Domain and Range/Functions
We read the domain from _____ to _____ and the range from the _______ to the ______.
Domain: from left to right
Range: from the bottom to the top
What is the highest degree of a quadratic function?
2 (you will not see any exponents larger than 2 in any quadratic function)
Write the equation of a polynomial in factored form with zeroes at 5, -8, and 13.
f(x) = (x-5)(x+8)(x-13)
A rational function is a ratio of two _________.
polynomials
Linear functions have a ________ rate of change while exponential functions have a _________ rate of change.
Exponential - multiplicative rate of change
Is the following set considered a function? Why or why not?
{(2,0) (1,1) (-5, 8) (-3, 6) (2, -1) (8,0)}
No, not a function. The x-value 2 appears more than once with a different output.
What are all three forms of a quadratic function AND their general equations?
Vertex form: y=a(x-h)2+k where (h,k) is the vertex
Standard form: y=ax2+bx+c where c is the y-intercept
Factored form: y=a(x-p)(x-q) where p & q are the roots
Describe what multiplicity is and describe in detail how the multiplicity tells us about graph behavior near an x-intercept.
Multiplicity is the number of times a factor occurs.
When multiplicity is even, the graph behavior BOUNCES off the x-intercept.
When multiplicity is odd, the graph either PASSES STRAIGHT THRU OR CURVES THRU the x-intercept.
Let the equation of a rational function be
g(x)= (5x3+4x2)/(6x3+8).
What is the horizontal asymptote?
Since the degree of the top and the degree of the bottom are EQUAL (both have a degree of 3), the horizontal asymptote is a ratio of the leading coefficients.
y=5/6
Ms. Dababneh decides to come up with a new cell phone policy for our classroom. The first time she catches you on your phone, she'll take it for 1 block period. The second time she catches you on your phone, she'll take it for 2 block periods. The third time she catches you on your phone, she'll take it for 4 block periods. The fourth time, 8 block periods, and so on. Is this policy linear or exponential? How do you know?
Exponential - the amount of blocks are doubling each time she catches you with your phone.
Let h(x)=3x3-4x2+8x-5.
Find h(-2).
3(-2)3-4(-2)2+8(-2)-5
= -61
Let f(x)=(x+4)2+5.
Identify the VERTEX and the AXIS OF SYMMETRY.
VERTEX: (-4,5)
AOS: x= -4
The end behavior will be in opposite directions because the degree of the function is 7, which is odd.
How can we tell if there is a hole in the graph of a rational function by examining its equation? Describe in detail what you must do to figure this out.
1. Make sure the function is fully factored.
2. Cancel out common factors.
3. Any factors that cancel out - there is a hole located at that x-value.
A brand new iPhone is worth $955. Every month that goes by, the iPhone's value decreases by 12%. Write an exponential function that models this situation. Then, find at what month the iPhone's value will be less than $500.
Function: V(m)=955(1-0.12)m where V is the value of the phone and m is the number of months
The iPhone's value will be less than $500 after ~5 months.
The vertex of a parabola is located at the point (-2, 3) and it is facing upwards. What is the domain and range of the graph?
Domain: (-∞,∞)
Range: (3, ∞)
Write the equation of a parabola, in vertex form, that has a vertex of (2, 4) and passes through the point (5, 7).
*hint - you need to solve for a to have the correct equation.*
f(x) = 1/3(x-2)2+4
The equation of a polynomial function is
f(x)=(x-2)3(x+1).
What are the x-intercepts and what are their multiplicities? What is the y-intercept?
x-intercepts: 2 with a multiplicity of 3 and -1 with a multiplicity of 1
y-intercept: f(0)=(0-2)3(0+1)= -8
Ms. Dababneh was feeling petty and decided to make a final exam review packet that is 150 pages long that is a summative grade. Students decided to boycott this by claiming that they should be able to divide the pages evenly amongst all of them to divide the work load. Write a function that gives the amount of pages each student would have to do given the number of students participating in the boycott.
P(s)=150/s where P is the number of pages and s is the number of students.
The y-intercept of an exponential growth model is located at the point (0, 2500). Describe the domain and range of the graph.
*hint: sketching a graph might be helpful.*
Domain: [0,∞)
Range: [2500,∞)
A Tesla has a 80 watt capacity and can travel approximately 15 miles per 1 watt of power. Write a function that models that gives miles traveled per watt AND find the domain and range.
D(p)=15p where D is the miles traveled and p is the number of watts
Domain: [0,80] *(watt capacity)
Range: [0, 1200] *(miles traveled with 0 watts to miles traveled with 80 watts)
The sum of two numbers is 20. Their product is 96. Write a system of equations that models this then solve for the two numbers.
Equations: xy=96, x+y=20
Two numbers: 8 and 12
Sketch the graph of a polynomial with zeroes at -3 (with a multiplicity of 3), -1 (with a multiplicity of 1), and 4 (with a multiplicity of 2), and a y-intercept at 6.
Then based on the end behavior, determine if this function is EVEN or ODD.
*Dababneh will sketch graph on board*
Ends going in same direction so it's EVEN.
Let the equation of a rational function be
h(x) = (x-8)/(x-6).
What is the domain and range?
*hint: need to find the HA and the VA to help describe the domain and range!*
Here our HA is y=1 (ratio of LC) and the VA is x=6 (set denominator to 0 and solve for x).
So our domain is (-∞,6)U(6,∞).
and our range is (-∞,1)U(1,∞).
A microorganism reproduces by binary fission, where every hour each cell divides into two cells. Given that there were 15,141 cells to begin with, determine how many cells there were after 5 hours.