Interpreting/Sketching Key Features of Functions - Algebra 2

1. Which interval is $f(x)=x^2-4$ positive on?

   $(-\infty,-2)\cup(2,\infty)$. Parabola is positive outside the roots.

2. What does it mean for a function to be increasing on an interval?

   As $x$ increases, $f(x)$ increases on that interval. The graph slopes upward from left to right.

3. Identify the average rate of change from $x=2$ to $x=5$ given $f(2)=1$ and $f(5)=7$.

   Average rate of change is $\frac{7-1}{5-2}=2$. Formula: $\frac{\text{change in output}}{\text{change in input}}$.

4. On which interval is $f(x)=-2(x+1)^2+7$ decreasing?

   Decreasing on $(-1,\infty)$. Right side of downward-opening parabola.

5. What is the vertex of a parabola written as $y=a(x-h)^2+k$?

   The vertex is $(h,k)$. The turning point of the parabola.

6. Identify the open interval where $f(x)=(x-2)(x-5)$ is negative.

   $ (2,5)$. Parabola is negative between its roots.

7. Which statement gives the end behavior of $f(x)=-3x^3+2$?

   As $x\to\infty$, $f(x)\to-\infty$; as $x\to-\infty$, $f(x)\to\infty$. Odd degree with negative leading coefficient.

8. What does it mean for a function to be odd?

   It satisfies $f(-x)=-f(x)$ and has origin symmetry. Point reflection through the origin.

9. On which interval is $f(x)=(x-3)^2+5$ increasing?

   Increasing on $(3,\infty)$. Right side of upward-opening parabola.

10. What is the meaning of end behavior of a function?

    How $f(x)$ behaves as $x\to\infty$ and as $x\to-\infty$. Describes the function's behavior at extreme $x$-values.

11. Identify the maximum value of $f(x)=-2(x+1)^2+7$.

    Maximum value is $7$. Vertex gives the maximum for downward parabola.

12. Identify the relative minimum point of $f(x)=(x-3)^2+5$.

    The relative minimum is at $(3,5)$. Parabola opens upward with vertex at $(3,5)$.

13. What does it mean for a function to be decreasing on an interval?

    As $x$ increases, $f(x)$ decreases on that interval. The graph slopes downward from left to right.

14. What does the statement "$f$ is positive on $(a,b)$" mean?

    For all $x$ in $(a,b)$, $f(x)>0$. All function values are above the $x$-axis.

15. What is a relative maximum point on a graph?

    A point where $f(x)$ is greatest compared to nearby $x$-values. A local peak on the graph.

16. Identify the $y$-intercept of $f(x)=2x-6$.

    The $y$-intercept is $(0,-6)$. Evaluate $f(0)=2(0)-6=-6$.

17. Identify the $x$-intercept of $f(x)=2x-6$.

    The $x$-intercept is $(3,0)$. Set $2x-6=0$ and solve for $x$.

18. Identify the $x$-intercepts of $f(x)=(x-1)(x+4)$.

    The $x$-intercepts are $(1,0)$ and $(-4,0)$. Set each factor equal to zero.

19. What is the midline of $y=3\sin(x)+2$?

    The midline is $y=2$. Vertical shift of the sine function.

20. Identify the horizontal asymptote of $f(x)=\frac{1}{x-3}$.

    Horizontal asymptote: $y=0$. Rational function approaches zero as $x$ grows.

21. What is the period of $y=\sin(bx)$ for $b>0$?

    The period is $\frac{2\pi}{b}$. Coefficient $b$ compresses the period by factor $b$.

22. What is the definition of periodicity for a function?

    There exists $T>0$ such that $f(x+T)=f(x)$ for all $x$. The function repeats its pattern every $T$ units.

23. Identify the open intervals where $f(x)=(x-2)(x-5)$ is positive.

    $(-\infty,2)\cup(5,\infty)$. Parabola is positive outside its roots.

24. Which interval is $f(x)=x^2-4$ negative on?

    $(-2,2)$. Parabola is negative between the roots.

25. What is the definition of the $y$-intercept of a function's graph?

    The point where $x=0$, so the value is $(0,f(0))$. Where the graph crosses the $y$-axis.

26. Identify the $y$-intercept of $f(x)=(x-1)(x+4)$.

    The $y$-intercept is $(0,-4)$. Evaluate $f(0)=(0-1)(0+4)=-4$.

27. What is the period of $y=\cos(x)$?

    The period is $2\pi$. Standard period for cosine function.

28. On which interval is $f(x)=-2(x+1)^2+7$ increasing?

    Increasing on $(-\infty,-1)$. Left side of downward-opening parabola.

29. Identify the period of $y=\sin(4x)$.

    The period is $\frac{\pi}{2}$. Period formula: $\frac{2\pi}{4}=\frac{\pi}{2}$.

30. What is a relative minimum point on a graph?

    A point where $f(x)$ is least compared to nearby $x$-values. A local valley on the graph.

31. Identify the vertical asymptote of $f(x)=\frac{1}{x-3}$.

    Vertical asymptote: $x=3$. Denominator cannot equal zero.

32. Identify the symmetry of $f(x)=\frac{1}{x}$.

    Odd; it has origin symmetry. Standard reciprocal function has origin symmetry.

33. Identify the period of $y=\cos(\frac{x}{3})$.

    The period is $6\pi$. Period formula: $\frac{2\pi}{1/3}=6\pi$.

34. What is the meaning of $f(a)>0$ in terms of the graph at $x=a$?

    The graph is above the $x$-axis at $x=a$. Positive function values mean points above the axis.

35. What is the end behavior of $f(x)=\frac{1}{x-3}$ as $x\to\pm\infty$?

    As $x\to\pm\infty$, $f(x)\to 0$. Rational function approaches horizontal asymptote.

36. What is the amplitude of $y=3\sin(x)$?

    The amplitude is $3$. Coefficient of sine function.

37. What is the axis of symmetry for a parabola written as $y=a(x-h)^2+k$?

    The axis of symmetry is $x=h$. Vertical line through the vertex of the parabola.

38. Identify the vertex of $f(x)=x^2-6x+10$.

    The vertex is $(3,1)$. From vertex form $(x-3)^2+1$.

39. Which key feature is identified by solving $f(x)=0$?

    The $x$-intercepts (zeros) of the function. Found by setting the function equal to zero.

40. Which value is the $y$-intercept of $f(x)=\sqrt{x+1}$?

    The $y$-intercept is $(0,1)$. Evaluate $f(0)=\sqrt{0+1}=1$.

41. Identify the domain restriction implied by the model $f(x)=\sqrt{x-4}$.

    Domain: $x\ge 4$. Expression under square root must be non-negative.

42. Which statement gives the end behavior of $f(x)=x^4-2x^2$?

    As $x\to\pm\infty$, $f(x)\to\infty$. Even degree with positive leading coefficient.

43. What is the definition of the $x$-intercept of a function's graph?

    A point where $y=0$ on the graph, so $f(x)=0$. Where the graph crosses the $x$-axis.

44. What is the meaning of $f(a)<0$ in terms of the graph at $x=a$?

    The graph is below the $x$-axis at $x=a$. Negative function values mean points below the axis.

45. What is the slope interpretation of average rate of change on $[a,b]$?

    It is the slope of the secant line: $\frac{f(b)-f(a)}{b-a}$. Geometric interpretation of average rate of change.

46. On which interval is $f(x)=(x-3)^2+5$ decreasing?

    Decreasing on $(-\infty,3)$. Left side of upward-opening parabola.

47. What does it mean for a function to be even?

    It satisfies $f(-x)=f(x)$ and is symmetric about the $y$-axis. Mirror image across the $y$-axis.

48. What is the period of $y=\sin(x)$?

    The period is $2\pi$. Standard period for sine function.

49. Which key feature is identified by evaluating $f(0)$?

    The $y$-intercept $(0,f(0))$. Found by evaluating the function at zero.

50. What does the point $(a,b)$ in a table for $f$ represent in context?

    When input is $a$, the output is $f(a)=b$. Input-output relationship in tabular form.

51. What is the period of $y=\cos(bx)$ for $b>0$?

    The period is $\frac{2\pi}{b}$. Coefficient $b$ compresses the period by factor $b$.

52. What does the statement "$f$ is increasing on $(a,b)$" mean using inequalities?

    If $x_1<x_2$ in $(a,b)$, then $f(x_1)<f(x_2)$. Definition using ordered pairs in the interval.

53. Identify the relative maximum point of $f(x)=-2(x+1)^2+7$.

    The relative maximum is at $(-1,7)$. Parabola opens downward with vertex at $(-1,7)$.

54. Identify whether $f(x)=x^2+x$ is even, odd, or neither.

    Neither. Neither even nor odd function.

55. Identify the axis of symmetry of $f(x)=x^2-6x+10$.

    The axis of symmetry is $x=3$. Complete the square: $(x-3)^2+1$.

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