Graphing Exponential Functions
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View: Exponentially Lyric Explanation
1st verse:
5 raised to the power of x equals y
Replace the x to find
The coordinates that define
An exponential line
Replace the with x with -2, -1, 0, 1, 2, 3, 4, 5
These ordered pairs define
A growing curve left along that line
Like (-2, 1/25) and (5, 3,125)
Chorus:
With a coefficient greater than 1
And a positive exponent
This results in growth
The curve explodes like a population
Coefficient between zero and 1
This results in decay
The curve starts high
And falls toward zero, exponentially
2nd verse:
1/5 raised to the power of x equals y
Replace the x to find
The ordered pairs that define an exponential line
Replace the x with -2, -1, 0, 1, 2, 3, 4, 5
The coordinates define
A decaying curve left along that line
Like (-2, 25) and (3, 1/125)
Bridge:
Growth with a coefficient greater than 1
And a positive exponent
Or coefficient between zero and 1
With a negative exponent
Decay - coefficient between zero and 1
And a positive exponent
Or coefficient that is greater than 1
With a negative exponent
Graphing Quadratic Functions
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View: U Lyric Explanation
1st verse:
ax^2 + bx + c = y
Set y to zero so we we we we we we can find
The x intercepts (x,0) and (x,0)
Because this U-shaped curve
Can cross the x line, two times
Chorus:
Negative b plus or minus the square root of
b squared minus 4ac
All over 2a
Give up the two x…
The two x intercepts
Where that quadratic curve
Crosses the x axis
Take the equation y = x^2 + 5x - 39
a is 1, b is positive 5
c is - 39, keep the sign
2nd verse:
x^2 - 5x + 4 = 0
a is 1, b -5, c is 4
It’s a free throw
To use the formula to get two x intercepts
A U-shaped curve hits the 1 and the 4
Along the x
Bridge:
If y = -x^2
It means the U is inverted
With a pretty wide world
If y = 10x^2
That means the U is quite normal
With a narrow curve
3rd verse:
-x^2 -5x -4 = 0
a -1, b -5, c -4
It’s for real
We use the formula to get the two x intercepts
Inverted U at -1 and -4
Along the x
System of Inequalities
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View: Area of Shade Lyric Explanation
Verse 1:
y > 3x + 5
Graph using y-intercept and slope
And shade above the dotted line
y ≤ x + 9
This goes on the same, same graph
Shade below the solid line
Graphing two linear inequalities
Do this one then that
Look for the shaded area that overlaps
That’s the solution for this system
This is what I’m telling you
Those are the coordinates that make both inequalities true
Chorus:
Two lines in a system of inequalities
Shade this one this way
And that one over there
Solve the system of inequalities
Find what’s in common
And what area of shade they share
Verse 2:
y < 3x + 5
Graph using y-intercept and slope
And shade below the dotted line
y ≥ x + 9
This goes on the same, same graph
Shade above the solid line
Graphing two linear inequalities
Do this one then that
Look for the shaded area that overlaps
That’s the solution for this system
This is what I’m telling you
Those are the coordinates that make both inequalities true
Bridge:
Parallel lines are just full of surprises
Shading in opposite directions on there
Like steak and Ramen
They’ve got nothing in common
No shaded solutions, just empty air
System of Equations
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View: In Common Lyric Explanation
Special guest: Dr. Anthony Dove - Background Vocals
Verse 1:
y = x + 4 and y = 2x - 1
A different y-intercept and slope
For each equation
See where the two lines cross
It’s what’s in common
Ordered pair (5,9) is the solution
Chorus:
Find the solution for this system
Take two linear equations
And solve what is both true and in common
It’s where they’re crossing
The shared solution is a coordinate or ordered pair
To get it we graph it, eliminate or use substitution
Verse 2:
Isolate the x or the y, y = x + 4
Substitute y in the other equation
With the value of y from before (2x - 1 = x + 4)
Solve for the x, value that variable
Plug in x to solve for y, then you’ll know
(5,9), hello!
Bridge:
Two lines with the same slope
And different y-intercepts
Those are parallel lines
Yea, they’ll never connect
There’s no solution they share
No point in common, in fact
Nothing in common along the lines
But if they have the same slope
And same y-intercept
Those are the same exact line
They will always connect
Every solution they share
All points in common, in fact
All things in common along one line
Verse 3:
y = x + 4 and y = 2x - 1
We line up the variables
And multiply one equation (-y = -2x + 1)
So when we add the two equations
One variable goes, elimination
Then we solve for x
And use some substitution
(5,9) is the solution
Linear Inequalities
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View: True Lyric Explanation
1st verse:
y = mx + b
Change the equal sign to an inequality
Showing a range of true possibilities
It’s not just numbers on a line
Y is greater than or greater than or equal to me
Or less than or less than or equal you’ll see
Y on the left and alone, think positively
Symbol turns, from a negative divide
Chorus:
Use the y-intercept and slope to draw a solid line
Like greater than or equal to
Or dotted line, like less than
Then shade above the line or below
We shade above with any greater than
Shade below with any less than
Including less than or equal to
We’re, shading the points that make this true
2nd verse:
If y is greater than, it’s a dotted line
We shade above to identify
All the ordered pairs, or coordinates I…
Continue to prepare
Y is less than or equal to - solid line
We shade below to identify
All the ordered pairs along that line
Or coordinates down there
Bridge:
Plug in any coordinate along the line
Or in a shaded region, that would be just fine
And check your work to find a true statement
Like 7 is greater than or equal to 5
Or pick an ordered pair on the shady spot
But not along the dotted line, we don’t work with that
Now check your work to find a true statement
Like 2 is less than 9
Factoring (GCF)
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View: Bite-Sized Pieces Lyric Explanation
Verse 1:
Find the greatest common factor
For this polynomial
What coefficients, variables, or terms
Are divisible
2x^2yz^2 extract monomial yet?
With the greatest common factor
Out front, what is next?
What’s left over after we factor out the GCF?
Chorus:
Break down complex polynomials
Into bite-sized pieces
Pull out or extract the common factors
In each new simplified nomial
The difficulty decreases
If you factor this
Find divisible factors if they exist
Pull them out and add them to a list
Bridge/Verse 2:
Two x squared, y cubed, z to the fourth
Plus eight x cubed y z squared, minus
Ten x to the fourth, y squared, z to the fifth
Lets make our G-G-G-GCF list…
Factor two x squared y z squared
Times the product
y squared z squared plus four x minus five
x squared y z to the third
A simplified expression
For a complex world