Partnership for Assessment of Readiness for College and Careers (PARCC) Mathematics: High School Algebra and Functions Flashcards

24 sample cards from the Mathematics: High School Algebra and Functions deck

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

1. State the three main laws of logarithms (product, quotient, power).

   log(MN) = log M + log N; log(M/N) = log M - log N; log(M^p) = p * log M.

2. What is the change of base formula for logarithms?

   log_b(x) = log_c(x) / log_c(b), for any valid base c (commonly 10 or e).

3. What is the natural logarithm, and what is its base?

   The natural logarithm, written ln(x), is the logarithm with base e (approximately 2.71828).

4. What are the values of log_b(1) and log_b(b) for any valid base b?

   log_b(1) = 0 and log_b(b) = 1.

5. Define sine, cosine, and tangent using the sides of a right triangle (SOH-CAH-TOA).

   sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent.

6. State the Pythagorean trigonometric identity.

   sin^2(x) + cos^2(x) = 1.

7. What are the amplitude and period of y = a*sin(bx)?

   Amplitude = |a|; period = 2*pi / |b|.

8. How do you convert between degrees and radians?

   Multiply degrees by pi/180 to get radians; multiply radians by 180/pi to get degrees. (180 degrees = pi radians.)

9. What are the exact values of sin(30 degrees), sin(45 degrees), and sin(60 degrees)?

   sin(30) = 1/2, sin(45) = (square root of 2)/2, sin(60) = (square root of 3)/2.

10. What is the difference between the domain and range of a function?

    The domain is the set of all valid input (x) values; the range is the set of all resulting output (y) values.

11. What does the average rate of change of a function over an interval [a, b] equal?

    (f(b) - f(a)) / (b - a), the slope of the secant line between the two points.

12. What is the difference between a relative (local) maximum and an absolute (global) maximum?

    A relative maximum is the highest point in a local neighborhood; an absolute maximum is the highest value over the entire domain.

13. What does it mean for a function to be even or odd, and the symmetry of each?

    Even: f(-x) = f(x), symmetric about the y-axis. Odd: f(-x) = -f(x), symmetric about the origin.

14. How do f(x) + k and f(x + k) transform the graph of f(x)?

    f(x) + k shifts the graph up by k (down if k negative); f(x + k) shifts the graph left by k (right if k negative).

15. How do a*f(x) and f(-x) transform a graph?

    a*f(x) vertically stretches (|a|>1) or compresses (|a|<1), and reflects over the x-axis if a is negative; f(-x) reflects the graph over the y-axis.

16. How do you find the inverse of a function algebraically?

    Swap x and y in the equation, then solve for y; the result is f inverse, valid when the original function is one-to-one.

17. What does it mean for two functions to be composed, written (f o g)(x)?

    (f o g)(x) = f(g(x)); you evaluate g at x first, then apply f to that result.

18. What is the general form of an arithmetic sequence's nth term?

    a_n = a_1 + (n - 1)d, where a_1 is the first term and d is the common difference.

19. What is the general form of a geometric sequence's nth term?

    a_n = a_1 * r^(n-1), where a_1 is the first term and r is the common ratio.

20. What does it mean to factor a quadratic, and what is the factored form's connection to roots?

    Factoring rewrites ax^2 + bx + c as a product like a(x - r1)(x - r2); the roots r1 and r2 are the x-intercepts (zeros) of the function.

21. State the difference of squares factoring pattern.

    a^2 - b^2 = (a + b)(a - b).

22. What is the process of completing the square for x^2 + bx?

    Add (b/2)^2 to form a perfect square trinomial: x^2 + bx + (b/2)^2 = (x + b/2)^2.

23. Why are polynomials said to be closed under addition, subtraction, and multiplication?

    Because adding, subtracting, or multiplying any two polynomials always produces another polynomial (the result stays within the system).

24. When creating an equation to model a real-world situation, what do the slope and y-intercept of a linear model typically represent?

    The slope represents the constant rate of change (per-unit cost, speed, etc.), and the y-intercept represents the initial or starting value when the input is zero.

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