SAT Mathematics Flashcards

24 sample cards from the Mathematics deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

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

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

2. 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: $$x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$$

3. 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.

4. 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 $x = h$.

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

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

6. 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 $r_1 \cdot r_2 = \frac{c}{a}$.

7. 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 $f$.

8. 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 $f$.

9. 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 $h<0$).

10. 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.

11. 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 $x = a$.

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

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

13. 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.

14. 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 $0 < b < 1$.

15. 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.

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

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

17. 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).

18. 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.

19. 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 $d = 2r$).

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

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

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

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

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

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

23. 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).

24. 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 $4$.

What this deck covers

Mathematics flashcards FAQ