)III. Repeat the process until all radicals have been removedIV. Check for extraneous solutions!

Why is it important to check for extraneous solutions to equations involving radicals?

How many solutions exist for the equation x = 3 ? __________________

Square both sides of the equation: ___________________________

What are the solutions to this equation?__________________________

See how ________________ is not a solution to the original equation?!

REMEMBER: (a + b)2 ≠ a2 + b2 (a + b)2 = _______________________

6. Solvea) 3 = x + 2x − 3


b) 8x +17 − 2x + 8 = 3

c) x + x − 5 = 1


Equations with Absolute Values

! ! Recall the definition of x =

7. Solve

a) x = 4

b) x − 3 = 2

c) 8 − x + 2 = 7


d) x +1 + 3= 2

e) x2 − x −12 = 8

! !


Equations with several variables

Equations in science and engineering often include many variables, and it is useful to be able to solve for one of the variables in terms of the others.

! ! Warm up: If 1x= 12+ 13

, then what is the value of x ?

8. The following equation comes from the physics of circuits 1Req

= 1R1

+ 1R2

+ 1R3

Solve for Req .


9. Ts = 2πmk

Solve for k .

10. Solve for y x = 3y2y − 5



1. Determine all real solutions of x + x − 4 = 4

2. Solve x = 10 − 9y3y − 5

for y .

3. From Dr. Lynch Fall 2016 Exam 1: Solve the equation A = 2π (tr + s) for r .


4. Determine all solutions of

a) x4 + x3 − 30x2 = 0

b) 18x3 + 9x2 − 2x − 3= 0

c) 6x2 +13x + 2x3 + 8

= 0

d) x4 + 6x2 − 27 = 0


3.2 - Solving InequalitiesTrue or False: 4 < 4 ! ! ! True or False: 4 ≤ 4

Also though 3 < 4 , -4 < -3 which could also be written -3 > -4.

Written more generally, if a < b , then _____________________

This leads to the “rule” that when you multiply or divide both sides of an inequality by a

negative number, you change the direction of the inequality.

1. Solve

a) x < 4

b) x ≥ 2


c) x + 2 <1

d) 2x − 4 ≥ 3

e) 5 − 2x > 4


f) 4 − x − 3 >1

Nonlinear Inequalities

Consider a,b ∈ ! ab > 0 , then either

or

If ab < 0 then either

or

To solve nonlinear inequalities:I. Move every term to one side (make one side zero).II. Factor the nonzero side.III. Find the critical values. (Critical values make the expression zero or undefined.)IV. Let the critical numbers divide the number line into intervals.V. Determine the sign of each factor in each interval.VI. Use the sign of each factor to determine the sign of the entire product or quotient.


2. Solve

a) x2 − x ≤ 6

TRUE or FALSE: 2 < 3

TRUE or FALSE: 2x < 3x

because when x < 0 , ______________

So don’t multiply both sides of an inequality by a variable that could be either positive or negative.

b) 2x<1


c) 2x + 3x + 3

> 6x +11− x

Consider the graph of f (x) = 2x + 3x + 3

It has a VA of ______________

the quotient of the leading terms is________

so the HA is _________________

the intercepts are ______________

Consider the graph of g(x) = 6x +11− x

It has a VA of ______________

the quotient of the leading terms is________

so the HA is _________________

the intercepts are ______________


Linear Inequalities

3. Solve

a) 1− x > x + 3

b) 1− 2x <17 − 4x ≤ 8 − x


c) 2x + 4 < x + 2 < x2+ 3

d) 4x +1≤ 4 − 2x ≤ 3− x



Solve the inequalities and express your answer in interval notation

a) 5 − x + 4 ≥ 3 b) 3x − 4 − 4 ≥ 7 (From Dr. Lynch’s Fall 2016 Exam 1.)

c) 65 − x

< 2 d) 5 + 2xx − 2

≥ 3 (From Dr. Lynch’s Fall 2016 Exam 1.)


5.4 - Quadratic Functionserinfry/math150/supporting_docs/notes/Math_15… · It is nice when a quadratic function is given in standard form _____ because it makes it easy to see that

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