Linear Functions
What is the general form of a linear function?
y=mx+b
What is the general form of an exponential function?
y=a⋅bx
What is the standard form of a quadratic function?
y=ax2+bx+c
What is the formula for a standard exponential function?
y= abx
Solve for x: 2x+3=7
x=2
How do you identify the slope of a linear function from its graph?
The slope of a linear function can be identified as the rise over run between two points on the graph, or the coefficient of x in the equation y=mx+b
How does the graph of an exponential function behave as x increases?
As x increases, the graph of an exponential function either grows rapidly (exponential growth) or decreases rapidly (exponential decay).
If (x-3)(x+5) = x2+bx-15, what is the value of b?
b=2
The population of the African country Zambia is growing continuously at a rate of about 3% per year.
In 2019, the population was 13.47 million.
Assuming this rate of growth will continue, what will Zambia’s population be in the year 2030?
Use the growth formula:
A=13.47(1+0.03)(2030-2019)
A=13.47(1.03)11
A=18.65 million
Factor x2-12x+36
(x-6)(x-6) or (x-6)2
If a linear function has a slope of 2 and passes through the point (1,3), what is its equation?
The equation of the linear function is y=2x+1.
If an exponential function grows by a factor of 3 for each unit increase in x, what is its equation?
y=3x
What is the shape of the graph of a quadratic function?
The shape of the graph of a quadratic function is a parabola.
Calculate the total amount using compound interest for a principal of $1000, an interest rate of 5%, compounded annually for 3 years.
$1157.63
Simplify the expression: 3(x−4)+2x
5x−12
Describe a real-world situation that can be modeled by a linear function.
Sample Response: A real-world situation that can be modeled by a linear function is calculating the total cost of items where each item has a fixed price.
Describe a real-world situation that can be modeled by an exponential function.
Sample Example: A real-world situation that can be modeled by an exponential function is population growth where the population doubles every fixed period.
Produce a different quadratic that shares a common expression with x2-4x-32.
First factor x2-4x-32: (x+4)(x-8).
To produce a different quadratic with a common expression, simply pick one of those factors and multiply it by a different factor. Let's say... (x+4)(x+3) = x2+7x+12
A population of bacteria grows continuously at a rate of 5% per hour. If you start with 100 bacteria, how many bacteria will there be after 3 hours?
A=116.183
Solve the system of equations: y=2x+1 and y=−x+4
x=1, y=3
What are the characteristics of the graph of a linear function?
The characteristics of the graph of a linear function include a straight line with a constant slope, and it can be increasing, decreasing, or horizontal.
What is the difference between a growth and decay exponential function?
The difference between a growth and decay exponential function is that in growth, the base b is greater than 1, while in decay, the base bb is between 0 and 1.
David Wright hits a baseball into left field. The height, h, of the baseball, in meters, is given by the equation h=-4.9t2+29.4t+1.5 where t is the number of seconds since the ball was hit.
What is the maximum height attained by the ball?
45.6 m
What number do you get when you keep increasing the number of times you compound interest? Give us the name of the constant and value.
Euler's number is about 2.718...
Solve the equation: 2x3−10x2+12x
The solutions are x=0, x=3, x=2