Exponential Functions
What is the equation for an exponential function?
Define the variables
y = abx
a = starting value (y-intercept)
b= growth/decay factor
What is the shape of the graph for a quadratic function? (name and description)
Parabola, u-shaped
When you factor a quadratic, what does your equation need to be equal to?
zero (zero product property)
Factor the trinomial:
x2 + 10x + 25
(x + 5)2
What is the quadratic formula and when can you use it?
Works for any quadratic equation
Given an equation, how can you tell if it is exponential growth/decay?
If b is greater than 1, it is growth.
If b is less than 1, it is decay.
The minimum or maximum of a parabola is located at this point.
The vertex
TWO PARTS:
1.) Factor the equation y = x2 + 10x + 21
2.) Solve using the zero product property
1.) y = (x + 7)(x + 3)
2.) 0 = x + 7 AND 0 = x + 3
x = -7 & -3
Find the possible values that could make this a perfect square trinomial, then factor.
x2 + _______x + 256
x2 + (-32)x + 256 -----> (x-16)2
OR
x2 + 32x + 256 -----> (x+16)2
Solve the following using the quadratic formula. Show your steps!
7x2+9x + 8 = 0
No solutions, the square root of a negative number is not a real number so there is no real solution
TWO PARTS:
1.) A new car costs $23,000. Each year, the value will depreciate by 15%. Write an equation to model this situation. Define your variables.
2.) How much will the car be worth in 7 years?
1.) y = 23000(0.85)x
2.) $7,373.27
What the 3 forms of quadratic equations that we have learned? Give an example of each.
Standard, Factored, Vertex
TWO PARTS:
1.) Factor: 8x2 = 30 + 43x
2.) Solve using the zero product property
1.) 0 = (8x + 5)(x - 6)
2.) x = -5/8 & 6
Fill in the blank with the value that would make this a perfect square trinomial, then factor it.
9x2 + ______x + 25
9x2 + 30x + 25
(3x + 5)2
How many solutions does this equation have? Rational or irrational?
2x2 − 7x − 13 = −10
2 irrational solutions
On the first day, there is 2.5 mm of mold on the bread. Each day, the mold increases by 7.25%. Write an equation to model the situation. Define your variables.
y = 2.5(1.0725)x
TWO PARTS:
1.) The solutions for quadratics have many names, what are they and where are they found?
2.) How many solutions can a quadratic function have?
1.) Zeros, roots, solutions, x-intercepts. Found where the parabola crosses the x-axis
2.) 2, 1, 0
Name 5 values of c which make the expression factorable:
x2 − 3x + c
Examples Include: 0, 2, -4, -10, -18
Solve by completing the square- you MUST show your work.
x2 − 12x + 23 = 0
Is the solution irrational or rational? Explain.
x ~ 9.61 & 2.39
Irrational
How many solutions will this equation have? Will they be rational or irrational? What are the solutions?
5x2 = 80
2 rational solutions
x = 4,-4
A business owner receives a $6,500 loan with 11.5% interest, compounded quarterly. If he pays nothing on the loan for 7 years, how much would he owe?
6500(1+ 0.11/4)28=$14,374.40
TWO PARTS:
1.) How can you tell if a parabola will open upwards or downwards?
2.) How can you tell if the parabola will be wide or narrow?
1.) Negative a will result in a downward parabola. Positive a will be upward.
2.) A values farther from zero will produce narrower parabolas. Ex: -9 is narrower than 1/2
For what 6 values of b is the expression factorable?
x2 + bx + 12
13, 8, 7, −13, −8, −7
Solve by completing the square- show ALL work!
2x2 +32x - 16 = 18
x = 1 OR x = -17
Rational
Solve using the quadratic formula. Rational or irrational solutions?
x2 − 31 − 2x = −6 − 3x2 − 2x
x = 2.5 & -2.5