Exponential Function 2a1
State the general form of an exponential function.
f(x)=bx, where b>0 and b≠1
What is the y-intercept of f(x)=2x?
(0,1)
How many degrees are in one radian?
57o
Define sine and cosine on the unit circle
sin(θ)=y, cos(θ)=x
How does the unit circle help explain why sine and cosine values repeat every 2π radians?
3b1
Because the circle’s circumference repeats positions after one full revolution, bringing (x, y) coordinates back to the same point.
What is the domain of all exponential functions?
All real numbers
How does changing the base affect the steepness of an exponentional base?
Larger bases grow faster; smaller (0 < b < 1) decay faster.
Convert 120° to radians.
2pi/3
What is sin(0°) and cos(0°)?
0 and 1
Why do mathematicians prefer radians over degrees when working with trigonometric functions?
Because radians directly relate arc length to radius, making formulas like s=rθand derivatives simpler and consistent.
Evaluate f(x)=3x when x=−2
1/9
What transformation occurs in f(x)=2x-3+1?
Shift 3 units right and up 1 unit.
What is a coterminal angle of 45° between 0° and 360°?
405° (or −315°)
What is sin(90°)
What is cos(180°)
1 and -1
How can you tell from a graph whether an exponential function represents growth or decay?
Growth curves rise as x increases (b > 1), decay curves fall as x increases (0 < b < 1).
Find a function passing through (0, 4) and (2, 16).
f(x)=4⋅2x
Compare f(x)=2x and g(x)=(½)x.
f(x) grows (increasing), g(x) decays (decreasing).
Find the length of an arc with radius 6 and central angle π/3.
2pi
Identify the domain and range of sine and cosine.
Domain: all real numbers; Range: [−1, 1]
Describe the range of an exponential function with a positive base.
(0,infintiy)
Write an exponential equation that passes through (2, 2) and (1, 6).
a=6/b
6/.33 (1/3)
=18(1/3)x
Create an equation that is shift right 3 units, down 2 units, and reflected across the x-axis.
f(x)= -2x-3-2
Describe what it means for an angle to be in standard position.
its vertex is at the origin and its initial side lies on the positive x-axis.
At what quadrantal angles is cosine equal to 0?
90° (π/2) and 270° (3π/2)
If each of the math concepts were characters in a movie, which one would be the “predictable hero,” the “mysterious twist,” "plot twist" and the “uncontrollable force”?
(Think: angles, sine/cosine, exponential functions, and graph transformations.)
Explain your choices based on how each behaves mathematically.
Angles = the reliable hero (always consistent)
Sine/Cosine = mysterious twist (cyclical ups and downs)
Exponential functions = unstoppable force (growth or decay)
Graph transformations = plot twist (change everything without changing the core story)