Precalc Unit 2 Test prep

Express f(x) in the form f(x) = (x - k)q(x) + r for the given value of k. f(x) = 3x^4 + 5x^3 - 10x^2 + 15, k = -1

3x^4 + 5x^3 - 10x^2 + 15 = (x+1)(3x^3 + 2x^2 - 12x + 12) + 3

What is the difference between a relation and a function?

A RELATION is a set of ordered pairs. A FUNCTION is a relation in wich, for each distinct value of the first component of the ordered pairs, there is exactly one value of the second component Example: using ordered pairs to represent gas prices-- (3, 10.5) indicates we pay $10.50 for 3 gallons of gas. the y-value (price) is dependent on the x-value (number of gallons). A set of ordered pairs in this example would be a function because each x-value is paired with exactly one y-value

Define zero of a polynomial function

A zero of a polynomial function f(x) is a number k such that f(k) = 0. Found through synthetic division and the remainder theorem-- if the remainder 0, f(k) = 0 and k is a zero of f(x); if the remainder does not = 0, then k is not a zero of f(x). A zero of f(x) is a root, or solution, of the equation f(x) = 0 Real number zeros are the x-values of the x-intercepts of the graph of the function.

Does this relation define a function? What's its domain? Range? y = x^8

It is a function DOMAIN: (-∞,∞) RANGE: [0,∞)

Midpoint equation

M = ((x1 + x2)/2), (y1+y2)/2) numbers are subscript for two different points

What makes a graph of a circle nonexistent? What makes a graph of a circle a point?

Nonexistent: r < 0 Point: r = 0

For the polynomial function, factor f(x) f(x) = 10x^3 - 63x^2 +87x +20

Rational zeros = 4, -1/5, 5/2 f(x) = (x-4)(5x+1)(2x-5)

What is the interval notation for the following sets: 1. {x | x > a} 2. {x | a < x < b} 3. {x | x < b} What type of interval are these?

1. (a, ∞) 2. (a, b) 3. (-∞, b) Open interval

Decided whether the relation defines y as a function of x. Domain? Range? y = x^(1/5)

Y is a function of x. DOMAIN: (-∞,∞) RANGE: (-∞,∞)

What is the center-radius form of the equation of a circle?

(x-h)^2 + (y-k)^2 = r^2 Center is (H,k), radius is r

Is the triangle with vertices A(7,2), B(0,6), and C(-8.-8) a right triangle

Yes

-x^2 - 6x - 7 < -2 (less of equal to) What is the solution set in interval notation?

(-∞, -5]∪[-1,∞)

What is the interval notation for the following set: {x | x < a or x > b} What type of interval is this?

(-∞, a) ∪ (b, ∞) Disjoint interval

What is the interval notation for the following set: {x | x is a real number}

(-∞, ∞) All real numbers

Find the center-radius form of the equation of the circle with center (-7,-6) and tangent to the x-axis.

(x+7)^2 + (y+6)^2 = 36

Decide whether or not the following equation has a circle as its graph. If it does, give the center and the radius. If it does not, describe the graph. 49x^2 + 49y^2 -14x +14y -14 = 0

(x^2 - 2/7x + 1/49) + (y^2 + 2/7y + 1/49) = (16/49) (x-1/7)^2 + (y+1/7)^2 = 16/49 Center: (1/7, -1/7) radius: 4/7

What is the interval notation for the following sets: 1. {x | x > a} or equal to 2. {x | a < x < b} x < or equal to b 3. {x | a < x < b} a < or equal to x 4. {x | x < b} or equal to What type of interval are these?

1. [a,∞) 2. (a, b] 3. [a, b) 4. (-∞, b] Other intervals

Determine the center-radius form equation and general form equation for the following endpoints of a circle: (2,4) and (8,4)

Center-radius form: (x-5)^2 + (y-4)^2 = 9 General form: x*2-10x+y^2-8y+32=0

The function g(x) = √x has what domain? What range?

DOMAIN: [0,∞) RANGE: [0,∞)

At how many points does a line tangent touch a circle?

Exactly one point

A baseball is hit so that its height, s, in feet after t seconds is s = -16t^2 + 60t +3 For what time period is the ball at least 47 ft above the ground? Is this time period inclusive or not inclusive?

For the time period between (and INCLUSIVE OF) 1 sec and 2.75 sec the ball will be at least 47 ft above the ground

Decide whether or not the following equation has a circle as its graph. If it does, give the center and the radius. If it does not, describe the graph. 4x^2 + 36x + 4y^2 - 20y - 150 = 0

Graph is a circle with center (-9/5, 5/2). Radius= 8

What is the remainder theorem?

If f(x) is divided by x-k, then the remainder is equal to f(k)

What is the rational zeros theorem

If p/q is a rational number written in lowest terms, and if p/q is a zero of f(x) (a polynomial function with integer coefficients), then p is a factor of the constant term and q is a factor of the leading coefficient f(p/q) = 0 EXAMPLE: f(x) 6x^4 + 7x^3 -12x^2 -3x + 2 p must be a factor of 2 (+/- 1, +/- 2) and q must be a factor of 6 (+/- 1, +/-2, +/-3, +/- 6) Thus possible rational zeros p/q are +/- 1, +/- 2, +/- 1/2, +/- 1/3, +/- 1/6, +/- 2/3

In an ordered pair, which variable is dependent and which is independent?

Independent variable = x Dependent variable = y

Determine range of relation {(6,1), (2,7), (11,13)}

Range = {1,13,7}

Solve the rational inequality. Write the solution in interval notation ((2x-3)(3x+7))/((x-8)^3) > 0 or equal to

Solution set is [-7/3, 3/2] ∪ (8,∞)

Does the relation define y as a function of x. Domain? Range? x + y < 4

Y is not a function of x, because one x-value can correspond to multiple y values (i.e., many numbers can be less than -x + 4 when x is equal to, say, 1) DOMAIN: (-∞,∞) RANGE: (-∞,∞)

Decide whether or not the following equation has a circle as its graph. If it does, give the center and the radius. If it does not, describe the graph. x^2+y^2-10x+2y = -10

The graph is a circle with center (5,-1). Radius = 4

Decide whether or not the following equation has a circle as its graph. If it does, give the center and the radius. If it does not, describe the graph. x^2 + y^2 + 4x - 10y + 33 = 0

The graph is nonexistent (radius < 0)

f(x) = √(4x - 1) and g(x) = 1/x. What is the domain of (f+g)(x)?

The intersection of the domain for f(x) and the domain for g(x)= (-∞, ∞) ∩ [1/4, ∞) = [1/4, ∞)

On a graph, if every vertical line intersects the graph of a relation in no more than one point, what does that mean?

The relation is a function (Vertical Line Test)

Does the equation define y as a function of x. Domain? Range? x = y^12

Y is not defined as a function of x, because y can be both positive and negative, giving one x -value two y-values (fails vertical line test) DOMAIN: [0,∞) RANGE: (-∞,∞)

For the following function, one zero is given. Find all the others. f(x) = x^3 - 11x^2 + 41x - 51; zero = 4 - i

The zeros of f(x) are 4 - i, 4 + i, and 3 (use synthetic division to find a polynomial that results from dividing f(x) by 4 - i. Then divide that polynomial by 4 + i (the natural other zero) to find the last zero for the f(x).

Decide whether the given k is a zero of f(x) f(x) = x^3 - 4x^2 +9x - 6, k = 1

Using synthetic division, the remainder is found to be 0. Because the remainder is 0, f(1) = 0 and 1 is a zero of f(x)

When can you use synthetic division?

When a given polynomial x is divided by a first degree binomial of the form x - k

In an inequality equation, when should you reverse the direction if the inequality symbol?

When multiplying or dividing by a negative number

Solve -3x + 5 > -7 Write solution in set builder notation and interval notation

X < 4 Set-builder notation {x | x <4} Interval notation (-∞, 4)

Decide whether the relation defines y as a function of x. Give the domain and range: y = 4x - 10

Y is a function of x. DOMAIN: (-∞, ∞) RANGE: (-∞, ∞)

Determine whether the three points are collinear (0,-5), (-3,-11), (2,-1)

Yes, they are collinear. Find distance from each point: d(A,B)= 3√5 d(B,C,)= 5√5 d(A,C)= 2√5 The sum of two distances is equal to the third distance, so the points are collinear

Does the relation define y as a function of x? Domain? Range? xy = 4

Yes. DOMAIN: (-∞,0) ∪ (0,∞) RANGE: (-∞,0) ∪ (0,∞)

Is y a function of x: y = 6/(x-8) Domain? Range?

Yes. DOMAIN: (-∞,8) ∪ (8,∞) RANGE: (-∞,0) ∪ (0,∞)

Is y defined as a function of x in the following equation: x = y^13 Domain? Range?

Yes. DOMAIN: (-∞,∞) RANGE: (-∞,∞)

Does the relation define y as a function of x? Domain? Range? y = (4x+3)^(1/2)

Yes. DOMAIN: [-3/4, ∞) RANGE: [0,∞) Find the domain by making solving 4x + 3 > 0 or equal to. x cannot be less than -3/4 (which makes the square = 0)

Solve x(x-3) < 18 or equal to. interval notation

[-3,6]

Solve the rational inequality. Write the solution in interval notation (2x-3)(5x+7)/(x-8)^3 > 0 or equal to

[-7/5,3/2] ∪ (8,∞)

Solve the rational inequality. Write the solution in interval notation (9x - 16)/(x^2 + 1) > 0 or equal to

[16/9,∞)

x^2 - 6x < -4 (less than or equal to) Write solution set in interval notation

[3 -√5, 3 + √ 5]

What is the interval notation for the following set: {x | a < x < b} a < x or equal to; x < or equal to b What type of interval is this?

[a, b] Closed interval

Difference quotient equation

[f(x+h)-f(x)]/h

Let f(x) = 2x^2 - 3x. Find the difference quotient and simplify

[f(x+h)-f(x)]/h [2(x+h)^2-3(x+h)]-(2x^2 - 3x)/h = [(2x^2+4xh+2h^2) - 3x+3h ] - 2x^2 - 3x/h = 4xh + 2h^2 - 3h/h = 4x + 2h - 3

Factor f(x) into linear factors given that k is a zero of f(x) f(x) = x^4 + 3x^3 - 20x^2 - 84x - 80; k = -2 (multiplicity 2)

[multiplicity 2 means that -2 is the zero of two polynomials in the function] f(x) = (x+2)^2(x-5)(x+4) Use synthetic division to find the polynomial that results from dividing f(x) by (x + 2). Then divide the resulting polynomial by (x+2). Factor the polynomial that results from that division.

A projectile is launched from ground level with an initial velocity of v0 feet per second. Neglecting air resistance, its height in feet t seconds after launch is given by s = -16t^2 + v0t. Find the time(s) that the projectile will (a) reach a height of 320 ft and (b) return to the ground when v0 is 64 feet per second

a) 360 = -16t^2 + 64v0t t = b) 0 = -16t^2 + 64v0t t =

Descartes' Rule of Signs Let f(x) define a polynomial function w/ real coefficients and a nonzero constant term, with terms in descending powers of x

a) the number of positive real zeros of f either equals the number of variations in sign occuring in the coefficients of f(x), or is less than the number of variations by a positive even integer b) The number of negative real zeros of f either equals the number of variations in sign occurring in the coefficients of f(-x) or is less than the number of variations by a positive even integer f(x) = x^4 - 6x^3 + 8x^2 + 2x - 1 f(x) has three variations in signs

For the point P(-17,-20) and Q(-10,-15), find the distance d(P,Q) and the coordinates of midpoint M of the segment PQ

d= √74 M= (-27/2,-35/2)

Use the remainder theorem and synthetic division to find f(k) f(x) = -2x^3 - 10x^2 - 13x + 4, k = -3

f(-3) = 7

Use the remainder theorem and synthetic division to find f(k) f(x) = x^2 - 5x + 5, k = 2 + i

f(2 + I) = -2 - I [remember, i^2 = -1]

Use the remainder theorem and synthetic division to find f(k) f(x) = 3x^5 - 8x^3 - 13x^2 - 30, k = 3

f(3) = 366

If (3,4) is on a graph of y = f(x), which of the following must be true? f(3) = 4 or f(4) = 3

f(3) = 4

Use the remainder theorem and synthetic division to find f(k) f(x) = 3x^2 + 60, k = 4i

f(4i) = 12 [remember, i^2 = -1]

Use synthetic division to decide whether the given number k is a zero of the polynomial function. If it is not, give the value of f(k). f(x) = x^2 - 10x + 26, k = 5 - i

f(5 - i) = 0

Use the remainder theorem and synthetic division to find f(k) k = -1; f(x) = x^2 - 2x + 4

f(k) = 7 7 is the remainder to (x^2 - 2x + 4)/(x+1) = (x+1)(x - 3) + 7

Express f(x) in the form f(x) = (x - k)q(x) + r for the given value of k. f(x) = 5x^3 + x^2 + x - 7, k = -1

f(x) = (x+1)(5x^2 - 4x + 5) - 12

Express f(x) in the form f(x) = (x - k)q(x) + r for the given value of k. f(x) = x^3 + 7x^2 + 13x + 6. k = -2

f(x) = (x+2)(x^2 + 5x +3)

Express f(x) in the form f(x) = (x - k)q(x) + r for the given value of k. f(x) = 5x^4 - 4x^3 - 17x^2 - x, k = 4

f(x) = (x-4)(5x^3 + 16x^2 + 47x + 187) + 748

What is the division algorithm?

f(x) = g(x)q(x) + r(x) where f(x) and g(x) are polynomials with g(x) of a lesser degree than f(x) and g(x) of degree 1 or more. r(x) = 0 or the degree of r(x) is less than the degree of g(x)

Find the midpoint of the diameter (the center of the circle), the radius, and the center-radius form for the following endpoints: (9,2) and (-5,6)

midpoint= ((-5+9)/2, (6+2)/2)= (2,4) radius= √((9-2)^2 + (2-4)^2) = √53 center-radius form: (x-2)^2 + (y-4)^2 = 53

What is the center-radius form of the equation of a circle when the center is (0,0)

x^2 + y^2 = r^2

What is the general form equation of a circle?

x^2-2hx+y^2-2ky+(h^2 + k^2 - r^2) = 0