Pre Calculus Quiz 1

refresh on everything

What is the variable for DEGREE

n = degree

What is the variable for leading coeficient?

a = leading coefficient

How do you write something in standard form?

when the terms of a expression is written by descending order of their exponents (largest to smallest)

What is the leading coefficient?

coefficient of the term with the highest degree

What is the degree of a polynomial?

the greatest exponent

How do REAL zeros look on a graph?

they will x intercepts (this also includes irrational ones)

What does the degree represent?

it represents ALL zeros, including the imaginary ones. So don’t think that zeros = x intercepts (cus its only the real ones).

What is special about irrational/imaginary zeros?

THEY COME IN PAIRS!! (±)

How many times can a function intersect the x axis? (ex: how many times does a x² cross the x axis?)

the number of times equals the number of degrees (aka x intercepts).

How do you know the maximum amount of extrema a graph can have?

n - 1 (one less than the degree). That doesn’t mean that the graph DEFINITELY WILL ALWAYS have one less but its more so of a cap.

What happens to a graph at an extrema?

it changes direction or behavior

How do you preform LCT Test?

degree determines the end behavior of your graph.

• if its even, then its going to act like a parabola: goes up or down in the same direction

• if its odd, then its going to act like a cubic function: goes up and down in different direction

coefficients determine its direction (negative or positive).

• In even function with a negative, its going to down in the same direction

• if its an odd function, the left side will now go up, but the right side will go down

even or odd? negative or positive?

even, positive

even or odd? negative or positive?

even, negative

even or odd? negative or positive?

odd, positive

even or odd? negative or positive?

odd, negative

What does an even positive function look like?

What does an even negative function look like?

What does an odd positive function look like?

What is a simple mulitplicity?

it means that it crosses the x-axis once (meaning there is no duplicates of the zero)

What is an even multiplicity?

it bounces of the x-axis. It means that there is at least two of the same zeros. (since this is even, two is least it can be, but it can also be 4, 6, etc)

What is an odd multiplicity?

it snakes through the x axis. This means that there is at least 3 of the same zeros. (since its odd, it could be 3,5,7, etc.)

How do you determine a degree based on a graph?

• count your multiplicities (how many zeros can this graph have based on the multiplicity

• determine if your graph is even or odd

• add an extra possible zero if the zeros from the multiplicities and degree don’t match. (your multiplicity gives you 5 possible zeros but its an even function? add an extra possible zero!)

what is the multiplicity?

there is one multiplicity (simple multiplicity)

What is the multiplicity?

its even (2 multplicities)

what is the multiplicity

its an odd multiplicity, it flattens, changes direction and goes steep again.

How do you quickly identify the y intercepts?

the constant

How do you determine if something has multiplicity algebraically?

fully factor and count how many factors you have (special case: -2x³, its implied that its 2(x-0)³ so its 0 with a multiplicity of 3)

How do you determine the degree/zeros of a function if its fully factored?

count the factors

How do you write the multiplicity for the quiz

x = ___ has a (simple, even,odd) multiplicity of ___, so f(x) (passes, bounces, snakes) through the x axis

How do you write the equation of a function?

• write the x intercepts as factors (there is a zero on 5 it would be x-5)

• f(x) = a(x-b)(x-c), you need to put the right a value for the y intercept. plug in an x intercept to find it. or you can think what number can be divided or multiplied by the constant to equal the y intercept.

What is an even function?

is symmetrical across the y axis

What is an odd function?

its symmetrical over the origin

How do you algebraically determine if a function is odd or even?

plug in -x (literally)

• if its the same, its even

• if ALL of the signs change then its odd

• if neither of these things FULLY happen, its neither

How do you determine if a function equation is even or odd by looking at it?

ALL the exponents should be even or odd. you can use this to make sure your algebra is correct!