Comparing Functions Represented in Different Ways - Algebra 1

1. A table shows $f(0)=2$ and $f(1)=5$. What is the average rate of change on $[0,1]$?

   $3$. Change in y divided by change in x: $(5-2)/(1-0)=3$.

2. Which property is found by evaluating $f(0)$ for any function $f$?

   The $y$-intercept. Substituting zero for x gives the y-coordinate where graph crosses y-axis.

3. Which is decreasing on all real numbers: $f(x)=\frac{3}{2}x+1$ or $g(x)=-2x+5$?

   $g(x)$. Negative slope means function decreases everywhere.

4. Find the maximum of $f(x)=-2(x-3)^2+5$.

   $5$. Since $a<0$, parabola opens down with maximum k-value.

5. In a table, how do you identify a constant rate of change?

   Equal $\Delta y$ for equal $\Delta x$. Linear functions have constant rate of change between points.

6. Which has the greater slope: $f(x)=-3x+2$ or $g(x)=\frac{1}{2}x-4$?

   $g(x)$. Compare slopes: $-3$ vs $\frac{1}{2}$, so $\frac{1}{2}>-3$.

7. Which quadratic opens downward: $f(x)=2(x-1)^2+3$ or $g(x)=-\frac{1}{2}(x+4)^2-1$?

   $g(x)$. Negative coefficient means parabola opens downward.

8. A table shows $f(2)=7$ and $f(6)=-1$. What is the average rate of change on $[2,6]$?

   $-2$. Change in y divided by change in x: $(-1-7)/(6-2)=-2$.

9. What is the range of $f(x)=a(x-h)^2+k$ when $a>0$?

   $y\ge k$. Upward parabola has minimum at vertex, range starts at k.

10. Which is increasing on all real numbers: $f(x)=5x-2$ or $g(x)=-x+7$?

    $f(x)$. Positive slope means function increases everywhere.

11. What is the y-intercept of $f(x)=ax+b$ in terms of $b$?

    $b$. In slope-intercept form, the constant term is the y-intercept.

12. Identify the domain of a quadratic polynomial function in Algebra $1$.

    All real numbers. Polynomial functions accept any real number as input.

13. Identify which has greater vertex $y$-value: $f(x)=(x-1)^2+2$ or $g(x)=(x+3)^2-1$.

    $f(x)$. Compare vertex y-values: $2>-1$.

14. What is the range of $f(x)=a(x-h)^2+k$ when $a<0$?

    $y\le k$. Downward parabola has maximum at vertex, range ends at k.

15. A function is described as "starts at $4$ when $x=0$ and decreases $2$ per $1$ unit." What is the slope?

    $-2$. Decreases 2 per unit means slope is $-2$.

16. Which representation gives the maximum of a downward parabola most directly?

    Vertex form $a(x-h)^2+k$ gives maximum $k$. Vertex form immediately shows the maximum value k.

17. Compare maxima: $f(x)=-(x-2)^2+6$ and $g(x)=-(x+1)^2+4$. Which is larger?

    $f(x)$. Compare maximum values: $6>4$.

18. Which has smaller minimum: verbal "minimum value $-5$" or $g(x)=(x+2)^2-3$?

    The verbal function. Compare minimum values: $-5<-3$.

19. Which has larger maximum: verbal "maximum value $8$" or $g(x)=-(x-1)^2+6$?

    The verbal function. Compare maximum values: $8>6$.

20. What key feature determines whether a quadratic has a smaller minimum than another?

    Compare vertex $y$-values when $a>0$. Upward parabolas achieve minimum at their vertex point.

21. Find the minimum of $f(x)=\frac{1}{2}(x+4)^2-7$.

    $-7$. Since $a>0$, parabola opens up with minimum k-value.

22. What does it mean if a function is decreasing on an interval?

    As $x$ increases, $f(x)$ decreases. Function values fall as x-values move from left to right.

23. What is the slope of $f(x)=ax+b$ in terms of $a$?

    $a$. In slope-intercept form, the coefficient of x is the slope.

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

    $x=h$. Vertex form shows the axis of symmetry passes through the vertex.

25. Identify the vertex of $f(x)=-3(x+2)^2+1$.

    $(-2,1)$. Vertex form gives coordinates $(h,k)$ directly.

26. What key feature determines whether a quadratic has a larger maximum than another?

    Compare vertex $y$-values when $a<0$. Downward parabolas achieve maximum at their vertex point.

27. Which has the greater maximum: $f(x)=-2(x-1)^2+3$ or $g(x)=-(x-1)^2+2$?

    $f(x)$. Compare maximum values: $3>2$.

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

    $\frac{y_2-y_1}{x_2-x_1}$. Rise over run formula for finding rate of change.

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

    $x=1$. Vertex form shows axis of symmetry as $x=h$.

30. Compare minima: $f(x)=(x-5)^2-1$ and $g(x)=2(x-1)^2-3$. Which is smaller?

    $g(x)$. Compare minimum values: $-3<-1$.

31. Which has smaller slope: table points $(1,6)$ and $(3,2)$ or $g(x)=-1x+0$?

    The table function. Table slope is $(2-6)/(3-1)=-2$, which is less than $-1$.

32. Identify which has greater value at $x=-1$: $f(x)=2x+5$ or $g(x)=x^2+1$.

    $f(x)$. Evaluate at $x=-1$: $f(-1)=3$ vs $g(-1)=2$.

33. Which has the larger $y$-intercept: $f(x)=2x-1$ or $g(x)=-x+3$?

    $g(x)$. Compare y-intercepts: $f(0)=-1$ vs $g(0)=3$.

34. In a table, what pattern suggests a quadratic function instead of linear?

    Constant second differences. Quadratics have changing first differences but constant second differences.

35. For $f(x)=a(x-h)^2+k$, what is the maximum or minimum value?

    $k$. The k-value represents the function's output at the vertex.

36. Which has greater slope: table points $(0,1)$ and $(2,9)$ or $g(x)=3x-4$?

    The table function. Table slope is $(9-1)/(2-0)=4$, which is greater than $3$.

37. If $a>0$ in $f(x)=a(x-h)^2+k$, does the parabola have a max or min?

    Minimum. Positive coefficient means parabola opens upward with minimum.

38. Which has vertex farther right: $f(x)=(x-4)^2$ or $g(x)=(x+2)^2$?

    $f(x)$. Compare h-values: $4>-2$, so vertex at $x=4$ is farther right.

39. What is the vertex of $f(x)=a(x-h)^2+k$?

    $(h,k)$. Vertex form directly reveals the vertex coordinates.

40. If $a<0$ in $f(x)=a(x-h)^2+k$, does the parabola have a max or min?

    Maximum. Negative coefficient means parabola opens downward with maximum.

41. Which has larger initial value: table shows $f(0)=-2$ or algebraic $g(x)=-5x+1$?

    $g(x)$. Compare initial values: $-2<1$, so $g(x)$ is larger.

42. Which has larger initial value: verbal $f(0)=7$ or algebraic $g(x)=2x+1$?

    $f$. Compare initial values: $7>1$.

43. Identify which has smaller vertex $y$-value: $f(x)=- (x-2)^2+5$ or $g(x)=-(x-2)^2+1$.

    $g(x)$. Compare vertex y-values: $1<5$.

44. A function is described as "starts at $-3$ when $x=0$ and increases $\frac{1}{2}$ per $1$ unit." What is $f(x)$?

    $f(x)=\frac{1}{2}x-3$. Starts at $-3$ with slope $\frac{1}{2}$ gives this form.

45. For $f(x)=-(x-7)^2+9$, what is the domain and range?

    Domain: all real; Range: $y\le 9$. Downward parabola has range ending at maximum k-value.

46. Which has larger value at $x=-2$: table gives $f(-2)=1$ or $g(x)=x^2$?

    $g(x)$. Compare values: $1<4$.

47. Which has larger average rate of change on $[1,3]$: $f(x)=2x+1$ or $g(x)=x^2$?

    $g(x)$. Linear has constant rate 2, quadratic's average rate is 4.

48. What is the average rate of change from $x_1$ to $x_2$ for $f$?

    $\frac{f(x_2)-f(x_1)}{x_2-x_1}$. Change in output divided by change in input over interval.

49. For $f(x)=3(x+1)^2-4$, what is the domain and range?

    Domain: all real; Range: $y\ge -4$. Upward parabola has range starting at minimum k-value.

50. Which has the smaller minimum: $f(x)=\frac{1}{2}(x-3)^2+1$ or $g(x)=(x-3)^2-2$?

    $g(x)$. Compare minimum values: $-2<1$.

51. Which has larger average rate of change on $[0,2]$: $f(x)=x^2$ or $g(x)=3x$?

    $g(x)$. Linear has constant rate 3, quadratic's average rate is 2.

52. What does it mean if a function is increasing on an interval?

    As $x$ increases, $f(x)$ increases. Function values rise as x-values move from left to right.

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