Mathematics Algebra Flashcards

24 sample cards from the Algebra deck

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

1. What is the domain of a function, and what is its range?

   The domain is the set of all permissible input values ($x$); the range is the set of all resulting output values ($y$ or $f(x)$).

2. What defines a function, and what test checks whether a graph is a function?

   A function assigns exactly one output to each input. The vertical line test: if any vertical line crosses the graph more than once, it is not a function.

3. What is the absolute value $|x|$, and how do you solve $|x| = a$ for $a > 0$?

   $|x|$ is the distance of $x$ from $0$: $|x| = x$ if $x \geq 0$, and $|x| = -x$ if $x < 0$. Solving $|x| = a$ gives $x = a$ or $x = -a$.

4. When you multiply or divide both sides of an inequality by a negative number, what must you do?

   Reverse the direction of the inequality sign. E.g., from $-2x < 6$ you get $x > -3$.

5. What are the vertex form of a parabola and the coordinates of its vertex?

   Vertex form: $y = a(x-h)^{2} + k$, with vertex at $(h, k)$. It opens upward if $a > 0$ and downward if $a < 0$.

6. Give the formula for the $x$-coordinate of the vertex of $y = ax^{2}+bx+c$.

   $x = -\dfrac{b}{2a}$; this is also the axis of symmetry. The $y$-coordinate is found by substituting this value back into the equation.

7. State the Factor Theorem.

   For a polynomial $P(x)$, $(x - a)$ is a factor of $P(x)$ if and only if $P(a) = 0$ (i.e., $a$ is a root).

8. State the Remainder Theorem.

   When a polynomial $P(x)$ is divided by $(x - a)$, the remainder equals $P(a)$.

9. What is a rational expression, and what value(s) must be excluded?

   A rational expression is a ratio of two polynomials $\dfrac{P(x)}{Q(x)}$. Any value making the denominator zero ($Q(x) = 0$) must be excluded from the domain.

10. How do you add two fractions $\dfrac{a}{b} + \dfrac{c}{d}$?

    Use a common denominator: $$\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$$

11. What is a direct proportion versus an inverse proportion?

    Direct: $y = kx$, so $y$ increases as $x$ increases ($k$ constant). Inverse: $y = \dfrac{k}{x}$, so $y$ decreases as $x$ increases.

12. State the formula for the determinant of a $2\times 2$ matrix.

    For $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, $\det(A) = ad - bc$.

13. What is the inverse of a $2\times 2$ matrix, and when does it exist?

    For $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, $$A^{-1} = \frac{1}{ad-bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix},$$ which exists only when $\det(A) = ad - bc \neq 0$.

14. State the commutative, associative, and distributive properties of real numbers.

    Commutative: $a+b = b+a$, $ab = ba$. Associative: $(a+b)+c = a+(b+c)$, $(ab)c = a(bc)$. Distributive: $a(b+c) = ab + ac$.

15. What are the additive identity, multiplicative identity, and their inverses?

    Additive identity: $0$ (since $a+0=a$); its inverse is $-a$. Multiplicative identity: $1$ (since $a\cdot 1=a$); its inverse is $\dfrac{1}{a}$ for $a\neq 0$.

16. How do you simplify a square root such as $\sqrt{50}$ (extracting factors)?

    Factor out perfect squares: $\sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}$. Use $\sqrt{ab} = \sqrt{a}\,\sqrt{b}$.

17. How do you rationalize the denominator of $\dfrac{1}{\sqrt{a}}$?

    Multiply numerator and denominator by $\sqrt{a}$: $\dfrac{1}{\sqrt{a}} = \dfrac{\sqrt{a}}{a}$.

18. What is function composition $(f \circ g)(x)$, and is it commutative?

    $(f \circ g)(x) = f(g(x))$: apply $g$ first, then $f$. It is generally not commutative, so $f(g(x)) \neq g(f(x))$ in general.

19. How do you find the inverse function $f^{-1}(x)$ of a one-to-one function?

    Replace $f(x)$ with $y$, swap $x$ and $y$, then solve for $y$. The result is $f^{-1}(x)$, satisfying $f(f^{-1}(x)) = x$.

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

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

21. Give the distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$.

    $$d = \sqrt{(x_2 - x_1)^{2} + (y_2 - y_1)^{2}}$$

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

    $$M = \left( \frac{x_1 + x_2}{2}, \; \frac{y_1 + y_2}{2} \right)$$

23. Classify polynomials by degree: name degrees 0 through 3.

    Degree 0: constant; degree 1: linear; degree 2: quadratic; degree 3: cubic. (Degree 4 is quartic, degree 5 quintic.)

24. What does the Fundamental Theorem of Algebra state?

    Every non-constant polynomial of degree $n$ with complex coefficients has exactly $n$ roots in the complex numbers, counting multiplicity.

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