Math · 21 · Advanced Math

Functions and Graphs

Notes

The graph is every point $(x,\ f(x))$ : height at $x$ is $f(x)$ .
x-intercepts (zeros): $f(x)=0$ . y-intercept: $f(0)$ .
Point $(a,b)$ on the graph means $f(a)=b$ — plug it in.
Solutions of $f(x)=k$ : where the graph meets the line $y=k$ .
Solutions of $f(x)=g(x)$ : x-coordinates of the intersection points.
Increasing: rises left to right. Decreasing: falls. Read intervals on x.
Max/min: the y-value is the maximum; the x-value is where it occurs.

Point on the graph

(a,b)\text{ on graph}\iff f(a)=b

y-intercept

\big(0,\ f(0)\big)

Zeros and factors

f(r)=0\iff(x-r)\text{ is a factor}

Polynomials only

Intersections

f(x)=g(x)

x-coordinates of the crossing points

Average rate of change

\dfrac{f(b)-f(a)}{b-a}

Slope between two graph points

Giving the x-value when asked for the maximum value (the y), or the reverse.
Reading points without checking the axis scale — gridlines are often not 1 unit.
Counting zeros of $f$ when the question asks how many times $f(x)=k$ .

“How many solutions to f(x) = k?” Add y = k in Desmos and count the crossings.

10 official SAT questions on Functions and Graphs, easiest first.