Core Ideas in Plain Terms

$f(x)$ means "the output when the input is $x$." Small changes to the formula move the graph in predictable ways: $f(x) + k$ shifts it up, $f(x - h)$ shifts it right, and a minus sign flips it. Learn the handful of "parent" shapes and everything else is a shifted, stretched, or flipped version.

Some equations have no real solution because you'd need the square root of a negative. Mathematicians define $i = \sqrt{-1}$, so $i^2 = -1$. A complex number $a + bi$ just has a real part and an "imaginary" part; you add and multiply them like binomials, using $i^2 = -1$.

$\log_b x = y$ is just another way to write $b^y = x$ — it asks "what power turns $b$ into $x$?" That is why logs are the tool for solving equations where the unknown is stuck in an exponent, and why they turn multiplication into addition.

Plugging $a$ into a polynomial gives the remainder when dividing by $(x - a)$: $P(a)$ is the remainder. If $P(a) = 0$, the division is exact and $(x - a)$ is a factor — that is how you hunt for roots.

A rational function blows up where its denominator hits $0$ (vertical asymptotes), and for huge $x$ it flattens toward the ratio of the leading terms (horizontal asymptote). $y = \dfrac{2x+1}{x-3}$: vertical at $x = 3$, horizontal at $y = 2$.

A scary equation is often a familiar one in disguise. $x^4 - 13x^2 + 36 = 0$ is a quadratic in $u = x^2$. Spot the pattern, substitute, solve the easy version, then translate back.