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You've done very well so far. You've found all there is to find except

• since f(-x) = -f(x), the graph will be symmetric about the origin.
• if you know Calculus, you can find
  • local maximum and minimum points, if any
  • where the graph is increasing and where it is decreasing
  by finding and using the first and second derivatives.
• you can build a table of values and plot the points.

Here's a graph:

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There are several way to graph this:

• If you know what the graph of looks like then the graph of will be the graph of shifted 4 units down (because of the -4).
• Use the fact that when and when . So for ,
  • when . This is a line that should be easy to graph. But just graph it where . In other words graph just from the y-axis and to the right.
  • when , . This is another line. Graph this line just where . In other words graph just to the left of the y-axis.
  So the graph of is composed of two "half-lines", one to the right of the y-axis and one to the left.
• Brute force. Just build a table of values and plot the points until you can see what is going on.

Domain.
There is no reason to exclude any values for x because any number can be put into an absolute value. So the domain is all Real numbers.

Range.
What are the possible vales of ? Of course the answer to this question will depend on the possible values of . The absolute value of any number can only be positive or zero. It can never be negative. Another way to say this is that absolute values are always greater than or equal to zero. Then must be greater than or equal to -4. So the range is all Real numbers that are greater than or equal to -4.

If the logic used in calculating the range is unclear to you, here's some Algebra that may explain it more clearly:

Add 4 to both sides:

Since all absolute values are greater than or equal to zero and since this equation tells us that y+4 equals an absolute value:

Subtract 4 from each side:

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So you are given the equation and the height and you've been asked to find the time. You just put the height into the equation and then use Algebra to solve for the time:

To solve this we need to recognize that, because of the , it is a quadratic equation.

To solve a quadratic equation you get one side equal to zero and then factor it of use the Quadratic formula. So we will start by subtracting 20 from each side (the quickest way to get one side to be zero):

(I've also rearranged the terms to make the factoring (or use of the Quadratic formula easier.) When factoring, always start by factoring out the Greates COmmon Factor (GCF) (unless it is a 1). The GCF here is 5:

We could factor the trinomial factor as it is but I like having a positive leading coefficient so I will factor out a -1. (I could have factored out a -5 earlier but I didn't want to confuse you.)

The trinomial is now more recognizably factorable. It fits the pattern: . So it factors into:

According to the Zero Product Property this product can be zero only if one of the factors is zero. 5 can't be zero and neither can -1. But t-2 can:
t-2 = 0
t = 2

So the cricket ball is 25 meters (metres in the UK) high after 2 seconds. (If you think about it, this must be the maximum height of the cricket ball. The cricket ball will be at the various heights once on the way up and once on the way down. So there should be two times for each height. The exception is the maximum height. The maximum height is achieved only once, in the middle of the flight. This is why we only got one time for this height.)

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Slope-Intercept Form?
In laymans terms, can you explain the simplest way I can learn how to identify the slope and y intercept?
Example 2x-y=12
I have no idea where to even start??

For any equation to be in slope-intercept form, it has to be in the form, y = mx + b, where m, or the coefficient of x is the slope, and " + b" is the y-intercept.

Now, let's take your example, 2x - y = 12

We have to first of all put this equation in slope-intercept form

2x - y = 12 -----> - y = - 2x + 12 (subtract 2x from both sides of equation)

- y = - 2x + 12 ------> y = 2x - 12 (divide each side of equation by - 1)

Now that we have the equation in slope-intercept form, y = 2x - 12 indicates that m or slope of the equation is , and " + b" or the y-intercept of the equation is

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The Quick Study reference guide has a very simple explanation and I quote it here. The breakdown into sentences is mine. Read each sentence and MAKE SURE YOU UNDERSTAND IT before you go on to the next.
"The slope of a line can loosely be described as the slant of the line.
If the line slants up on the right end of the line, then the slope will be a positive number.
If the line slants up on the left end of the line, then the slope will be a negative number.
If the line is horizontal, then the slope is zero.
If the line is vertical, then the line has no slope; it is undefined."
Now... once you understand that, then go on to this formula:
If the line is NOT vertical, then slope (indicated by the letter "m") can be found using two distinct points: A and B. Coordinates are written (x,y) with x being the horizontal point and y being the vertical point. For instance, if A's coordinates (x) are (5,6) and B's coordinates (y) are (7,3), then you would plug them into this formula:
m = (y2 - y1) / (x2 - x1) = (change in y) / (change in x) = rise / run
This is a complicated formula to write here, but Google "slope of a line formula" and you should see a better rendition of the formula.
In the case I stated above, you would plug A's coordinates in:
m = (3 - 6) over or divided by (7 - 5) = (change in y) over or divided by (change in x) = "rise" over or divided by "run"
I'm sorry I cannot give you a better explanation here, but this is one of those concepts that is difficult to explain sometimes. Here is a link on this site that I use for my students (I am a tutor) that might help explain it better.
http://www.algebra.com/algebra/homework/Graphs/Graphing-by-Slope-Intercept-Method.lesson
I hope this helps. :)

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1019/3500
=.2911...=29.11%
.
Ed

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Too many equal signs! Perhaps you should correct and re-state this one.

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None!! If the vertex is in Q III and opens down, it doesn't touch the x axis. NO INTERCEPTS!!

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida
Altamonte Springs Campus

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You are correct!!

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida
Altamonte Springs Campus

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Let Tom = x
James = 1/2 x
Tony = 3/4 x
Total lenght of sandwich = 18 inches
.
Our equation now becomes:
Since your denominators are 1, 2, and 4... the LCD is 4.
The new equation with common denominators becomes:



Then add numerators:

Now multiply by the fraction's reciprocal on both sides to determine the value of x.



.
Tom's sandwich is 8 inches long.
James's sandwich is (1/2)*8=4 inches long.
Tony's sandwich is (3/4)*8=6 inches long.
.
If we add all pieces, we get:

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This is unanswerable as not enough information is given. No range of years was given, and the answer will depend on what year you start and when time ends, which no one knows.

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16

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In order for a fraction to be undefine, its denominator would equal to zero. So, let the denominator (by itself) equal to zero and find the solution(s).
.


Take the square root on both sides of the equation to determine the value of n.

Since the square root of a number raised to the 2nd power cancel each other, we can re-write the equation like this:

This will yield two possible answers:
n=2 and n=-2
.
Check:

.

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A poker hand consists of five cards from a standard deck of 52.
Find the number of different poker hands of straight (five cards of consecutive denominations: A, 2, 3, 4, 5 up through 10, J, Q, K, A, not all of the same suit). (Note that the Ace counts either as a 1 or as the denomination above
King.
---------
Pick a lowest starting card type: 10 ways
Pick that lowest card and each of the next 4: 4^5 = 1024 ways
----
Total number of straights: 10*1024= 10,240
--------------
Cheers,
stan H.

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Take the log base 10 of both sides






The interest rate has to be greater than 7%

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49 and 50
50 and 51
99 and 100
100 and 101
----------
Any multiple of 50 and the integer adjacent to it.

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too many equal signs

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(-1/8+1/8i)(4+4i)
= -1/2 - 1/2
= -1

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The height about the ground of an object t seconds after it is launched vertically upward is h(t)=100t-5t^2 meters
What is the velocity of the object after 3 seconds
---------------
h'(t) = 100 - 10t
h'(3) = 70 m/sec
-----------------
Did you mean to ask the height? If so,
h(3) = 100*3 - 5*9
h(3) = 255 meters, going up at 70 meters/sec

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This website has a great explanation on sets. Please check it out.
.
http://www.mathsisfun.com/sets/sets-introduction.html

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one fourth of a number, decreased by 10, is -50
.




.
Done! :)

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2x²-11x-2=0
------------

Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)

Quadratic equation (in our case ) has the following solutons: For these solutions to exist, the discriminant should not be a negative number. First, we need to compute the discriminant : . Discriminant d=137 is greater than zero. That means that there are two solutions: . Quadratic expression can be factored: Again, the answer is: 5.67617497767991, -0.176174977679906. Here's your graph:

------
= (11/4) ±sqrt(137)/4

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You drove 56 miles on a service call for your company and it took 10 minutes longer than the return trip when you drove an average of 8 miles per hour faster. What was your average speed on the return trip?
---------
Service Call DATA:
distance = 56 miles ; rate = x mph ; time = d/r = 56/x hrs
================================
Return trip DATA:
distance = 56 miles ; rate = (x+8) mph ; time = 56/(x+8) hrs
------------------------------------
Equation:
go time - return time = 1/6 hr
56/x - 56/(x+8) = 1/6
1/x - 1/(x+8) = 1/336
Multiply thru by 336x(x+8) to get:
336(x+8) - 336x = x(x+8)
336*8 = x^2+8x
x^2 + 8x - 2688 = 0
(x-48)(x+56) = 0
Positive solution:
x = 48 mph (rate on the service call)
x+8 = 56 mph (rate on the return)
==============================================
Cheers,
Stan H.

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If you need to write this equation in its standard form, this is what we would do.
.



.
Done! :)

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Hello!
.
If Kelly ran x number of races, we can then say Kim ran 2x-4=18.
So we now solve for x to determine how many races Kelly ran.
.





.
Since Kelly is x, she ran 11 races.
.
Let's check for Kim's number of races.



... it checks!
.
Done! :)

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Compute the discriminant. Then determine the number and type of solutions for the given equation.
4x²-2x+1=0
a = 4 ; b = -2 ; c = 1
What is the discriminant?
----------------------------
discriminant = b^2-4ac
----------------------------
Your Problem:
D = (-2)^2 - 4*4*1 = 4-16 = -12
--------------
Since the discriminant is negative
the quadratic has two complex number solutions.
===================================================
Cheers,
Stan H.

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evaluate (-6c)^2 and 6c^2 when c = 10
-----
(-6c)^2 = (-6*10)^2 = (-60)^2 = 3600
-------------------------------
6c^2 = 6*(10)^2 = 6*100 = 600
================================
Cheers,
Stan H.

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That is not a trinomial.
Cheers,
Stan H.

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how long would it take a car traveling 60 miles an hour to go 250 miles?
----
Time = distance/rate
time = 250/60 = 4 1/6 hrs or 4 hours 10 minutes
================================================
Cheers,
Stan H.

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simply divide 250 by 60
.
250/60 = 4.167 hours

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x^2 - 64
can be rewritten as:
x^2 - 8^2
this is a "special case" -- "difference of squares"
which factors out to
(x-8)(x+8)

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(x + 7) (x - 5) = 0

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N + 5 = 3(N - 3)

N + 5 = 3N - 9

14 = 2N ___ 7 = N