ENEM Mathematics Syllabus

Define a function $f: A \to B$.

A rule associating to each element $x \in A$ (domain) exactly one element $f(x) \in B$ (codomain).

Give the general form and graph shape of a first-degree (affine) function.

$f(x) = ax + b$ with $a \neq 0$; its graph is a straight line with slope $a$ and $y$-intercept $b$.

How do you find the vertex $x$-coordinate of the parabola $f(x)=ax^{2}+bx+c$?

$x_{v} = -\dfrac{b}{2a}$, and $y_{v} = -\dfrac{\Delta}{4a}$.

What is the defining property of an exponential function $f(x)=a^{x}$?

With $a>0, a\neq 1$, equal increments in $x$ multiply $f(x)$ by a constant factor; it satisfies $f(x_1+x_2)=f(x_1)f(x_2)$.

State the definition of the logarithm $\log_{a} b$.

$\log_{a} b = x \iff a^{x} = b$, with $a>0, a\neq 1, b>0$.

Give the logarithm product and power rules.

$\log_{a}(xy) = \log_{a}x + \log_{a}y$ and $\log_{a}(x^{n}) = n\log_{a}x$.

What is the general term of an arithmetic progression (AP)?

$a_{n} = a_{1} + (n-1)r$, where $r$ is the common difference.

Give the formula for the sum of the first $n$ terms of an AP.

$S_{n} = \dfrac{(a_{1}+a_{n}) \cdot n}{2}$.

What is the general term of a geometric progression (GP)?

$a_{n} = a_{1} \cdot q^{\,n-1}$, where $q$ is the common ratio.

Give the sum of an infinite convergent GP with $|q|<1$.

$S = \dfrac{a_{1}}{1-q}$.

State the area of a triangle given base $b$ and height $h$, and Heron's formula.

$A = \dfrac{b \cdot h}{2}$; Heron: $A = \sqrt{s(s-a)(s-b)(s-c)}$ where $s = \dfrac{a+b+c}{2}$.

State the Pythagorean theorem.

In a right triangle, $a^{2} + b^{2} = c^{2}$, where $c$ is the hypotenuse and $a, b$ the legs.

Give the formulas for the circumference and area of a circle of radius $r$.

Circumference $C = 2\pi r$; area $A = \pi r^{2}$.

What is the sum of the interior angles of a convex polygon with $n$ sides?

$S = (n-2) \cdot 180^{\circ}$.

Give the volume formulas for a rectangular box (prism) and a cylinder.

Box: $V = a \cdot b \cdot c$ (product of edges). Cylinder: $V = \pi r^{2} h$.

Give the volume formulas for a cone and a sphere.

Cone: $V = \dfrac{1}{3}\pi r^{2} h$. Sphere: $V = \dfrac{4}{3}\pi r^{3}$.

State the volume of a pyramid.

$V = \dfrac{1}{3} \cdot A_{\text{base}} \cdot h$, one-third of the base area times the height.

Give the distance formula between two points in the plane.

$d = \sqrt{(x_{2}-x_{1})^{2} + (y_{2}-y_{1})^{2}}$.

Give the midpoint formula for the segment joining $(x_1,y_1)$ and $(x_2,y_2)$.

$M = \left(\dfrac{x_{1}+x_{2}}{2}, \dfrac{y_{1}+y_{2}}{2}\right)$.

What is the slope-intercept equation of a line and how is slope computed from two points?

$y = mx + b$, where $m = \dfrac{y_{2}-y_{1}}{x_{2}-x_{1}}$ is the slope.

Give the standard equation of a circle with center $(a,b)$ and radius $r$.

$(x-a)^{2} + (y-b)^{2} = r^{2}$.

Define the sine, cosine, and tangent ratios in a right triangle.

$\sin\theta = \dfrac{\text{opposite}}{\text{hypotenuse}}$, $\cos\theta = \dfrac{\text{adjacent}}{\text{hypotenuse}}$, $\tan\theta = \dfrac{\text{opposite}}{\text{adjacent}}$.

State the fundamental trigonometric identity.

$\sin^{2}\theta + \cos^{2}\theta = 1$.

State the Law of Sines and the Law of Cosines.

Law of Sines: $\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}$. Law of Cosines: $a^{2} = b^{2} + c^{2} - 2bc\cos A$.

Give the exact values of $\sin$, $\cos$, and $\tan$ for $30^{\circ}$.

$\sin 30^{\circ} = \dfrac{1}{2}$, $\cos 30^{\circ} = \dfrac{\sqrt{3}}{2}$, $\tan 30^{\circ} = \dfrac{\sqrt{3}}{3}$.

How do the mean, median, and mode differ as measures of central tendency?

Mean is the arithmetic average $\bar{x}=\dfrac{\sum x_i}{n}$; median is the middle value of ordered data; mode is the most frequently occurring value.

Define variance and standard deviation of a data set.

Variance $\sigma^{2} = \dfrac{1}{n}\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}$ (mean squared deviation); standard deviation $\sigma = \sqrt{\sigma^{2}}$.

State the classical (Laplace) definition of the probability of an event.

$P(A) = \dfrac{\text{number of favorable outcomes}}{\text{number of possible outcomes}}$, for equally likely outcomes, with $0 \leq P(A) \leq 1$.

Give the addition rule for the probability of $A$ or $B$.

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$; for mutually exclusive events $P(A \cap B)=0$.

Give the formulas for the number of arrangements (permutations $P_n$) and combinations $C_{n,p}$.

$P_{n} = n!$ orderings of $n$ elements; $C_{n,p} = \dbinom{n}{p} = \dfrac{n!}{p!\,(n-p)!}$ chooses $p$ from $n$ ignoring order.