Solving Nonlinear Functions - SAT Math

1. Identify the axis of symmetry for $y = x^2 + 6x + 5$.

   $x = -3$. Use formula $x = -\frac{b}{2a} = -\frac{6}{2(1)} = -3$.

2. Convert $y = x^2 + 6x + 9$ to vertex form.

   $y = (x+3)^2$. Perfect square: $(x+3)^2 = x^2 + 6x + 9$.

3. What is the solution to $4x^2 - 4x + 1 = 0$?

   $x = \frac{1}{2}$. Perfect square: $(2x-1)^2 = 0$ gives $x = \frac{1}{2}$.

4. Find the roots of $x^2 - 5x + 6 = 0$ using factoring.

   $x = 2, x = 3$. Factors as $(x-2)(x-3) = 0$, so $x = 2$ or $x = 3$.

5. State the relationship between roots and factors of a quadratic.

   Roots are solutions; factors are $(x - r_1)(x - r_2)$. If roots are $r_1, r_2$, then factors are $(x-r_1)(x-r_2)$.

6. What is the discriminant in the quadratic formula?

   $b^2 - 4ac$. Expression under the square root that determines root types.

7. Convert $y = x^2 + 4x + 4$ to vertex form.

   $y = (x+2)^2$. Complete the square: $(x+2)^2 = x^2 + 4x + 4$.

8. What is the solution for $4x^2 = 16$?

   $x = 2$ or $x = -2$. Divide by 4: $x^2 = 4$, so $x = \pm 2$.

9. What is the vertex of $y = x^2 - 2x + 1$?

   $(1, 0)$. Perfect square: $(x-1)^2$ has vertex at $(1,0)$.

10. What is a nonlinear function?

    A function that is not a straight line. Graph is not a straight line (degree > 1).

11. Find the discriminant of $x^2 - 4x + 4 = 0$.

    $0$. Calculate $b^2 - 4ac = 16 - 16 = 0$.

12. What is the sum of the roots of $ax^2 + bx + c = 0$?

    $-\frac{b}{a}$. Vieta's formula for sum of roots.

13. Identify the discriminant of $x^2 + 4x + 4 = 0$.

    $0$. Calculate $b^2 - 4ac = 16 - 16 = 0$.

14. Determine the nature of roots for $x^2 + 2x + 5 = 0$.

    Complex. Discriminant $4 - 20 = -16 < 0$ means complex roots.

15. Find the vertex of $y = -x^2 + 6x - 9$.

    $(3, 0)$. Perfect square: $-(x-3)^2$ has vertex at $(3,0)$.

16. State the quadratic formula used to solve $ax^2 + bx + c = 0$.

    $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Derived by completing the square on the general quadratic form.

17. What is the vertex of $y = -2(x+3)^2 + 4$?

    $(-3, 4)$. Vertex form shows $(h,k) = (-3,4)$ directly.

18. What is the standard form equation of a quadratic function?

    $ax^2 + bx + c = 0$. General form where $a \neq 0$ defines any quadratic equation.

19. State the quadratic formula.

    $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Standard formula for solving quadratic equations.

20. Find the roots of $x^2 - 5x + 6 = 0$.

    $x = 2$ or $x = 3$. Factor: $(x-2)(x-3) = 0$ gives roots.

21. Which form is $y = a(x-p)(x-q)$?

    Factored form. Shows roots $p$ and $q$ directly as factors.

22. Solve for $x$: $x^2 - 9 = 0$.

    $x = 3$ or $x = -3$. Difference of squares: $(x-3)(x+3) = 0$.

23. What are the roots of $x^2 - 1 = 0$?

    $x = 1$ or $x = -1$. Difference of squares: $(x-1)(x+1) = 0$.

24. Solve for $x$: $x^2 + 6x + 9 = 0$ using the square root method.

    $x = -3$. Perfect square trinomial: $(x+3)^2 = 0$, so $x = -3$.

25. Identify the roots of $x^2 - 6x + 9 = 0$.

    $x = 3$. Perfect square: $(x-3)^2 = 0$ has repeated root.

26. What is the product of the roots of $ax^2 + bx + c = 0$?

    $\frac{c}{a}$. Vieta's formula for product of roots.

27. Convert the quadratic $y = 2x^2 + 8x + 6$ to vertex form.

    $y = 2(x+2)^2 - 2$. Complete the square: factor out 2, then add/subtract 4.

28. Which method can solve $x^2 + 4x + 4 = 0$ besides factoring?

    Completing the square. Transforms equation to perfect square trinomial form.

29. What does a positive discriminant indicate about a quadratic's roots?

    Two distinct real roots. When $b^2 - 4ac > 0$, the parabola crosses the x-axis twice.

30. What is the standard form of a quadratic equation?

    $ax^2 + bx + c = 0$. General form with degree 2 polynomial set to zero.

31. Solve for $x$: $x^2 - 4 = 0$.

    $x = 2$ or $x = -2$. Factor as difference of squares: $(x-2)(x+2) = 0$.

32. What is the degree of a quadratic function?

    2. Highest power of variable is 2.

33. What is the effect of $a < 0$ in $y = a(x-h)^2 + k$?

    Parabola opens downward. Negative coefficient makes parabola open downward.

34. If $y = x^2 + 2x + 1$, what is the axis of symmetry?

    $x = -1$. Perfect square: $(x+1)^2$, so axis is $x = -1$.

35. Convert $y = x^2 - 4x + 6$ to vertex form.

    $y = (x-2)^2 + 2$. Complete the square: $(x-2)^2 = x^2 - 4x + 4$.

36. Find the vertex of $y = 2(x - 3)^2 + 4$.

    Vertex: $(3, 4)$. In vertex form, $(h,k)$ gives the vertex coordinates directly.

37. What type of solutions does a quadratic have if the discriminant is negative?

    Complex solutions. Negative discriminant means no real roots.

38. Identify the maximum or minimum value of $y = (x-2)^2 + 5$.

    Minimum: $5$. Since $a > 0$, parabola opens up with minimum at vertex.

39. Which form is $y = a(x-h)^2 + k$?

    Vertex form. Form that shows vertex $(h,k)$ and transformations.

40. Find the value of $x$ in $x^2 + 4x + 4 = 0$.

    $x = -2$. Perfect square: $(x+2)^2 = 0$ gives $x = -2$.

41. Find the axis of symmetry for $y = 2x^2 + 4x + 1$.

    $x = -1$. Use $x = -\frac{b}{2a} = -\frac{4}{2(2)} = -1$.

42. Solve for $x$: $x^2 + 9 = 0$.

    $x = 3i$ or $x = -3i$. Take square root: $x^2 = -9$, so $x = \pm 3i$.

43. For $y = (x-1)^2 + 3$, identify the vertex.

    $(1, 3)$. Vertex form shows $(h,k) = (1,3)$ directly.

44. For $y = 3(x-4)^2 - 7$, what is the vertex?

    $(4, -7)$. Vertex form shows $(h,k) = (4,-7)$ directly.

45. What is the vertex form of a quadratic function?

    $y = a(x-h)^2 + k$. Shows vertex $(h,k)$ and vertical shifts/stretches.

46. What does the discriminant $b^2 - 4ac$ indicate about roots?

    Number and type of roots. Determines number of real roots and their nature.

47. What is the effect of $a > 0$ in $y = a(x-h)^2 + k$?

    Parabola opens upward. Positive coefficient makes parabola open upward.

48. Identify the vertex form of a quadratic function.

    $y = a(x-h)^2 + k$. Shows vertex $(h,k)$ and stretch factor $a$ explicitly.

49. For $y = x^2 - 4x + 4$, determine the axis of symmetry.

    $x = 2$. Axis of symmetry is $x = -\frac{b}{2a} = -\frac{-4}{2(1)} = 2$.

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