Objective 18 Quadratic Functions

The simplest quadratic function is f(x) = x2.

x

y

Objective 18b Quadratic Functions in (h, k) formApplying all of Obj 14 (reflections and translations) to the function.

f(x) = a(x− h)2 + k vertex

a > 0 parabola opens

a < 0 parabola opens

Objective 18a Quadratic Functions in Standard form

f(x) = ax2 + bx + c

What’s the vertex? We could and put it in (h, k) form.

Good news:

f(x) = ax2 + bx + c vertex

a > 0 parabola opens

a < 0 parabola opens

For either quadratic form:

To find x-intercepts, let solve for x.

To find y-intercepts, let solve for y.

Sometimes we ask “How many x-intercepts are there?”

For Obj 18a, You can use the

For Obj 18b, Just

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Objective 18c Max/Min of Quadratic Function

x

y

x

y

Ob 18a example The information included in this example would be asked in separate on lineproblems.

f(x) = −12x2 − 4x+ 1

Opens Up/Down

x-coordinate of vertex =

How many x-intercepts?

Max/Min is

Max/Min is at x =

Find all intercepts. (For on line problems: Enter them in any order separated by a comma.)

Which of the following most closely resembles the graph of f(x) = −12x2 − 4x+ c?y

x x

y

x

y

x

y

2

Ob 18b example The information included in this example would be asked in separate on lineproblems.

f(x) = a(x+ 1)2 − 8, a > 0

Opens Up/Down

vertex =

How many x-intercepts?

Max/Min is

Max/Min is at x =

Find all intercepts for f(x) = 2(x + 1)2 − 8. (For on line problems: Enter them in any orderseparated by a comma.)

Find all intercepts for f(x) = 2(2x + 1)2 − 10. (For on line problems: Enter them in any orderseparated by a comma.)

Which of the following most closely resembles the graph of f(x) = a(x+ 1)2 − 2, a > 0?y

x x

y

x

y

x

y

3

Ob 18c example Studies have found that the relationship between advertising dollars, a, in thou-sands, and revenue, R, can be modeled by a quadratic function.

If R(a) = −4a2+364a+2569.5, how many thousands of advertising dollars should be spent in orderto maximize revenue?

(Enter number answer - integers or exact decimals; mathematical operators are not allowed. Forexample, 15/2 must be entered as 7.5. Don’t type any dollar signs, commas, or units. The functiongiven does not represent the results of an actual study.)

Ob 18c example A large swimming pool is treated regularly to control the growth of harmfulbacteria. If the concentration of bacteria, C (per cubic centimeter), t days after treatment, is givenby C(t) = 0.4t2 − 4.4t+ 30.1, What is the minimum concentration of bacteria? (Same cautions asin previous example.)

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Objective 19 Power Functions

f(x) = xn, where n is an integer, n ≥ 2

The power functions are classified into 2 groups:

f(x) = xn, where n is an even positive integer, n ≥ 2

For example: f(x) = x2, f(x) = x4, f(x) = x6, ... f(x) = x58, ...

x

y

f(x) = xn, where n is an odd positive integer, n ≥ 3

For example: f(x) = x3, f(x) = x5, f(x) = x7, ... f(x) = x59, ...

x

y

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Objective 20 Solving Polynomial and Rational Inequalities

***************Plan of Attack

1. Factor, if needed. (Watch for Diff of Squares and Factoring out GCF.)

2. Find Partitioning Points. These are points that make the expression (from factorsin the numerator) or make the expression (from factors in the denominator).Set each factor equal to 0 to find these Partitioning Points.

3. Mark these on a number line (must be in number-line order).

4. Make a Sign-Chart. Select a value in each interval that’s created by these partitioning points(don’t use one of end-points). Plug this value into each separate factor and record whether theresult is + or −. Consolidate the signs from all the factors.***************

Solve.9− 4x2x2 − 5x ≤ 0

Solve. (x2 + 1)(4− x2)(x+ 1)2 < 0

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Solve. (x2 + 36)2(x− 1)(4− x) < 0

Solve.x3(x+ 1)

x− 3 ≥ 0 Solve.(x− 30)(1− 2x)(100− x)(3x− 5) > 0

Solve. x5 > 3x4

Solve. x5 ≥ 3x4

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Objective 20c How many partitioning points would be needed to solve?

***************Plan of Attack Obtain a single fraction on one side with 0 on the other. That is, make theproblem ready to be solved by the sign-chart method.

NOTE: You can’t or multiply by any expression that contains the variable becauseyou won’t know if the inequality sign should be reversed.

NOTE: If the denominators are constants, then you are allowed to cross-multiply (multiply byLCM).

***************3

6x+ 1≤ 2

2x− 3x

x− 2 >4

x− 2

x− 53

<x

4

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Objective 20d Find the domain when a sign-chart is needed.

Recall Obj 10c. Give the domain for each.

f(x) =√3− 2x f(x) = 3

√

x+ 1

x− 2 f(x) =3√x− 2

Objective 20d Give the domain.

f(x) =√x+ 3x2 − 10x3

f(x) = 4√

1

3 + 5x− 2x2

f(x) = 3√

1

3 + 5x− 2x2 f(x) =5√3 + 5x− 2x2

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These are special cases you may see in your Practice (or Hmwk Quiz) problems, but won’t encounterin the Lab Quiz or Test problems. We need to be aware that an even-root radical can have “Domainall Reals”.

f(x) =√x2 + 25 f(x) =

√−x2 − 25

—————————————————————————————————————–

Objective 21 Inverse Functions

Illustrate the idea of inverse functions.

f(x) = x2+1

x

y

f(x) =√x− 1

x

y

—————————————————————————————————————–

Two one-to-one functions are inverses of each other if (f ◦ g)(x) = for all x in the domainof g, and (g ◦ f)(x) = for all x in the domain of f .

We write f−1 to denote the inverse function.

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Objective 21b How are the graphs of f and f−1 related?

x

y

If (a, b) is on the graph of y = f(x), then is on the graph of y = f−1(x).

Objective 21b Example Select the graph of y = f−1(x).

x

y

x

y

x

y

x

y

x

y

A function can be its own inverse. Consider

x

y

11

Objective 21a Does every function have an inverse?

x

y

x

y

Graph of a function must pass the to be be the graph of a one-to-one function.

Which are one-to-one functions?

{(1, 2), (1, 3), (5, 4)} {(1, 2), (3, 2), (4, 5)} {(1, 2), (3, 3), (4, 5)}

If a function is not one-to-one, restrict the domain in order to define an inverse function. (Recallintro to Obj 21.)

x

y

12

Objective 21d Given a function, find the function rule for f−1.

***************Plan of Attack 1. Write y for f(x) (to simplify the notation).

2. Solve for x. For applied mathematicians, when units are usually associated with the variables,you have the inverse function.

3. For our College Algebra course, we will interchange x and y to write the inverse as a function ofx.

4. Write f−1(x) for y (to return to function notation).

***************

Find the function rule for f−1 for f(x) = 3x− 5

Find the function rule for f−1 for f(x) = (4− x)5 + 7

Find the function rule for f−1 for f(x) = 3√x− 2 + 7

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Find the function rule for f−1 for f(x) =x+ 1

x− 3

Find the function rule for f−1 for f(x) =2x− 35− x

—————————————————————————————————————–

Objective 22 Exponential Functions. f(x) = ax, a > 0, a 6= 1

Does this define a function?

Don’t allow base to be negative because could be for some x. i.e.

Don’t allow base to be 1 because , graph would be linear, not exponential.

What’s the domain? “All reals?” If so, we have to define what’s meant by irrational exponents.

For example: 4√2 or 4π We haven’t worked with irrational exponents.

Good News: The limiting processes of calculus guarantee that irrational exponents are defined, and“line up” as we want. (See example below.)

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The exponential functions are classified into 2 groups, depending on the base.

f(x) = ax, a > 1 f(x) = ax, 0 < a < 1

We will consider two specific cases to develop the concept. This is not an on line problem example;you will not be making tables of values - you will not be plotting points.

Consider f(x) = 4x Consider f(x) =(

1

4

)x

for an example of a > 1 for an example of 0 < a < 1

x y

−50

−3

−2

−1

0

12

1

√2

2

52

3

π

4

50

x

y

x

y

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Objective 22a Properties and Graphs of Exponential Functions

f(x) = ax, a > 1 f(x) = ax, 0 < a < 1

x

y

x

y

—————————————————————————————————————–

Objective 22b Graphing Exponential Functions with Reflections or Translations

Don’t . Don’t . Use Obj 14!

Select the graph that best represents the graph of each of the following.

f(x) = −5x f(x) =(

1

4

)−x

y

x

y

x

1

y

x

1

y

x

Which function best describes the graph shown? Which function best describes the graph shown?y

x

1

y

x

f(x) = (2.5)−x f(x) = −(2.5)x f(x) = (2.5)−x f(x) = −(2.5)x

f(x) = (0.4)−x f(x) = −(0.4)x f(x) = (0.4)−x f(x) = −(0.4)x

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More Objective 22b Graphing Exponential Functions with Translations

Don’t . Don’t . Use Obj 14!

Select the graph that best represents the graph of each of the following.

f(x) = 4x − 3 f(x) =(

15

)x+2

y

x

y

x

y

x1

y

x

1

y

x

y

x

1

y

x

y

x

1

Which function best describes the graph shown? Which function best describes the graph shown?y

x

1

y

x

f(x) = 6x+3 f(x) = 6x + 3 f(x) =(

52

)x+2f(x) =

(

52

)x−2

f(x) = (0.6)x+3 f(x) = (0.6)x + 3 f(x) =(

25

)x−2f(x) =

(

25

)x+2

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Objective 22c “The” exponential function is f(x) = ex because of so many areas of application.

e ≈ 2.71828 e = limn→∞(

1 +1

n

)n

Graph f(x) = ex

Evaluate ex on a scientific calculator (the Mac calculator in lab class).

Strontium 90 is a radioactive material that decays over time. The amount, A, in grams of Strontium90 remaining in a certain sample can be approximated with the function A(t) = 225e−0.037t , wheret is the number of years from now. How many grams of Strontium 90 will be remaining in thissample after 7 years?

$8,000 is invested in a bond trust that earns 5.9% interest compounded continuously. The accountbalance t years later can be found with the function A = 8000e0.059t. How much money will be inthe account after 6 years?

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Objective 22d Solving exponential equations when we can obtain the same base.

Exponential functions are one-to-one; that means:

if and only if

Rewrite each side (if needed) in terms of a common base; use the smallest base possible. Be sureto replace equals.

Solve 52x+1 = 253−x Solve(

4

9

)x−4

=(

27

8

)3x

Solve(

1125

)4x−1= 57x+5

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Objective 23 Logarithmic Functions

Consider an exponential function y = ax

What’s the inverse function?

There is no algebraic operation to solve for x.

We must define a new function. y = loga x

Objective 23a Evaluate Logarithmic Functions

log2 8 =

log25 5 =

log1/16 2 =

log2 2 =

log2 1 =

Which are defined? (Be careful, sometimes ask “Which are undefined?”)

log1/2 1 log1/4 4 log1/2(−4) log1/2 0

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Objective 23b Properties and Graphs of Logarithmic Functions f(x) = loga x, a > 0, a 6= 1

The logarithmic functions are classified into two groups comparable to the exponential functions.

Recall Obj 22a

x

y

x

y

y = ax, a > 1 y = ax, 0 < a < 1

x

y

x

y

y = loga x, a > 1 y = loga x, 0 < a < 1

Objective 23c Graphing Logarithmic Functions with Reflections or Translations

Don’t . Don’t . Use Obj 14!

Select the graph that best represents the graph of each of the following.

f(x) = − log4 x f(x) = log1/4(−x)

1

y

x 1

y

x 1

y

x 1

y

x

21

Which function best describes the graph shown? Which function best describes the graph shown?

1

y

x 1

y

x

f(x) = − log(5/2)(x) f(x) = log(5/2)(−x) f(x) = − log(5/2)(x) f(x) = log(5/2)(−x)

f(x) = − log(2/5)(x) f(x) = log(2/5)(−x) f(x) = − log(2/5)(x) f(x) = log(2/5)(−x)

More Objective 23c Graphing Logarithmic Functions with Translations

Don’t . Don’t . Use Obj 14!

Select the graph that best represents the graph of each of the following.

f(x) = log3(x) + 2 f(x) = log1/3(x+ 2)

1

y

x

1

y

x

y

x

y

x

1

y

x

1

y

x

y

x

y

x

22

Which function best describes the graph shown? Which function best describes the graph shown?

1

y

x

y

x

f(x) = log(5/2)(x) + 2 f(x) = log(5/2)(x)− 2 f(x) = log(5/2)(x+ 2) f(x) = log(5/2)(x− 2)

f(x) = log(2/5)(x) + 2 f(x) = log(2/5)(x)− 2 f(x) = log(2/5)(x+ 2) f(x) = log(2/5)(x− 2)

—————————————————————————————————————–

Objective 23d Domain of Logarithmic Functions (not by graphing)Give the domain.f(x) = logb(4− 5x)

f(x) = 15− logb(3x)

f(x) = logb

(

x+ 1

x− 3

)

f(x) = log3(4− x2)

f(x) = log3(x2 + 4)

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Objective 24 Properties of Logarithmic Functions

As used below: a > 0, a 6= 1, b > 0, b 6= 1, M > 0, N > 0, x > 0, y and r represent any real number

Definition - Obj 24a means

Common Logarithms are logarithms base 10; we write instead of .

Natural Logarithms are logarithms base e; we write instead of .

Objective 24a Example Which of the following is equivalent to ln 5 = x?

A) 5e = x B) ex = 5 C) x5 = e

Properties of Logarithms - Obj 24b

Product Rule logb(MN) =

Must Note: logb(MN) 6=

Must Note: logb(M +N) 6=

Quotient Rule logb

(

M

N

)

=

Must Note: logb

(

M

N

)

6=

Must Note: logb(M −N) 6=

Power Rule logb Mr =

When Base and Result Match logb b =

When Result is 1 logb 1 =

24

Inverse Function Properties - Obj 24c Recall Obj 21: (f ◦ f−1)(x) = x and (f−1 ◦ f)(x) = x

aloga M = loga ar =

Objective 24c Examples

Solve for x if 5log5(3x) = 15 Solve for x if ln e15x = 3

Objective 24b Example Which of the following is equivalent to logb(x− y)?

A) logb

(

x

y

)

C) both A and B

B) logb x− logb y D) none is equivalent

Applying Log Properties - Objective 24d

Expand using log properties. logb

(

x2

yz

)

Expand using log properties. logb

(

x2y

z(w + 3)

)

Which of the following is equivalent tologb(x

2y)

logb(z(w + 3))

A)2 logb x+ logb y

logb z + logb(w + 3)

B) 2logbx+ logb y − logb z − logb(w + 3)

C) A and B are the same

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another Objective 24d Example

Write as a single logarithm 2 logb x− logb y + 12 logb z

A) logbx2

y√z

B) logbx2√z

y

another Objective 24d Example

Write as a single logarithm. 2 logb(z − w)− logb w + 3 logb z + logb x− logb(x+ w)

another Objective 24d Example

If logb 2 = l and logb 5 = m, express logb 100 in terms of l and m.

another Objective 24d Example

If logb 2 = l and logb 5 = m, express (logb 4) · (logb 25) in terms of l and m.

Copyright c©2010-present, Annette Blackwelder, all rights are reserved. Reproduction or distribu-tion of these notes is strictly forbidden.

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