Lesson 1-8 Interpreting graphs of functions
y intercept- where the line crosses the y axis
x intercept- where the line crosses the x axis
intercepts- points where graph intersects an axis
As labeled below:
x intercept- where the line crosses the x axis
intercepts- points where graph intersects an axis
As labeled below:
line symmetry- when a vertical line is drawn and either side of the graph matches
Look at the graph below:
Look at the graph below:
positive- where the graph lies above the x axis
negative- where the graph lies below the x axis
As seen below:
negative- where the graph lies below the x axis
As seen below:
increasing- when the graph goes up viewed from left to right
decreasing- when the graph goes down viewed from left to right
For example:
decreasing- when the graph goes down viewed from left to right
For example:
extrema- relatively high or low function values
relative minimum- point that is the lowest y coordinate on the graph (can be more than 1)
relative maximum- point that is the highest y coordinate on the graph (can be more than 1)
See the graph below
relative minimum- point that is the lowest y coordinate on the graph (can be more than 1)
relative maximum- point that is the highest y coordinate on the graph (can be more than 1)
See the graph below
end behavior- describes how the ends of the graphs behave
For example: x increases as y increases; x decreases as y decreases
from the graph below
For example: x increases as y increases; x decreases as y decreases
from the graph below
You need to describe and label all of these parts to interpret a graph unless asked.
lesson 2-9 weighted averages
Weighted average- found by multiplying each data value by its weight and then finding the mean of the new data set.
Mixture Problem- Problems in which two or more parts are combined into a whole. Solve these using weighted averages.
Example=A tea company sells blended tea for $25 per pound. To make blackberry tea, dried blackberries that cost $10.50 per pound are blended with black tea that costs $35 per pound. How many pounds of black tea should be added to 5 pounds of dried blackberries to make blackberry tea?
To solve this, you need to first find the total price of the dried blackberries. This is the cost times the amount: 10.5(5)
Next you need the price of the black tea. That's 35 (the cost) times the price per unit. We don't know this yet, so it's going to be 35t.
And finally we need the total price. That's (price)(units). So the final would be 25(5+t).
So it's 10.5(5)+35t= 25(5+t)
Which makes t=7.25
Next we have Percent Mixture Problems. An Example of this is shown below:
Albert has 16 cups of punch that is 3% pineapple juice. He also has a punch that is 33% pineapple juice. How many cups of the 33% punch will he need to add to the 3% punch in order to obtain a punch that is 20% pineapple juice?
Using the same format as the mixture problem, the equation is
0.03(16) + 0.33w = 0.2(16+w)
Solving this you get w to equal about 20.9 of 33% punch.
Next is Uniform Motion Problems, or rate problems. These problems involve an object moving at a certain speed or rate. Ever heard of
d=rt, or distance is equal to rate times time? That's going to come into use.
Her'e's a problem involving only one object.
It took Nick 40 minutes to skate 5 miles. The return trip took them 30 minutes. What was their average speed for the trip?
First of, the problem is asking about the weighted average speed for both trips. So let's use this d=rt.
The way we solve this is finding the rate of the going multiplied by the time of the going plus the rate of return times the time of return divided by time of going plus the time of return.
It looks like this:
(rate of going)(time of going)+(rate of return)(time of return)/ time of going+time of return
Got that?
To find the rate of going/return, you need to plug it into r=d/t
All together this equals about 8.6 miles per hour
If that was bad then you're in luck because this next part is easy.
Now we look at two objects at different speeds.
Two trains are 550 miles apart heading toward each other on parallel tracks. Train A is traveling east at 35 mph, while Train B travels west at 45 mph. When will the trains pass each other?
All you need to do is Train A's rate times time and Train B's rate times time and put them equal to the total distance. Because it stays the same.
So it's 35t + 45t = 550
Now t= 6.875 hours. ALWAYS REMEMBER TO LABEL
Next up is...... Writing equation in slope intercept for! How fun.
Write an equation in slope intercept form from (2,1) with a slope of 3
First find the y intercept by plugging in the numbers into the formula
1=3(2)+b
1=6+b
subtract six and you get
b=-5
BUT THAT'S NOT ALL!!!
You plug it back in to the slope. So your answer would be y=2x-5
When you have two points, such as (3,1) and (2,4) you need to find the slope first.
That's y2-y1/x2-x1
so, plugging that in, the slope would be -3. Now use the same steps you did to create the y-intercept and you get y=-3x+10
Mixture Problem- Problems in which two or more parts are combined into a whole. Solve these using weighted averages.
Example=A tea company sells blended tea for $25 per pound. To make blackberry tea, dried blackberries that cost $10.50 per pound are blended with black tea that costs $35 per pound. How many pounds of black tea should be added to 5 pounds of dried blackberries to make blackberry tea?
To solve this, you need to first find the total price of the dried blackberries. This is the cost times the amount: 10.5(5)
Next you need the price of the black tea. That's 35 (the cost) times the price per unit. We don't know this yet, so it's going to be 35t.
And finally we need the total price. That's (price)(units). So the final would be 25(5+t).
So it's 10.5(5)+35t= 25(5+t)
Which makes t=7.25
Next we have Percent Mixture Problems. An Example of this is shown below:
Albert has 16 cups of punch that is 3% pineapple juice. He also has a punch that is 33% pineapple juice. How many cups of the 33% punch will he need to add to the 3% punch in order to obtain a punch that is 20% pineapple juice?
Using the same format as the mixture problem, the equation is
0.03(16) + 0.33w = 0.2(16+w)
Solving this you get w to equal about 20.9 of 33% punch.
Next is Uniform Motion Problems, or rate problems. These problems involve an object moving at a certain speed or rate. Ever heard of
d=rt, or distance is equal to rate times time? That's going to come into use.
Her'e's a problem involving only one object.
It took Nick 40 minutes to skate 5 miles. The return trip took them 30 minutes. What was their average speed for the trip?
First of, the problem is asking about the weighted average speed for both trips. So let's use this d=rt.
The way we solve this is finding the rate of the going multiplied by the time of the going plus the rate of return times the time of return divided by time of going plus the time of return.
It looks like this:
(rate of going)(time of going)+(rate of return)(time of return)/ time of going+time of return
Got that?
To find the rate of going/return, you need to plug it into r=d/t
All together this equals about 8.6 miles per hour
If that was bad then you're in luck because this next part is easy.
Now we look at two objects at different speeds.
Two trains are 550 miles apart heading toward each other on parallel tracks. Train A is traveling east at 35 mph, while Train B travels west at 45 mph. When will the trains pass each other?
All you need to do is Train A's rate times time and Train B's rate times time and put them equal to the total distance. Because it stays the same.
So it's 35t + 45t = 550
Now t= 6.875 hours. ALWAYS REMEMBER TO LABEL
Next up is...... Writing equation in slope intercept for! How fun.
Write an equation in slope intercept form from (2,1) with a slope of 3
First find the y intercept by plugging in the numbers into the formula
1=3(2)+b
1=6+b
subtract six and you get
b=-5
BUT THAT'S NOT ALL!!!
You plug it back in to the slope. So your answer would be y=2x-5
When you have two points, such as (3,1) and (2,4) you need to find the slope first.
That's y2-y1/x2-x1
so, plugging that in, the slope would be -3. Now use the same steps you did to create the y-intercept and you get y=-3x+10
Lesson 3-6 Proportional and nonproportional relationships
A Proportional Relationship is if the relationship between the domain and range of a relation is linear, the relationship can be described by a linear equation.
Here is a example of a proportional relationship
Here is a example of a proportional relationship
As you can see, y=8x
You can prove this by imputing an x-value in the place of x in the equation, and a y-value in place of y. For example, 16= 8(2)
Now try this problem on a nonproportional relationship
You can prove this by imputing an x-value in the place of x in the equation, and a y-value in place of y. For example, 16= 8(2)
Now try this problem on a nonproportional relationship
The correct answer is J, y= 5x+3
You can check by replacing the y and x variables with real numbers from the graph,
33=5(6)=3 = 33=33
One easy way to tell if a relationship is use cross products.
You can check by replacing the y and x variables with real numbers from the graph,
33=5(6)=3 = 33=33
One easy way to tell if a relationship is use cross products.
Look at the first two boxes. I grouped 2 and 8 together and 4 and 4.
Now cross multiply those pairs.
2 x 8=16
4 x 4=16
16=16. This shows that the table above is proportional. If they do not equal each other, then it i unproportional.
Now cross multiply those pairs.
2 x 8=16
4 x 4=16
16=16. This shows that the table above is proportional. If they do not equal each other, then it i unproportional.