The graph of a function has no place where a vertical line intersects (touches) the graph more than once.
If a vertical line intersects (touches) the graph of a relation more than once, the graph fails the vertical line test and it is not a function.
If there is no place on the graph of a relation where a vertical line intersects (touches) the graph more than once, the graph passes the vertical line test and it is a function.
Monday, October 31, 2011
Friday, October 28, 2011
Prepare for Quiz on Tuesday 11-1
Given a function with a table of values, write the ordered pairs and draw the mapping.
Given a function with a mapping, write the ordered pairs and write the table.
Given a function with ordered pairs, write the table of values and draw the mapping.
Given the graph of a function or relation, write the table of values, the ordered pairs, or draw the mapping.
Given the graph of a function, write the domain and range using set notation with { }
Know when to use a description or a list in set notation.
Match the terms domain, input, output, range, x, and y.
Explain the difference between a function and a non-function. You may use the boy-girl rule.
Given a function and a domain that is a list, evaluate the function.
Given a function with a mapping, write the ordered pairs and write the table.
Given a function with ordered pairs, write the table of values and draw the mapping.
Given the graph of a function or relation, write the table of values, the ordered pairs, or draw the mapping.
Given the graph of a function, write the domain and range using set notation with { }
Know when to use a description or a list in set notation.
Match the terms domain, input, output, range, x, and y.
Explain the difference between a function and a non-function. You may use the boy-girl rule.
Given a function and a domain that is a list, evaluate the function.
10-28 Class notes
Set notation: If you are asked to describe or list a domain or range, use curly brackets { }
For the example we did in class from page 76, item 3a, the domain is { -5, 0, 4 }
For the example on page 78, item 1a, the domain is { x > -5 and x < 5 }.
The range is { y > -4 and y < 4 }.
There will be a quiz on Tuesday 11-1.
For the example we did in class from page 76, item 3a, the domain is { -5, 0, 4 }
For the example on page 78, item 1a, the domain is { x > -5 and x < 5 }.
The range is { y > -4 and y < 4 }.
There will be a quiz on Tuesday 11-1.
Wednesday, October 26, 2011
Class notes 10-26
To illustrate the difference between a function and a non-function, we used the example of a rule for boys asking girls to go out.
A boy can only ask ("------>" symbol in a mapping diagram) one girl to go out. That's a function.
It's OK for one girl to be asked out by more than one boy.
If a boy asks ("------>" symbol in a mapping diagram) more than one girl to go out, that's not a function.
It's not OK for two different girls to be asked out by the same boy.
Homework for Friday 10-28 - because of the trip to see the play - is page 72, item 15, all four parts.
You can remember the proper sequence of pairs of terms by looking at the alphabeticl order of the first letters.
Domain first, Range second. D comes before R in the alphabet
x first, y second. X comes before Y in the alphabet
Input first, Output second. I comes before O in the alphabet
A boy can only ask ("------>" symbol in a mapping diagram) one girl to go out. That's a function.
It's OK for one girl to be asked out by more than one boy.
If a boy asks ("------>" symbol in a mapping diagram) more than one girl to go out, that's not a function.
It's not OK for two different girls to be asked out by the same boy.
Homework for Friday 10-28 - because of the trip to see the play - is page 72, item 15, all four parts.
You can remember the proper sequence of pairs of terms by looking at the alphabeticl order of the first letters.
Domain first, Range second. D comes before R in the alphabet
x first, y second. X comes before Y in the alphabet
Input first, Output second. I comes before O in the alphabet
Tuesday, October 18, 2011
10-14 Absolute value
To evaluate a number inside absolute value bars, such as | -4 |,
If what's inside the bars is positive, you take away the bars and the sign stays positive.
If what's inside the bars is negative, you take away the bars and the sign is changed to positive.
In this case, what's inside the bars is negative, so you change it to positive 4 when you remove the bars.
| -4 | = 4
If you don't know the sign of what's inside the bars because there's a variable (letter) you don't know,
You have to consider the two cases where what's inside could be positive and what's inside could be negative, depending on the value of the variable.
| x | = x, positive x, if x is positive
| x | = -x, the opposite of x, if x is negative
| x + 3 | = x + 3, if x > -3
| x + 3 | = -x - 3, the opposite, if x < -3
This means that when you are solving absolute value equations or inequalities, once you remove the bars you will have to work on two parts of the problem. The final answer will be the joining or junction of two graphs. It could be two dots, two split arrows, or the overlapping tails of two arrows.
If what's inside the bars is positive, you take away the bars and the sign stays positive.
If what's inside the bars is negative, you take away the bars and the sign is changed to positive.
In this case, what's inside the bars is negative, so you change it to positive 4 when you remove the bars.
| -4 | = 4
If you don't know the sign of what's inside the bars because there's a variable (letter) you don't know,
You have to consider the two cases where what's inside could be positive and what's inside could be negative, depending on the value of the variable.
| x | = x, positive x, if x is positive
| x | = -x, the opposite of x, if x is negative
| x + 3 | = x + 3, if x > -3
| x + 3 | = -x - 3, the opposite, if x < -3
This means that when you are solving absolute value equations or inequalities, once you remove the bars you will have to work on two parts of the problem. The final answer will be the joining or junction of two graphs. It could be two dots, two split arrows, or the overlapping tails of two arrows.
10-18 Graping an absolute value inequality
Homework for Wednesday and Thursday is to do review problems, page 59, 60, 61;
Items 7 through 49 odds. Do the work on a separate sheet with your name on it, and hand it in Thursday.
Today bell-ringer and inclass: Solve and graph |x-3| < 2
When a problem has absolute value bars | |, you need to remove them and follow the rules for solving two parts of the problem.
Remove the absolute value bars and write what's left twice:
Items 7 through 49 odds. Do the work on a separate sheet with your name on it, and hand it in Thursday.
Today bell-ringer and inclass: Solve and graph |x-3| < 2
When a problem has absolute value bars | |, you need to remove them and follow the rules for solving two parts of the problem.
Remove the absolute value bars and write what's left twice:
x-3 < 2 x-3 < 2
One of these is the original and you leave it alone. For the other, introduce a negative sign (to create the opposite) AND change the direction of the inequality:
(original) x-3 < 2 (opposite) x-3 > -2
add 3 to each side add 3 to each side
x < 5 x > 1
position is 5 position is 1
dot is closed dot is closed
thick arrow points left thick arrow points right
Combine the graphs into one segment from 1 to 5 where the arrow tails overlap.
Wednesday, October 12, 2011
10-11 Compound inequalities - class notes
A compound inequalitiy is two inequality statements joined by the words "and" or "or".
Refer to the notes on page 47, item 20
x > -1 and x < 4
x < 3 or x > 6
The numbers that satisfy the compound inequality and the graph of the compound inequality are
the junction of two sets.
Conjunction goes with the word "and". It means the numbers must satisfy both inequalities. The graph is where the tails of the two graph arrows overlap. The graph looks like a segment. The graph could have open or closed dots at its ends.
Disjunction goes with the word "or". It means the numbers only have to satisfy one of the inequalities. The graph is two arrows pointing in opposite directions. The graph has a gap between the arrows. The arrows are split.
Homework for 10-12: page 48, items a, b, c, d, e.
Hint: Some exercises are counted only with whole numbers. Use the idea of "between" to form the graphs.
Refer to the notes on page 47, item 20
x > -1 and x < 4
x < 3 or x > 6
The numbers that satisfy the compound inequality and the graph of the compound inequality are
the junction of two sets.
Conjunction goes with the word "and". It means the numbers must satisfy both inequalities. The graph is where the tails of the two graph arrows overlap. The graph looks like a segment. The graph could have open or closed dots at its ends.
x > -1 and x < 4
Disjunction goes with the word "or". It means the numbers only have to satisfy one of the inequalities. The graph is two arrows pointing in opposite directions. The graph has a gap between the arrows. The arrows are split.
x < 3 or x > 6
Homework for 10-12: page 48, items a, b, c, d, e.
Hint: Some exercises are counted only with whole numbers. Use the idea of "between" to form the graphs.
Thursday, October 6, 2011
10-6 Class notes
To solve an inequality, simplifiy using the properties of equality (using the same operation on both sides of the inequality sign) until you get the variable by itself.
2x < 12, then divide both sides by 2 to get
x < 6
This gives us three things to graph: A position (6), a dot (open because there's no equality line under the "less than" symbol), and a direction (to the left, the same way that the "less than" sign points).
When you are solving an inequality and you multiply or divide by a negative number, the negative sign causes the position and the direction to become opposites. Multiplying or dividing by the negative sign "flips" the sign of the number and it also "flips" the direction of the thick bar and arrow in the graph.
-2x < 12, then divide both sides by -2 to get
x > 6, having changed the direction of the inequality from pointing left to pointing right.
2x < 12, then divide both sides by 2 to get
x < 6
This gives us three things to graph: A position (6), a dot (open because there's no equality line under the "less than" symbol), and a direction (to the left, the same way that the "less than" sign points).
When you are solving an inequality and you multiply or divide by a negative number, the negative sign causes the position and the direction to become opposites. Multiplying or dividing by the negative sign "flips" the sign of the number and it also "flips" the direction of the thick bar and arrow in the graph.
-2x < 12, then divide both sides by -2 to get
x > 6, having changed the direction of the inequality from pointing left to pointing right.
Wednesday, October 5, 2011
10-5 Homework quiz
Wednesday 10-5 homework quiz
p37 11a, 11b, 12b
p38 1, 5
p44 a, b
Homework for Thursday 10-7
page 46; items a, b, c
p37 11a, 11b, 12b
p38 1, 5
p44 a, b
Homework for Thursday 10-7
page 46; items a, b, c
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