Graph Square Root and Piecewise Functions - Algebra 1

1. What is the vertex of $f(x)=|x-6|+4$?

   $(6,4)$. Vertex form parameters: $(h,k) = (6,4)$.

2. What is the parent function for a cube root graph?

   $f(x)=\sqrt[3]{x}$. Basic cube root function before any transformations.

3. What is the $y$-intercept of $f(x)=\sqrt{x}$?

   $(0,0)$. Evaluate at $x=0$: $f(0)=\sqrt{0}=0$.

4. Identify the horizontal shift of $f(x)=|x+7|$ compared to $|x|$.

   Shift left $7$. Inside absolute value, $+7$ shifts left by $7$ units.

5. Identify the reflection in $f(x)=-\sqrt{x}$ compared to $\sqrt{x}$.

   Reflect over the $x$-axis. Negative sign in front flips graph over the $x$-axis.

6. What is the transformation form for a cube root function used for graphing?

   $f(x)=a\sqrt[3]{x-h}+k$. Standard form showing inflection point $(h,k)$ and vertical stretch/compression $a$.

7. What does a closed circle at an endpoint mean on a piecewise graph?

   Endpoint is included (uses $\le$ or $\ge$). Inequality symbols determine endpoint inclusion.

8. Identify the horizontal shift of $f(x)=\sqrt{x-5}$ compared to $\sqrt{x}$.

   Shift right $5$. Inside the radical, $-5$ shifts right by $5$ units.

9. What point is the vertex of $f(x)=a|x-h|+k$ on its graph?

   $(h,k)$. Absolute value function has its corner at this translated point.

10. What is the range of the parent cube root function $f(x)=\sqrt[3]{x}$?

    $(-\infty,\infty)$. Cube root produces all real number outputs.

11. What is the parent function for a square root graph?

    $f(x)=\sqrt{x}$. Basic square root function before any transformations.

12. What is the domain of $f(x)=|x-4|+1$?

    $(-\infty,\infty)$. Absolute value functions accept all real inputs.

13. Identify the horizontal shift of $f(x)=\sqrt[3]{x+2}$ compared to $\sqrt[3]{x}$.

    Shift left $2$. Inside the radical, $+2$ shifts left by $2$ units.

14. What is the vertex of the parent absolute value function $f(x)=|x|$?

    $(0,0)$. V-shape opens at the origin.

15. What is the range of $f(x)=\sqrt{x}+5$?

    $y\ge 5$. Adding $5$ shifts all output values up by $5$.

16. What is the vertex form of an absolute value function used for graphing?

    $f(x)=a|x-h|+k$. Standard form showing vertex $(h,k)$ and vertical stretch/compression $a$.

17. Identify the reflection in $f(x)=-|x|$ compared to $|x|$.

    Reflect over the $x$-axis. Negative sign in front flips graph over the $x$-axis.

18. What is the range of $f(x)=-|x+2|-3$?

    $y\le -3$. Negative with vertex at $(-2,-3)$ gives maximum value $-3$.

19. Identify the reflection in $f(x)=-\sqrt[3]{x}$ compared to $\sqrt[3]{x}$.

    Reflect over the $x$-axis. Negative sign in front flips graph over the $x$-axis.

20. What is the range of $f(x)=-\sqrt{x}+2$?

    $y\le 2$. Negative flips and adding $2$ gives maximum value $2$.

21. What is the definition of a piecewise function in terms of rules and intervals?

    A function defined by different rules on different intervals. Each piece has a specific domain interval.

22. What is the domain of the parent square root function $f(x)=\sqrt{x}$?

    $x\ge 0$. Square root requires non-negative input values.

23. What point is the center point of $f(x)=a\sqrt[3]{x-h}+k$ on its graph?

    $(h,k)$. Cube root function passes through this translated point.

24. What is the transformation form for a square root function used for graphing?

    $f(x)=a\sqrt{x-h}+k$. Standard form showing starting point $(h,k)$ and vertical stretch/compression $a$.

25. What is the range of $f(x)=|x-4|+1$?

    $y\ge 1$. Vertex at $(4,1)$ gives minimum output value $1$.

26. What is the range of the parent square root function $f(x)=\sqrt{x}$?

    $y\ge 0$. Square root output is always non-negative.

27. What point is the starting point of $f(x)=a\sqrt{x-h}+k$ on its graph?

    $(h,k)$. Square root function begins at this translated point.

28. What is the output of the ceiling function $\lceil x\rceil$ in words?

    The least integer greater than or equal to $x$. Always rounds up to the nearest integer.

29. What do you use to decide which rule to apply in a piecewise function?

    The input $x$ and the stated interval conditions. Check which interval contains the input value.

30. What is the center point of $f(x)=\sqrt[3]{x-8}+2$?

    $(8,2)$. Transformation parameters: $(h,k) = (8,2)$.

31. What is the domain of $f(x)=\sqrt{2x+6}$?

    $x\ge -3$. Square root requires $2x+6\ge 0$, so $x\ge -3$.

32. Evaluate $\lfloor -1.2\rfloor$.

    $-2$. Greatest integer $\le -1.2$ is $-2$.

33. Evaluate $\lceil 2.1\rceil$.

    $3$. Least integer $\ge 2.1$ is $3$.

34. What is the starting point of $f(x)=\sqrt{x+1}-3$?

    $(-1,-3)$. Transformation parameters: $(h,k) = (-1,-3)$.

35. What is the domain of the parent cube root function $f(x)=\sqrt[3]{x}$?

    $(-\infty,\infty)$. Cube root accepts all real number inputs.

36. Identify whether $x> -1$ should have an open or closed circle at $x=-1$.

    Open circle at $x=-1$. The $>$ symbol excludes the endpoint.

37. Identify the vertical shift of $f(x)=\sqrt[3]{x}-4$ compared to $\sqrt[3]{x}$.

    Shift down $4$. Subtracting outside the function shifts down by $4$ units.

38. What is the domain of $f(x)=\sqrt{x-9}$?

    $x\ge 9$. Square root requires $x-9\ge 0$.

39. Identify whether $x\le 2$ should have an open or closed circle at $x=2$.

    Closed circle at $x=2$. The $\le$ symbol includes the endpoint.

40. Identify the reflection in $f(x)=\sqrt[3]{-x}$ compared to $\sqrt[3]{x}$.

    Reflect over the $y$-axis. Negative inside the radical flips graph over the $y$-axis.

41. Identify the vertical shift of $f(x)=|x|-2$ compared to $|x|$.

    Shift down $2$. Subtracting outside the function shifts down by $2$ units.

42. Evaluate $\lceil -1.2\rceil$.

    $-1$. Least integer $\ge -1.2$ is $-1$.

43. What is the $x$-intercept of $f(x)=\sqrt{x-4}$?

    $(4,0)$. Set $f(x)=0$ and solve: $\sqrt{x-4}=0$ gives $x=4$.

44. Evaluate $\lfloor 2.9\rfloor$.

    $2$. Greatest integer $\le 2.9$ is $2$.

45. What is the step function notation for the greatest integer function?

    $f(x)=\lfloor x\rfloor$. Floor function symbol for step functions.

46. What is the parent function for an absolute value graph?

    $f(x)=|x|$. Basic absolute value function before any transformations.

47. Evaluate $\lfloor -3\rfloor$.

    $-3$. Greatest integer $\le -3$ is $-3$ itself.

48. What does an open circle at an endpoint mean on a piecewise graph?

    Endpoint is not included (uses $<$ or $>$). Inequality symbols determine endpoint inclusion.

49. What is the output of the greatest integer function $\lfloor x\rfloor$ in words?

    The greatest integer less than or equal to $x$. Always rounds down to the nearest integer.

50. Identify the vertical shift of $f(x)=\sqrt{x}+3$ compared to $\sqrt{x}$.

    Shift up $3$. Adding outside the function shifts up by $3$ units.

51. What are two symmetric line pieces that define $|x|$ as a piecewise function?

    $|x|=\begin{cases}x&x\ge 0\\-x&x<0\end{cases}$. Absolute value splits into two linear pieces at $x=0$.

Study these flashcards