Comparing Functions Represented in Different Ways - Algebra 2

1. For $f(x)=a(x-h)^2+k$, when does the quadratic have a minimum value?

   When $a>0$, minimum value is $k$. When the parabola opens upward, the vertex gives the lowest point.

2. Identify which has greater output at $x=-1$: $f(x)=x^2+2$ or $g(x)=3x+1$.

   $f(-1)=3$ is greater than $g(-1)=-2$. Substitute $x=-1$ into each function and compare results.

3. If $f(x)=x^2$ and $g(x)=(x-4)^2$, how do their minimum values compare?

   They are equal; both minima are $0$. Horizontal shifts don't change the minimum value, only its location.

4. Identify the maximum value of $f(x)=-4(x+1)^2+9$.

   Maximum value is $9$. Negative $a$ means downward opening, so vertex gives maximum at $y=k$.

5. Which has the larger value at $x=3$: $f(x)= \frac{1}{x-1}$ or $g(x)= \frac{1}{x+1}$?

   $f(3)= \frac{1}{2}$ is larger than $g(3)= \frac{1}{4}$. Substitute $x=3$ into each function and compare the results.

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

   $f(x)=a(x-h)^2+k$. Standard form that directly shows the vertex coordinates.

7. What property is compared by checking which function has the larger $y$-value at each $x$?

   Relative size: compare outputs $f(x)$ and $g(x)$ for the same $x$. Compare function values at the same input to see which produces larger outputs.

8. What is the definition of an $x$-intercept (zero) of a function $f$?

   An $x$-intercept is an $x$ where $f(x)=0$. Where the graph crosses the $x$-axis; solve for when the output equals zero.

9. Identify the axis of symmetry of $f(x)=-3(x-4)^2+1$.

   $x=4$. Read the $h$-value directly from the vertex form.

10. A table gives $f(1)=4$, $f(2)=4$, $f(3)=4$; what is the best comparison of $f$ over $[1,3]$?

    $f$ is constant on $[1,3]$. All function values are identical, so the function is constant.

11. What is the end behavior of a quadratic $f(x)=ax^2+bx+c$ if $a<0$?

    As $x\to\pm\infty$, $f(x)\to-\infty$. Negative leading coefficient means the parabola opens downward at both ends.

12. Identify the function with larger value at $x=0$: $f(x)=-2x^2+3$ or $g(x)=x+1$.

    $f(0)=3$ is larger than $g(0)=1$. Evaluate both functions at $x=0$ and compare the outputs.

13. A function is described as "starts at $y=3$ when $x=0$ and rises $2$ per $1$ right"; what is its equation?

    $y=2x+3$. Linear function with slope $2$ and $y$-intercept $3$.

14. Identify the minimum value of $f(x)= \frac{1}{2}(x-6)^2-2$.

    Minimum value is $-2$. Positive $a$ means upward opening, so vertex gives minimum at $y=k$.

15. Identify the range of $f(x)=-2(x+1)^2+8$.

    Range: $y \leq 8$. Downward-opening parabola with vertex at $y=8$ gives range $y \leq 8$.

16. What is the end behavior of a quadratic $f(x)=ax^2+bx+c$ if $a>0$?

    As $x\to\pm\infty$, $f(x)\to\infty$. Positive leading coefficient means the parabola opens upward at both ends.

17. Which is larger: maximum of $f(x)=-2(x-1)^2+3$ or maximum of $g(x)=-(x+2)^2+5$?

    $g$ has the larger maximum ($5>3$). Compare the $k$-values from vertex form; $5>3$.

18. What is the average rate of change of $f$ from $x=a$ to $x=b$?

    $\frac{f(b)-f(a)}{b-a}$. The slope of the secant line between two points on the function.

19. Identify the vertex of $f(x)=2(x-3)^2-5$.

    $(3,-5)$. Read the vertex coordinates directly from the vertex form.

20. Identify the $x$-intercept(s) of $f(x)=(x-1)(x+3)$.

    $x=1$ and $x=-3$. Set each factor equal to zero and solve for the $x$-values.

21. Which has larger slope: a line with points $(0,1)$ and $(2,9)$ or a line with slope $3$?

    The first line, since slope = $\frac{9-1}{2-0} = 4$. Calculate the slope and compare with the given slope of $3$.

22. Which has the larger maximum: $f(x)=-x^2+1$ or $g(x)=-2x^2+1$?

    They are equal; both maxima are $1$. Both have the same vertex $y$-coordinate despite different shapes.

23. In $f(x)=a(x-h)^2+k$, what is the axis of symmetry?

    Axis of symmetry is $x=h$. The vertical line through the vertex that divides the parabola symmetrically.

24. Identify the $y$-intercept of $f(x)=3x^2-2x+4$.

    $f(0)=4$. Substitute $x=0$ into the function to find where it crosses the $y$-axis.

25. Which average rate of change is larger: $f$ with $\frac{f(5)-f(1)}{5-1}=2$ or $g$ with $\frac{g(5)-g(1)}{5-1}=-1$?

    $f$ is larger ($2>-1$). Compare the computed rates directly; $2>-1$.

26. If $f(x)$ is shifted to $g(x)=f(x)+5$, how do their maximum values compare?

    Maximum of $g$ is $5$ more than maximum of $f$. Vertical shifts add the same amount to all function values.

27. Identify the end behavior of $f(x)=-x^2+6x-1$ as $x\to\infty$.

    $f(x)\to-\infty$. Negative leading coefficient causes the parabola to go down as $x$ increases.

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

    As $x$ increases, $f(x)$ increases on that interval. The function's output values rise as the input values increase.

29. What is the definition of the $y$-intercept of a function $f$?

    The $y$-intercept is $f(0)$, the point $(0,f(0))$. Where the graph crosses the $y$-axis; substitute $x=0$ into the function.

30. What transformation is represented by $-f(x)$ compared to $f(x)$?

    Reflection across the $x$-axis. Negating the output flips the graph over the horizontal axis.

31. Which has the larger value at $x=3$: $f(x)= \frac{1}{x-1}$ or $g(x)= \frac{1}{x+1}$?

    $f(3)= \frac{1}{2}$ is larger than $g(3)= \frac{1}{4}$. Substitute $x=3$ into each function and compare the results.

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

    As $x$ increases, $f(x)$ decreases on that interval. The function's output values fall as the input values increase.

33. In $f(x)=a(x-h)^2+k$, what is the vertex?

    Vertex is $(h,k)$. The turning point where the parabola changes direction.

34. For $f(x)=a(x-h)^2+k$, when does the quadratic have a maximum value?

    When $a<0$, maximum value is $k$. When the parabola opens downward, the vertex gives the highest point.

35. Which line is steeper: $f(x)=3x+1$ or $g(x)=-5x+2$?

    $g$ is steeper because $| -5 |> |3|$. Compare absolute values of slopes; steeper means larger absolute slope.

36. Identify the range of $f(x)=(x-2)^2+5$.

    Range: $y\ge^5$. Upward-opening parabola with vertex at $y=5$ gives range $y \geq 5$.

37. Choose which has more zeros: $f(x)=x^2+1$ or $g(x)=(x-2)(x+2)$.

    $g$ has more zeros ($2$ zeros; $f$ has $0$ real zeros). Count real solutions; $f$ has no real zeros, $g$ has two.

38. What is the slope between points $(x_1,y_1)$ and $(x_2,y_2)$?

    $m= \frac{y_2-y_1}{x_2-x_1}$. Rise over run; the change in $y$ divided by the change in $x$.

39. Which has the larger $y$-intercept: $f(x)=2x-1$ or $g(x)=-x^2+4$?

    $g$ has the larger $y$-intercept ($4>-1$). Find each $y$-intercept by setting $x=0$ and compare.

40. A function decreases from $x=-2$ to $x=4$; which is larger, $f(-2)$ or $f(4)$?

    $f(-2)$ is larger. On decreasing intervals, smaller $x$-values produce larger function values.

41. A function increases from $x=1$ to $x=5$; which is larger, $f(1)$ or $f(5)$?

    $f(5)$ is larger. On increasing intervals, larger $x$-values produce larger function values.

42. What transformation is represented by $f(x-c)$ compared to $f(x)$?

    Horizontal shift right $c$ units (left if $c<0$). Subtracting from the input moves the graph horizontally opposite direction.

43. Identify which is larger: $f(2)=7$ or $g(2)=5$.

    $f(2)$ is larger. Compare the function values directly to determine which is greater.

44. If $f(x)=x^2$ and $g(x)=x^2-6$, how do their minimum values compare?

    Minimum of $g$ is $6$ less than minimum of $f$. Vertical shifts change the minimum by the same amount as the shift.

45. A quadratic has vertex $(-3,-1)$ and opens upward; what is its minimum value?

    Minimum value is $-1$. For upward-opening parabolas, the vertex $y$-coordinate is the minimum.

46. What transformation is represented by $f(-x)$ compared to $f(x)$?

    Reflection across the $y$-axis. Negating the input flips the graph over the vertical axis.

47. What transformation is represented by $f(x)+c$ compared to $f(x)$?

    Vertical shift up $c$ units (down if $c<0$). Adding to the output moves the graph vertically.

48. Identify the domain of $f(x)= \frac{1}{x-3}$.

    Domain: all real $x$ with $x\ne^3$. The denominator cannot be zero, so exclude $x=3$.

49. Which has the larger minimum: $f(x)=(x-1)^2-4$ or $g(x)=2(x+3)^2-1$?

    $g$ has the larger minimum ( $-1 > -4$ ). Compare the $k$-values from vertex form; $-1 > -4$.

50. If a line has equation $y=mx+b$, what do $m$ and $b$ represent?

    $m$ is slope; $b$ is $y$-intercept. Slope-intercept form where $m$ determines steepness and $b$ is the starting value.

51. Find the slope of the line through $ (1,2) $ and $ (5,10) $.

    $m = \frac{10-2}{5-1} = 2$. Use the slope formula with the two given points.

52. If $g(x)=f(x-2)$, how do the $x$-coordinates of corresponding key points compare?

    Key points shift right $2$ units (add $2$ to $x$). Horizontal shifts move all points the same distance along the $x$-axis.

53. Identify which has larger maximum: $f(x)=-x^2+4$ or $g(x)=-(x-1)^2+2$.

    $f$ has the larger maximum ($4>2$). Compare the $k$-values from vertex form; $4>2$.

54. Find the slope of the line through $(1,2)$ and $(5,10)$.

    $m= \frac{10-2}{5-1}=2$. Use the slope formula with the two given points.

55. What is the domain of a function in words?

    The set of all allowed input values (all $x$-values). All $x$-values for which the function is defined.

56. What is the range of a function in words?

    The set of all possible output values (all $y$-values). All $y$-values that the function can produce as outputs.

57. A table shows $f(0)=2$ and $f(3)=11$; find the average rate of change on $[0,3]$.

    $\frac{11-2}{3-0}=3$. Use the average rate formula with the table values over the interval.

58. A quadratic has vertex $(2,7)$ and opens downward; what is its maximum value?

    Maximum value is $7$. For downward-opening parabolas, the vertex $y$-coordinate is the maximum.

59. A quadratic is described as "opens down with vertex at $(1,6)$"; what is its maximum value?

    Maximum value is $6$. For downward-opening parabolas, the vertex gives the maximum value.

60. Compute the average rate of change of $f$ from $x=1$ to $x=4$ if $f(1)=3$ and $f(4)=15$.

    $\frac{15-3}{4-1}=4$. Apply the average rate of change formula with the given values.

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