SAT Mathematics Syllabus

On a scatterplot, what does a line of best fit represent, and what does its slope indicate?

The line of best fit models the trend of the data; its slope indicates the predicted change in the $y$-variable per one-unit change in the $x$-variable.

What is the difference between a positive and a negative correlation on a scatterplot?

Positive correlation: as $x$ increases, $y$ tends to increase (upward trend). Negative correlation: as $x$ increases, $y$ tends to decrease (downward trend).

How do you calculate the mean (arithmetic average) of a data set?

$$\text{mean} = \frac{\text{sum of all values}}{\text{number of values}}$$

How do you find the median of a data set, and what do you do when there is an even number of values?

Order the values; the median is the middle value. With an even count, average the two middle values.

What is the range of a data set, and how does it differ from the standard deviation as a measure of spread?

Range $=$ maximum $-$ minimum. Standard deviation measures the typical distance of values from the mean, using all data points rather than just the extremes.

How is the probability of an event $E$ defined for equally likely outcomes?

$$P(E) = \frac{\text{number of favorable outcomes}}{\text{total number of possible outcomes}}$$

For independent events $A$ and $B$, what is the probability that both occur?

$P(A \text{ and } B) = P(A) \times P(B)$.

What is the standard form of a quadratic equation, and what is the quadratic formula for its solutions?

Standard form: $ax^{2} + bx + c = 0$. Solutions: $$x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$$

What is the discriminant of a quadratic, and what does each sign tell you about the roots?

The discriminant is $b^{2} - 4ac$. If it is positive there are two real roots; if zero, one real (repeated) root; if negative, two complex roots.

What is the vertex form of a quadratic function, and where is the vertex?

$f(x) = a(x - h)^{2} + k$, with vertex at $(h, k)$; the axis of symmetry is $x = h$.

For a quadratic $ax^{2} + bx + c$, what is the $x$-coordinate of the vertex?

$$x = -\frac{b}{2a}$$

State the relationship between the sum and product of the roots $r_1, r_2$ of $ax^{2}+bx+c=0$ and its coefficients.

$r_1 + r_2 = -\frac{b}{a}$ and $r_1 \cdot r_2 = \frac{c}{a}$.

In function notation, what does $f(3) = 7$ mean?

When the input $x = 3$, the output of the function $f$ is $7$; the point $(3, 7)$ lies on the graph of $f$.

What is a composite function $(f \circ g)(x)$, and how is it evaluated?

$(f \circ g)(x) = f(g(x))$: evaluate $g$ at $x$ first, then use that result as the input to $f$.

How does the graph of $f(x) + k$ compare to the graph of $f(x)$, and how does $f(x + h)$ compare?

$f(x) + k$ shifts the graph vertically by $k$ (up if $k>0$). $f(x + h)$ shifts it horizontally by $h$ units to the left (right if $h<0$).

According to the Factor Theorem, what does it mean if $(x - r)$ is a factor of a polynomial $p(x)$?

It means $p(r) = 0$; equivalently, $r$ is a root (zero) of the polynomial and $(r, 0)$ is an $x$-intercept.

What does the Remainder Theorem state about dividing a polynomial $p(x)$ by $(x - a)$?

The remainder equals $p(a)$, the value of the polynomial evaluated at $x = a$.

Factor the difference of squares $a^{2} - b^{2}$.

$$a^{2} - b^{2} = (a - b)(a + b)$$

How does the degree of a polynomial relate to its maximum number of real roots and end behavior?

A degree-$n$ polynomial has at most $n$ real roots. End behavior depends on degree parity and leading coefficient sign: even degree gives both ends the same direction, odd degree gives opposite directions.

What is the general form of an exponential function modeling growth or decay?

$f(x) = a \cdot b^{x}$, where $a$ is the initial value and $b$ is the growth factor: growth if $b > 1$, decay if $0 < b < 1$.

How do you write an exponential model for a quantity that grows at rate $r$ per period versus decays at rate $r$?

Growth: $A = A_0(1 + r)^{t}$. Decay: $A = A_0(1 - r)^{t}$, where $A_0$ is the initial amount and $t$ is time in periods.

What is the compound interest formula for principal $P$ compounded $n$ times per year at annual rate $r$ for $t$ years?

$$A = P\left(1 + \frac{r}{n}\right)^{nt}$$

What is the difference between linear growth and exponential growth?

Linear growth adds a constant amount each period (constant difference), while exponential growth multiplies by a constant factor each period (constant ratio).

State the Pythagorean Theorem and when it applies.

For a right triangle with legs $a, b$ and hypotenuse $c$: $a^{2} + b^{2} = c^{2}$. It applies only to right triangles.

What are the area and circumference formulas for a circle of radius $r$?

Area $= \pi r^{2}$ and circumference $= 2\pi r$ (equivalently $\pi d$, where $d = 2r$).

What is the equation of a circle with center $(h, k)$ and radius $r$?

$$(x - h)^{2} + (y - k)^{2} = r^{2}$$

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

$$(n - 2) \times 180^{\circ}$$

State the SOH-CAH-TOA definitions of sine, cosine, and tangent for a right triangle.

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

What is the Pythagorean trigonometric identity, and how are complementary angle sine and cosine related?

$\sin^{2}\theta + \cos^{2}\theta = 1$, and $\sin\theta = \cos(90^{\circ} - \theta)$ (co-function relationship).

What is the imaginary unit $i$, and what are its powers up to $i^{4}$?

$i = \sqrt{-1}$, so $i^{2} = -1$, $i^{3} = -i$, and $i^{4} = 1$; the powers cycle with period $4$.