Interpreting/Sketching Key Features of Functions - Algebra 1

1. For $f(x)=-x^3$, what is the end behavior as $x\to\infty$ and $x\to-\infty$?

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

2. For $f(x)=(x-1)(x-5)$, on which intervals is $f(x)$ positive?

   $(-\infty,1)\cup(5,\infty)$. Parabola opens upward, so it's positive outside zeros.

3. For $f(x)=x^3$, what is the end behavior as $x\to\infty$ and $x\to-\infty$?

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

4. What is end behavior describing for a function's graph?

   What happens to $f(x)$ as $x\to\infty$ and $x\to-\infty$. Behavior as $x$ approaches positive and negative infinity.

5. Which symmetry does $f(x)=x^3$ have: $y$-axis, origin, or none?

   Origin symmetry (odd). Since $f(-x)=(-x)^3=-x^3=-f(x)$, it's odd.

6. What is the $y$-intercept of a function in terms of the graph?

   The point where the graph crosses the $y$-axis, at $x=0$. Occurs when the input equals zero.

7. For $f(x)=x^2$, what is the end behavior as $x\to\pm\infty$?

   $f(x)\to\infty$ as $x\to\infty$ and as $x\to-\infty$. Upward-opening parabola goes to infinity both ways.

8. What is the meaning of the statement "$f(x)$ is decreasing on $(a,b)$"?

   For $a<x_1<x_2<b$, $f(x_1)>f(x_2)$. Larger inputs give smaller outputs throughout the interval.

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

   A point higher than nearby points (local highest value). A peak on the graph, higher than surrounding points.

10. For $f(x)=(x-2)^2+5$, what is the relative minimum value?

    $5$. The vertex form shows $k$ is the minimum value.

11. For $f(x)=x^2-4$, on which intervals is the function positive?

    $(-\infty,-2)\cup(2,\infty)$. Function is positive outside its zeros at $x=\pm 2$.

12. For $f(x)=(x-1)(x-5)$, on which interval is $f(x)$ negative?

    $(1,5)$. Parabola opens upward, so it's negative between zeros.

13. Identify the $y$-intercept from the table point $(0,7)$.

    $7$. The $y$-intercept is the output value when input is zero.

14. For $g(x)=f(x)-3$, how does the $y$-intercept change from $f(0)$?

    It becomes $f(0)-3$. Subtracting $3$ from $f(0)$ gives the new intercept.

15. Which interval shows $f(x)=2x+1$ is increasing?

    All real numbers, $(-\infty,\infty)$. Linear functions with positive slope increase everywhere.

16. For $g(x)=f(x+2)$, how does the graph shift relative to $f(x)$?

    Shift left $2$ units. Adding inside parentheses shifts left, not right.

17. What is the $y$-intercept of $f(x)=3x-6$?

    $-6$. Substitute $x=0$ into $f(x)=3(0)-6=-6$.

18. For $f(x)=-x^2$, what is the end behavior as $x\to\pm\infty$?

    $f(x)\to-\infty$ as $x\to\infty$ and as $x\to-\infty$. Downward-opening parabola goes to negative infinity both ways.

19. What does a table tell you about the $y$-intercept?

    It is the output when $x=0$, the ordered pair $(0,f(0))$. Find the row where input is zero.

20. If $f(x)$ has period $6$, what is $f(x+6)$ equal to?

    $f(x)$. Since period is $6$, $f(x+6)=f(x)$ by definition.

21. For $f(x)=x^2-4$, what are the $x$-intercepts?

    $x=-2$ and $x=2$. Set $f(x)=0$: $x^2-4=0$, so $x=\pm 2$.

22. For $f(x)=x^2-4$, what is the $y$-intercept?

    $-4$. Substitute $x=0$: $f(0)=0^2-4=-4$.

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

    A point lower than nearby points (local lowest value). A valley on the graph, lower than surrounding points.

24. Which transformation occurs when $f(x)$ becomes $f(x)+k$?

    Vertical shift up $k$ units (down if $k<0$). Adds constant to all output values.

25. What is the axis of symmetry of $f(x)=(x-h)^2+k$?

    $x=h$. Vertical line through the vertex of a parabola.

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

    As $x$ increases, $f(x)$ decreases on that interval. The function values fall as you move right.

27. What does it mean for a function to be negative on an interval?

    $f(x)<0$ on that interval (graph is below the $x$-axis). All function values are less than zero.

28. Which transformation occurs when $f(x)$ becomes $f(x-h)$?

    Horizontal shift right $h$ units (left if $h<0$). Replaces $x$ with $x-h$ in the function.

29. For $f(x)=(x-2)^2+5$, what is the $x$-value of the relative minimum?

    $2$. The vertex form shows $h$ is the $x$-coordinate.

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

    As $x$ increases, $f(x)$ increases on that interval. The function values rise as you move right.

31. For $f(x)=-(x+1)^2+4$, what is the relative maximum value?

    $4$. Negative coefficient creates downward parabola with maximum $k$.

32. What is the meaning of the statement "$f(x)$ is increasing on $(a,b)$"?

    For $a<x_1<x_2<b$, $f(x_1)<f(x_2)$. Larger inputs give larger outputs throughout the interval.

33. What does it mean for a function to be even in terms of symmetry?

    Symmetric about the $y$-axis; $f(-x)=f(x)$. Folding across $y$-axis gives identical graph.

34. What is the axis of symmetry of $f(x)=-(x+4)^2+9$?

    $x=-4$. From $(x+4)^2$, the axis is at $x=-4$.

35. What does it mean for a function to be odd in terms of symmetry?

    Symmetric about the origin; $f(-x)=-f(x)$. Rotating $180°$ about origin gives identical graph.

36. Identify an $x$-intercept from the table point $(-3,0)$.

    $-3$. The $x$-intercept is the input value when output is zero.

37. For $f(x)=-(x+1)^2+4$, what is the $x$-value of the relative maximum?

    $-1$. The vertex occurs at $x=h$ from the form.

38. What is the $x$-intercept of $f(x)=3x-6$?

    $2$. Set $f(x)=0$: $3x-6=0$, so $x=2$.

39. What does it mean for a function to be positive on an interval?

    $f(x)>0$ on that interval (graph is above the $x$-axis). All function values are greater than zero.

40. What key features must be labeled when sketching from a verbal description?

    Intercepts, extrema, increasing/decreasing, sign, end behavior, period. Essential elements for complete function analysis.

41. What is the definition of period $P$ for a periodic function?

    The smallest $P>0$ such that $f(x+P)=f(x)$. Adding the period returns the same output.

42. Which interval shows $f(x)=-2x+1$ is decreasing?

    All real numbers, $(-\infty,\infty)$. Linear functions with negative slope decrease everywhere.

43. Which symmetry does $f(x)=x^2-1$ have: $y$-axis, origin, or none?

    $y$-axis symmetry (even). Since $f(-x)=(-x)^2-1=x^2-1=f(x)$, it's even.

44. What is periodicity for a function?

    A repeating pattern with some positive period $P$. Function values repeat at regular intervals.

45. What does a table tell you about an $x$-intercept?

    It occurs where the output is $0$, at a point $(x,0)$. Find the row where output is zero.

46. What is the $x$-intercept of a function in terms of the graph?

    A point where the graph crosses the $x$-axis, where $y=0$. Occurs when the output equals zero.

47. For $f(x)=(x-1)(x-5)$, what are the $x$-intercepts?

    $x=1$ and $x=5$. Set each factor to zero: $x-1=0$ or $x-5=0$.

48. What is the axis of symmetry of $f(x)=(x-3)^2-1$?

    $x=3$. From $(x-3)^2$, the axis is at $x=3$.

49. If $f(x)$ has period $4$, what is $f(x+12)$ equal to?

    $f(x)$. Since $12=3\times 4$, we get $f(x+12)=f(x)$.

50. For $f(x)=x^2-4$, on which interval is the function negative?

    $(-2,2)$. Function is negative between its zeros at $x=\pm 2$.

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