Algebra 1
Unit 2
Polynomials & Rational Functions
Polynomials & Rational Functions
Unit 3
Complex Numbers
& Rational Exponents
Complex Numbers
& Rational Exponents
Unit 4
Exponential Functions and Equations
Exponential Functions and Equations
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100
What is the solution of this equation?
x + 2 = 0
Subtract 2 from both sides.
x = -2
100
What are the solutions of the following equation
(x + 5)(x - 4) = 0
Set factors equal to zero: x + 5 = 0 and x - 4 = 0
Solve for x: x = -5 and x = 4
100
What is an example of a complex number?
It is a number of the form a + bi, a and b are real numbers and i is imaginary.
100
What is the form of an exponential Function?
f(x) = a(b)x
100
What is the value of e to the thousandths place?
2.718
200
What is the solution of this equation?
2x + 2 = 0
Subtract 2 from both sides: 2x = -2
Divide both sides by 2: x = -1
200
What are the solutions to
5/(x -1) = x + 3
if we cross multiply we get: 5 = (x - 1)( x+ 3)
Multiply the Factors: 5 = x2 + 2x - 3
Subtract 5 from both sides: x2 + 2x - 8 = 0
Rewrite in Factor Form:(x + 4)(x - 2) = 0
Solve for x , x = -4 and x = 2
200
What is the value of i?
sqrt(-1)
200
Exponential Functions are written in the form
f(x) = a(b)x, What do the constant a and the base b represent in the equation?
a represent the starting or initial value.
b represents the common factor.(Growth or Decay)
200
The point (0,1) ends up at (-3, -3) on the graph of g. Write an equation for g in function notation that represents the transformation.
g(x) = f(x +3) - 4
300
What inequality produces this graph?
x < y
300
Based on the graph what are the solutions to
f(blue) = g(red)
The solutions are x = -1, x = 0 and x = 1
300
The expression (b2 – 4ac) is called the discriminant. How can you use the discriminant to determine if a quadratic equation has real or non real solutions?
If the discriminant is > 0 if will have at least 1 real solution if not 2.
If the discriminant is < 0 it will have no real solutions.
300
What is the exponential form of ln(1) = 0?
e0 = 1
300
Solve 2x2 + 12x + 4 = 0 by completing the square; express the result in simplest form.
2x2 + 12x + 4 = 0
Divide both sides by 2: x2 + 6x + 2 = 0
Subtract 2 from both sides: x2 + 6x = -2
add (b/2)2 to both side: x2 + 6x + 9 = 7
Factor form: (x + 3)2 = 7
Square Root Both sides: x + 3 = +/- sqrt(7)
Solve for x: x = -3 - sqrt(7) and x = sqrt(7) - 3
400
What inequalities produce this graph?
a. x + y > 5 b. x - y > 5 c. 2x - y < 4
d. 2x - y > 5 e. 2x + y < 4 f. x - y > -4
a. x + y > 5 and c. 2x - y < 4
400
A train traveled m miles in t hours, while traveling at a rate of s.
Create an equation that represents how much time it took the train to travel that distance.
t = m/s
400
Which of the following equations have no real solutions?
a. x2 + 6x - 3 = 0
b. 4x2 - 2x + 9 = 0
c. x2 - 6 = 0
Only equation b will have no real solutions because it is the only equation that would have a discriminant which is less than 0.
b2 – 4ac
(-2)2 - 4(4)(9)
4 - 144 = -140
400
f(x)= 9350845(4/3)x/2
Is this function representing exponential growth or decay?
What is the factor?
Exponential growth.
The growth factor here is (4/3)(1/2) or sqrt(1.33).
400
Let h(x) = g(x) + f(x) What is the algebraic equation of h(x), describe it's graph?
Based on the graphs:
g(x) = (x - 3)2 - 1 and f(x) = -(x - 3)2 + 6
h(x) = (x2 - 6x + 9) -1 - (x2 - 6x + 9) + 6
h(x) = x2 - 6x + 9 -1 - x2 + 6x - 9 + 6
h(x) = 5, the graph will look like a horizontal line crossing the y-axis at 5.
500
The length of a rectangle exceeds its width by 4 inches. Find the dimensions of the rectangle if its area is 96 square inches.
length = x + 4, width = x
x(x+4) = 96, x2 + 4x - 96 = (x + 12)(x - 8)
x + 12 = 0 and x - 8 = 0, x = -12 and x = 8
x = 8, so 8 by 12
500
rewrite
in the form
c(x) = y = x + 8 + (26/x-3)
500
Which of the following equations have a real solution?
a. sqrt(x) + 1 = 0
b. sqrt(x) + 10 = 8
c. 4sqrt(x) - 9 = 0
d. sqrt(5x) - 3 = - 1
equations c and d have real solutions because those equations have a sqrt(x) equaling a positive value.
c. sqrt(x) = 9/4
d. sqrt(5x) = 2
500
f(x) = (6/7)2x
Is this function representing exponential growth or decay?
What is the factor?
The function is representing exponential decay.
The decay factor here is (36/49).
500
A plot of land for sale has a width of x ft., and a length that is 8ft less than its width. A farmer will only purchase the land if it measures 240 square feet. What value of x will cause the farmer to purchase the land?
width = x, length = x - 8
x(x - 8) = 240, x2 - 8x - 240 = 0
(x - 20)( x + 12) = 0
x = 20 and x = -12, so a value of 20.