SAT (Scholastic Assessment Test) Math: Advanced Math Flashcards

24 sample cards from the Math: Advanced Math deck

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

1. If the discriminant b² − 4ac > 0, how many real solutions does the quadratic have?

   Two distinct real solutions (the parabola crosses the x-axis at two points).

2. If the discriminant b² − 4ac = 0, how many real solutions are there?

   Exactly one real solution (a repeated/double root); the parabola is tangent to the x-axis.

3. If the discriminant b² − 4ac < 0, how many real solutions are there?

   No real solutions (two complex solutions); the parabola does not cross the x-axis.

4. What is the standard form of a parabola (quadratic function)?

   y = ax² + bx + c, where c is the y-intercept and a controls direction/width.

5. What is the vertex form of a parabola, and what does it reveal?

   y = a(x − h)² + k, where (h, k) is the vertex. It directly shows the maximum or minimum point.

6. What is the factored (intercept) form of a parabola, and what does it reveal?

   y = a(x − p)(x − q), where p and q are the x-intercepts (roots/zeros).

7. How do you find the x-coordinate of a parabola's vertex from standard form?

   x = −b/(2a). This is also the axis of symmetry.

8. What does the sign of 'a' tell you about a parabola?

   If a > 0 the parabola opens upward (vertex is a minimum); if a < 0 it opens downward (vertex is a maximum).

9. What is the axis of symmetry of a parabola?

   The vertical line x = −b/(2a) (or x = h in vertex form) that passes through the vertex and mirrors the two sides.

10. In a quadratic model like h = −16t² + v₀t + h₀ for projectile height, what does the vertex represent?

    The vertex gives the maximum height reached and the time at which it occurs.

11. In a quadratic revenue/profit model, what do the x-intercepts typically represent?

    The zeros/break-even points where the quantity (revenue or profit) equals zero.

12. What is the general form of an exponential function?

    y = a·b^x, where a is the initial value (y-intercept) and b is the constant growth/decay factor (base).

13. In y = a·b^x, how do you tell growth from decay?

    If b > 1 it's exponential growth; if 0 < b < 1 it's exponential decay.

14. How do you write the growth factor for an exponential increasing by r percent each period?

    b = 1 + r (as a decimal), so y = a(1 + r)^x. For 5% growth, b = 1.05.

15. How do you write the decay factor for an exponential decreasing by r percent each period?

    b = 1 − r (as a decimal), so y = a(1 − r)^x. For 5% decay, b = 0.95.

16. What is the key difference between linear and exponential change?

    Linear changes by a constant amount (addition) each step; exponential changes by a constant factor (multiplication/percent) each step.

17. What does function notation f(x) mean, and how do you evaluate f(3)?

    f(x) is the output of function f for input x. To find f(3), substitute 3 for every x in the function and simplify.

18. How does f(x) + k transform the graph of f(x)?

    It shifts the graph vertically: up by k if k > 0, down by k if k < 0.

19. How does f(x + h) transform the graph of f(x)?

    It shifts the graph horizontally: left by h if h > 0, right by h if h < 0 (opposite of the sign inside).

20. How does −f(x) versus f(−x) transform a graph?

    −f(x) reflects over the x-axis (flips vertically); f(−x) reflects over the y-axis (flips horizontally).

21. How does a·f(x) with a > 1 versus 0 < a < 1 affect a graph?

    a > 1 stretches the graph vertically (taller/narrower); 0 < a < 1 compresses it vertically (shorter/wider).

22. What does the end behavior of a polynomial depend on?

    Its leading term: the degree (even vs odd) and the sign of the leading coefficient determine the direction of both ends.

23. What is a vertical asymptote of a rational function and how do you find it?

    A vertical line the graph approaches but never crosses; it occurs where the denominator equals zero (after simplifying) but the numerator does not.

24. How do you typically solve a nonlinear system like y = x² and y = x + 2?

    Use substitution: set the expressions equal (x² = x + 2), rearrange to x² − x − 2 = 0, solve for x, then find y. Solutions are the intersection points of the graphs.

What this deck covers

Math: Advanced Math flashcards FAQ