Knowt
AP Pre-Calc (2025)
Unit 1
1.1 Change in Tandem
Domain is all the x values of a function
Range is all y values of a function
This function is increasing on the intervals x < -0.5 and x > 2
This function is decreasing on the interval -0.5 < x < 2
The function is positive on the intervals -1.5 < x < 1.2 and x > 2.2
The function is negative on the intervals x < -1.5 and 1.2 < x < 2.2
1.2 Rates of Change
Rate of change is the slope of a function
Found by (y1 - y2)/(x1-x2) on a linear function
1.3 Rates of Change in Linear and Quadratic Functions
The average rate of change for a linear function will always be constant
The rate of change of the rate of change should be zero
insert image here.
Average ROC is NOT constant over a quadratic function. The rate of change of the rate of change should be constant.
Increasing ROC | Decreasing ROC | |
Positive ROC | Function is increasing Concave up | Function is decreasing Concave up |
Negative ROC | Function is increasing Concave down | Function is decreasing Concave down |
1.4 Polynomial Functions and Rates of change
axn+bxn-1+โฆ+c
axn = Leading term
n = Degree
a = Leading coefficient
Even degree | Odd degree | |
Positive leading coefficient | As x โ โ, f(x) โ โ As x โ -โ, f(x) โ โ | As x โ โ, f(x) โ โ As x โ -โ, f(x) โ -โ |
Negative leading coefficient | As x โ โ, f(x) โ -โ As x โ -โ, f(x) โ -โ | As x โ โ, f(x) โ -โ As x โ -โ, f(x) โ โ |
Absolute or global minimum/maximum: The highest or lowest point on the entire graph
Relative or local minimum/maximum: The highest or lowest point within an interval or the point of inflection
1.5 Polynomial Functions and Complex Zeros
If f(a) = 0, then a is a zero of the function f(x)
Factor to find zeros
f(x) = (xยฒ - 2x - 3)
f(x) = (x - 2)(x - 1)
x โ 2, 1
2 and 1 are the zeros of the function f(x)
The green line is the first difference in the above function, where the purple line is the second difference. All of the differences in the second difference is even, which means the function is to the second degree.
Even functions are reflected across the y axis. This means f(-x) = f(x). The first degree must be even.
Odd functions are reflected about the origin. This means f(x) = -f(x). The first degree must be odd.
1.6 Polynomial Functions and End Behavior
Even degree | Odd degree | |
Positive leading coefficient | Limxโ-โ f(x) โ โ Lim xโโ f(x) โ โ | As x โ โ, f(x) โ โ As x โ -โ, f(x) โ -โ |
Negative leading coefficient | Limx โ โ, f(x) โ -โ Limx โ -โ, f(x) โ -โ | As x โ โ, f(x) โ -โ As x โ -โ, f(x) โ |
Same thing as 1.4, however it is a different format.
1.7 Rational Functions and End Behavior
End behavior of a rational function is
Unit 2
2.1
Geometric Sequence is a series of numbers where every number is the previous, multiplied by a common ratio.
Example: 6, 12, 24, 48โฆ