Graphing Functions - PSAT Math

1. What does the vertical line test determine for a graph in the coordinate plane?

   Whether the graph represents a function. Functions have at most one y-value per x-value.

2. Identify the domain of the rational function $y=\frac{1}{x-a}$ for constant $a$.

   All real $x$ except $x\ne a$. Division by zero is undefined, excluding $x=a$ from the domain.

3. What is the range of a function $f$ as shown on its graph?

   All $y$-values of points on the graph. The vertical extent of the graph shows all possible outputs.

4. What is the effect on the graph of replacing $f(x)$ with $-f(x)$?

   Reflect across the $x$-axis. Negating outputs flips all $y$-values to opposite signs.

5. Identify the slope and $y$-intercept of $y=-3x+5$.

   Slope $-3$; $y$-intercept $(0,5)$. Read directly from $y=mx+b$ form: $m=-3$, $b=5$.

6. Identify the transformation: $y=f(-x)$ compared to $y=f(x)$.

   Reflection across the $y$-axis. Negative input flips all points horizontally across the $y$-axis.

7. What is the $y$-intercept of the line $y=-2x+7$?

   $(0,7)$. Substitute $x=0$: $y=-2(0)+7=7$.

8. Identify the transformation: $y=af(x)$ for $a>1$ compared to $y=f(x)$.

   Vertical stretch by factor $a$. Multiplying output by $a>1$ stretches graph vertically.

9. What is the definition of the $y$-intercept of a graph of $y=f(x)$?

   The point where $x=0$, so $y=f(0)$. The $y$-intercept occurs where the graph crosses the $y$-axis.

10. Identify the domain and range of the parent function $y=\sqrt{x}$.

    Domain: $x\ge 0$; Range: $y\ge 0$. Square root requires non-negative input and produces non-negative output.

11. What is the vertical line test used to determine when a graph represents $y$ as a function of $x$?

    No vertical line intersects the graph more than once. Functions have only one $y$-value per $x$-value.

12. What transformation occurs to $y=f(x)$ when graphing $y=f(x)+k$ for constant $k$?

    Vertical shift by $k$ units (up if $k>0$, down if $k<0$). Adding $k$ to outputs shifts the entire graph vertically.

13. What transformation does $y=-f(x)$ represent on the graph of $y=f(x)$?

    Reflection across the $x$-axis. Negating the output flips all $y$-values across the $x$-axis.

14. What transformation occurs to $y=f(x)$ when graphing $y=f(x-h)$ for constant $h$?

    Horizontal shift by $h$ units (right if $h>0$, left if $h<0$). Subtracting $h$ from inputs shifts the graph horizontally.

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

    The vertical line $x=h$. Parabolas are symmetric about the vertical line through the vertex.

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

    $x=h$. Vertical line through vertex where parabola folds.

17. What is the vertical line test for deciding whether a graph represents a function?

    A graph is a function if every vertical line hits it at most once. Functions have only one $y$-value per $x$-value.

18. For $y=a(x-h)^2+k$, what is the axis of symmetry?

    Axis of symmetry is $x=h$. The parabola is symmetric about the vertical line through vertex.

19. What transformation occurs to $y=f(x)$ when graphing $y=f(-x)$?

    Reflection across the $y$-axis. Negating inputs flips the graph left-to-right.

20. What is the slope between points $(x_1,y_1)$ and $(x_2,y_2)$ on a graph?

    $\frac{y_2-y_1}{x_2-x_1}$. Slope equals rise over run: change in $y$ divided by change in $x$.

21. What are the vertical and horizontal asymptotes of $y=\frac{1}{x-a}$?

    Vertical: $x=a$; Horizontal: $y=0$. The function approaches these lines but never reaches them.

22. What is the $y$-intercept of the line given by $y=mx+b$?

    The point $(0,b)$. When $x=0$, $y=m(0)+b=b$, giving the point $(0,b)$.

23. What is the definition of an $x$-intercept of a function $y=f(x)$ on its graph?

    A point where $y=0$, so it satisfies $f(x)=0$. The $x$-intercept occurs where the graph crosses the $x$-axis.

24. What does it mean on a graph if $(a,b)$ lies on $y=f(x)$?

    $f(a)=b$. The point satisfies the function equation.

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

    $(0,4)$. Substitute $x=0$: $f(0)=2(0)^2-3(0)+4=4$.

26. What is the definition of an $x$-intercept (zero) of $y=f(x)$?

    Any $x$ where $f(x)=0$, giving a point $(x,0)$. Zeros occur where the graph crosses the $x$-axis.

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

    The vertical line $x=h$. Parabolas are symmetric about the vertical line through vertex.

28. Identify the end behavior of $y=x^2$ as $x\to\pm\infty$.

    $y\to\infty$ as $x\to\infty$ and as $x\to-\infty$. The parabola opens upward, so both ends go to infinity.

29. What is the range of a function given its graph?

    All $y$-values attained by points on the graph. The range includes all $y$-coordinates that appear on the graph.

30. What is the definition of the $x$-intercept of a graph of $y=f(x)$?

    A point where $y=0$, so $f(x)=0$. The $x$-intercept occurs where the graph crosses the $x$-axis.

31. Identify the vertex of $y=(x-4)^2-7$.

    $(4,-7)$. In vertex form $y=a(x-h)^2+k$, vertex is $(h,k)$.

32. What does it mean for a relation to be a function in terms of $x$-values and outputs?

    Each $x$-value corresponds to exactly one output $y$. This is the definition of a function - no two outputs for one input.

33. What is the $y$-intercept of $y=f(x)$ in terms of $f$?

    The point $(0, f(0))$. Occurs where the graph crosses the $y$-axis, when $x=0$.

34. What is the definition of the $y$-intercept of a function $y=f(x)$ on its graph?

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

35. What is the slope-intercept form of a linear function and what does each parameter represent?

    $y=mx+b$; slope $m$, $y$-intercept $b$. Linear graphs have constant slope $m$ and cross $y$-axis at $(0,b)$.

36. What is the $x$-intercept of $y=x-7$?

    $(7,0)$. Set $y=0$: $0=x-7$ gives $x=7$, so the intercept is $(7,0)$.

37. Identify the vertical asymptote of $f(x)=\frac{3}{x+5}$.

    $x=-5$. The function approaches infinity when the denominator $x+5=0$.

38. What is the effect on the graph of replacing $f(x)$ with $a f(x)$ for $a>1$?

    Vertical stretch by factor $a$. Multiplying outputs by $a>1$ stretches graph away from $x$-axis.

39. Identify the axis of symmetry of $y=-2(x+3)^2+1$.

    $x=-3$. Axis passes through vertex; here $h=-3$ gives $x=-3$.

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

    The point where $x=0$, so $y=f(0)$. The $y$-intercept occurs where the graph crosses the $y$-axis.

41. Identify the transformation: $y=f(x-k)$ compared to $y=f(x)$.

    Horizontal shift right $k$ units. Subtracting $k$ from input shifts graph right (opposite direction).

42. What is the effect on the graph of $y=-f(x)$?

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

43. What is the effect on the graph of $y=f(x)+k$ for a constant $k$?

    Vertical shift up $k$ units. Adding $k$ to outputs moves the graph up $k$ units.

44. Identify the vertex of the parabola $y=(x-3)^2-5$.

    $(3,-5)$. In vertex form, $(h,k)=(3,-5)$ directly.

45. Identify the transformation: $y=-f(x)$ compared to $y=f(x)$.

    Reflection across the $x$-axis. Negative sign flips all $y$-values across the $x$-axis.

46. Find the $x$-intercept of $y=2x-8$ as a point on the coordinate plane.

    $(4,0)$. Set $y=0$: $0=2x-8$, so $x=4$.

47. What transformation does $y=af(x)$ represent for $a>1$ compared to $y=f(x)$?

    Vertical stretch by factor $a$. Multiplying output by $a>1$ stretches the graph vertically.

48. What is the meaning of the range of a function $y=f(x)$ from its graph?

    All $y$-values attained by points on the graph. Range is the set of possible output values.

49. What does the vertical line test determine for a graph in the $xy$-plane?

    Whether the graph represents a function of $x$. If any vertical line crosses more than once, it's not a function.

50. What is the vertical line test used to determine about a graph?

    Whether the graph represents a function of $x$. A function has at most one $y$-value for each $x$-value.

51. What is the location of the vertex of $y=(x-3)^2-5$?

    $(3,-5)$. In vertex form $(x-h)^2+k$, the vertex is at $(h,k)$.

52. What is the effect on the graph of replacing $f(x)$ with $f(x)+k$?

    Shift up $k$ units. Adding $k$ to outputs increases all $y$-values by $k$.

53. What are the $x$-intercepts of $y=f(x)$ in terms of an equation involving $f$?

    Solutions to $f(x)=0$, written as points $(x,0)$. These are where the graph crosses the $x$-axis.

54. What is the effect on the graph of replacing $f(x)$ with $f(x-h)$?

    Shift right $h$ units. Replacing $x$ with $x-h$ delays the function by $h$ units.

55. What is the effect on the graph of replacing $f(x)$ with $f(ax)$ for $a>1$?

    Horizontal compression by factor $a$. Multiplying inputs by $a>1$ squeezes horizontally.

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

    Vertex is $(h,k)$. The parabola's turning point is at these coordinates.

57. What is the slope of a line passing through $(x_1,y_1)$ and $(x_2,y_2)$?

    $m=\frac{y_2-y_1}{x_2-x_1}$. This formula gives the rate of change between two points.

58. What is the effect on the graph of replacing $f(x)$ with $af(x)$ for $a>1$?

    Vertical stretch by factor $a$. Multiplying outputs by $a>1$ stretches vertically.

59. What is the effect on the graph of $y=f(x)$ when graphing $y=f(-x)$?

    Reflect across the $y$-axis. Replacing $x$ with $-x$ flips all points horizontally.

60. Identify the $y$-intercept of $f(x)=3x-5$.

    $(0,-5)$. Substitute $x=0$: $f(0)=3(0)-5=-5$.

61. What transformation does $y=f(-x)$ represent on the graph of $y=f(x)$?

    Reflection across the $y$-axis. Negating the input flips all $x$-values across the $y$-axis.

62. Identify the vertex form of a quadratic and state its vertex in terms of $h$ and $k$.

    $y=a(x-h)^2+k$; vertex is $(h,k)$. This form directly shows the parabola's turning point.

63. What is the meaning of $a$ in $y=a(x-h)^2+k$ for the parabola's opening?

    $a>0$ opens up; $a<0$ opens down. Sign of $a$ determines if parabola opens upward or downward.

64. What transformation does $y=f(x)+k$ represent on the graph of $y=f(x)$?

    A vertical shift up $k$ units (down if $k<0$). Adding $k$ to the output shifts every point up by $k$ units.

65. Identify the transformation in $y=-f(x)$ relative to $y=f(x)$.

    Reflection across the $x$-axis. Negative sign flips all y-values across the x-axis.

66. What are the $x$-intercepts of $y=(x-1)(x+4)$?

    $(-4,0)$ and $(1,0)$. Set $y=0$ and solve: $(x-1)(x+4)=0$ gives $x=1$ or $x=-4$.

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

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

68. Identify the axis of symmetry of $y=-2(x+1)^2+7$.

    Axis of symmetry is $x=-1$. In vertex form, $h=-1$ so axis of symmetry is $x=-1$.

69. Identify the $y$-intercept of $y=3x-4$.

    $y$-intercept is $(0,-4)$. Substitute $x=0$: $y=3(0)-4=-4$.

70. Identify the vertex of $y=(x-4)^2+7$.

    $(4,7)$. In vertex form $(x-h)^2+k$, vertex is $(h,k)$.

71. Identify the $x$-intercept of $y=2x+6$.

    $x$-intercept is $(-3,0)$. Set $y=0$ and solve: $0=2x+6$, so $x=-3$.

72. Identify the domain of $f(x)=\frac{1}{x-2}$.

    All real $x$ with $x\ne 2$. The function is undefined when the denominator equals zero at $x=2$.

73. What is the effect on the graph of replacing $f(x)$ with $f(x+h)$?

    Shift left $h$ units. Replacing $x$ with $x+h$ advances the function by $h$ units.

74. Identify the $y$-intercept of the line $y=-3x+7$ as an ordered pair.

    $(0,7)$. When $x=0$, $y=-3(0)+7=7$.

75. For $y=a(x-h)^2+k$, what are the vertex coordinates?

    Vertex is $(h,k)$. The parabola's turning point is shifted to $(h,k)$ from origin.

76. Find the slope of the line passing through $(2,5)$ and $(6,1)$.

    $m=-1$. Using slope formula: $\frac{1-5}{6-2}=\frac{-4}{4}=-1$.

77. What is the domain of a function $f$ as shown on its graph?

    All $x$-values of points on the graph. The horizontal extent of the graph shows all possible inputs.

78. What is the domain of $f(x)=\sqrt{x-5}$?

    $x\ge 5$. Square roots require non-negative inputs: $x-5\ge 0$.

79. What transformation does $y=f(ax)$ represent for $a>1$ compared to $y=f(x)$?

    Horizontal compression by factor $a$ (scale factor $\frac{1}{a}$). Multiplying input by $a>1$ compresses the graph horizontally.

80. What is the domain of a function given its graph?

    All $x$-values for which the graph has at least one point. The domain includes all $x$-coordinates that appear on the graph.

81. What is the effect on the graph of $y=f(x)$ when graphing $y=f(x)+k$?

    Shift up $k$ units (down if $k<0$). Adding $k$ to the output shifts the entire graph vertically.

82. What is the slope of the line given by $y=mx+b$?

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

83. What is the domain of a function in a graphing context?

    All $x$-values for which $f(x)$ is defined. The horizontal extent of the graph.

84. What is the definition of the $y$-intercept of a function $y=f(x)$ on its graph?

    The point where $x=0$, which is $(0,f(0))$. The y-intercept occurs when the input variable equals zero.

85. What is the effect on the graph of $y=f(x)$ when graphing $y=af(x)$ for $a>1$?

    Vertical stretch by factor $a$. Multiplying output by $a>1$ stretches graph away from $x$-axis.

86. What is the effect on the graph of $y=f(x-k)$ for a constant $k$?

    Horizontal shift right $k$ units. Subtracting $k$ from inputs shifts right $k$ units.

87. What is the vertex form of a parabola, and what point is the vertex?

    $y=a(x-h)^2+k$ with vertex $(h,k)$. This form makes the vertex coordinates immediately visible.

88. What is the effect on the graph of $y=f(ax)$ when $a>1$?

    Horizontal compression by factor $a$. Multiplying inputs by $a>1$ compresses horizontally.

89. What is the slope-intercept form of a line, and what do $m$ and $b$ represent?

    $y=mx+b$; $m$ is slope and $b$ is the $y$-intercept. Standard form showing slope and where line crosses $y$-axis.

90. What is the slope-intercept form of a line, and what do its parameters represent?

    $y=mx+b$; $m$ is slope and $b$ is the $y$-intercept. This form directly shows how the line rises and where it crosses the $y$-axis.

91. What is the slope of the line given by $y=mx+b$?

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

92. What is the effect on the graph of $y=af(x)$ when $a>1$?

    Vertical stretch by factor $a$. Multiplying outputs by $a>1$ stretches vertically.

93. Identify the vertex of $y=(x-3)^2-5$.

    Vertex is $(3,-5)$. Compare to $y=a(x-h)^2+k$ to identify $h=3$ and $k=-5$.

94. What is the effect on the graph of replacing $f(x)$ with $f(x)-k$?

    Shift down $k$ units. Subtracting $k$ from outputs decreases all $y$-values by $k$.

95. Identify the transformation in $y=f(-x)$ relative to $y=f(x)$.

    Reflection across the $y$-axis. Negating input flips all x-values across the y-axis.

96. Identify the vertex of $y=(x-3)^2+5$.

    $(3,5)$. Vertex form $y=(x-h)^2+k$ has vertex at $(h,k)$.

97. What is the $y$-intercept of $y=f(x)$ in terms of function notation (when it exists)?

    The point $(0,f(0))$. Found by substituting $x=0$ into the function.

98. What is the definition of the graph of a function $y=f(x)$ in the coordinate plane?

    The set of all points $(x,f(x))$ for allowed $x$. Each input $x$ maps to exactly one output point $(x,f(x))$.

99. Find the slope of the line through $(2,1)$ and $(6,9)$.

    $m=2$. Using slope formula: $m=\frac{9-1}{6-2}=\frac{8}{4}=2$.

100. What is the effect on the graph of replacing $f(x)$ with $f(x-k)$?

     Horizontal shift right $k$ units. Subtracting $k$ from inputs shifts the graph right.

101. What transformation does $y=f(x-k)$ represent on the graph of $y=f(x)$?

     A horizontal shift right $k$ units (left if $k<0$). Subtracting $k$ from input shifts the graph right (opposite direction).

102. What is the slope of a line through points $(x_1,y_1)$ and $(x_2,y_2)$?

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

103. Identify the transformation in $y=f(x-k)$ relative to $y=f(x)$.

     Horizontal shift right $k$ units. Subtracting $k$ from input shifts graph to the right.

104. Find the $x$-intercept of $y=2x-8$ as an ordered pair.

     $(4,0)$. Set $y=0$: $0=2x-8$, so $x=4$.

105. Identify the slope of the line $y=mx+b$ in terms of $m$.

     Slope $=m$. In slope-intercept form $y=mx+b$, the coefficient $m$ is the slope.

106. What is the definition of the $x$-intercept(s) of a function $y=f(x)$ on its graph?

     Point(s) where $y=0$, so $f(x)=0$. X-intercepts are where the graph crosses the x-axis.

107. What are the $x$-intercepts of $y=f(x)$ in terms of solutions to an equation?

     Points where $f(x)=0$, that is, $(x,0)$ for solutions of $f(x)=0$. The graph crosses the $x$-axis where the function equals zero.

108. What is the $y$-intercept of $y=f(x)$ in terms of $f(0)$ (when defined)?

     The $y$-intercept is $(0,f(0))$. The $y$-intercept occurs when $x=0$, giving the point $(0,f(0))$.

109. What is the range of a function in a graphing context?

     All possible output values $y=f(x)$. The vertical extent of the graph.

110. What is the definition of the graph of a function $f$ in the coordinate plane?

     The set of points $(x, f(x))$ for all $x$ in the domain. Each input $x$ pairs with output $f(x)$ to form coordinate pairs.

111. What is the vertex form of a quadratic, and what is its vertex?

     $y=a(x-h)^2+k$; vertex is $(h,k)$. Shows parabola's turning point directly.

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

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

113. What is the equation of a line with slope $m$ and $y$-intercept $b$?

     $y=mx+b$. Standard form where $m$ is slope and $b$ is $y$-intercept.

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

     The vertical line $x=h$. The parabola is symmetric about this vertical line through the vertex.

115. What transformation occurs to $y=f(x)$ when graphing $y=-f(x)$?

     Reflection across the $x$-axis. Negating outputs flips all $y$-values to opposite signs.

116. 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 formula calculates the rate of change.

117. What is the effect on the graph of $y=f(x)$ when graphing $y=-f(x)$?

     Reflect across the $x$-axis. Multiplying output by $-1$ flips all $y$-values to opposite sign.

118. Identify the transformation in $y=f(x)+k$ relative to $y=f(x)$.

     Vertical shift up $k$ units. Adding $k$ to the output shifts the graph upward.

119. What does the sign of $a$ tell you for $y=a(x-h)^2+k$?

     If $a>0$ it opens up; if $a<0$ it opens down. The sign determines whether the parabola opens upward or downward.

120. Identify the domain and range of the parent function $y=x^2$.

     Domain: all real $x$; Range: $y\ge 0$. The parabola extends infinitely left/right but only upward from vertex.

121. What is the effect on the graph of replacing $f(x)$ with $f(-x)$?

     Reflect across the $y$-axis. Negating inputs reverses the graph horizontally.

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

     $m=\frac{y_2-y_1}{x_2-x_1}$. Rise over run formula calculates the rate of change.

123. What is the vertex form of a quadratic and what point does it reveal immediately?

     $y=a(x-h)^2+k$; vertex $(h,k)$. The parabola's turning point is at $(h,k)$.

124. Identify the $x$-intercepts of $y=f(x)$ using an equation involving $f(x)$.

     $x$-intercepts occur where $f(x)=0$. These are the points where the graph crosses the $x$-axis.

125. What is the meaning of the domain of a function $y=f(x)$ from its graph?

     All $x$-values for which the graph has points. Domain is the set of possible input values.

126. What is the axis of symmetry of $y=ax^2+bx+c$?

     Axis of symmetry is $x=-\frac{b}{2a}$. Formula finds the x-coordinate of the vertex.

127. What is the effect on the graph of $y=f(x)$ when graphing $y=f(x-h)$?

     Shift right $h$ units (left if $h<0$). Subtracting $h$ from input shifts graph horizontally (opposite direction).

128. What is the vertical line test for deciding whether a graph represents a function $y=f(x)$?

     It is a function if no vertical line intersects the graph more than once. Functions have only one output per input, so vertical lines hit once max.

129. What is the effect on the graph of $y=f(-x)$?

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

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

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

131. Identify the horizontal asymptote of $f(x)=\frac{4}{x-1}$.

     $y=0$. As $x$ approaches $pminfty$, the fraction approaches zero.

132. What is the effect on the graph of $y=f(x)$ when graphing $y=f(ax)$ for $a>1$?

     Horizontal compression by factor $a$. Multiplying input by $a>1$ squeezes graph toward $y$-axis.

133. Identify the slope and $y$-intercept of the line $y=-3x+5$.

     Slope $m=-3$ and $y$-intercept $(0,5)$. Read directly from slope-intercept form: coefficient of $x$ and constant term.

134. What does it mean (graphically) if a point $(a,b)$ lies on $y=f(x)$?

     It means $f(a)=b$. The point satisfies the function equation with $x=a$ and $y=b$.

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

     $m=\frac{y_2-y_1}{x_2-x_1}$. Rise over run between two points gives the slope.

136. What are the $x$-intercepts of $y=f(x)$ in terms of an equation you solve?

     Solutions to $f(x)=0$ (points $(x,0)$). Where the graph crosses the $x$-axis, so $y=0$.

137. What is the vertex form of a quadratic function used for graphing?

     $y=a(x-h)^2+k$. Shows the parabola's vertex at $(h,k)$ directly.

138. Identify the axis of symmetry of $y=-2(x+3)^2+1$.

     $x=-3$. For $(x-h)^2$ form, axis is $x=h$; here $h=-3$.

139. Identify the transformation: $y=f(ax)$ for $a>1$ compared to $y=f(x)$.

     Horizontal compression by factor $a$. Multiplying input by $a>1$ compresses graph horizontally.

140. What is the effect on the graph of replacing $f(x)$ with $f(bx)$ for $b>1$?

     Horizontal compression by factor $b$. Multiplying inputs by $b>1$ squeezes graph toward $y$-axis.

141. What is the point-slope form of a line through $(x_1,y_1)$ with slope $m$?

     $y-y_1=m(x-x_1)$. This form uses a known point and the line's slope.

142. Identify the transformation: $y=f(x)+k$ compared to $y=f(x)$.

     Vertical shift up $k$ units. Adding $k$ to the output shifts the entire graph up by $k$ units.

143. Identify the $y$-intercept of $y=f(x)$ in terms of function notation.

     $y$-intercept is $(0,f(0))$. Substitute $x=0$ into the function to find where graph crosses $y$-axis.

144. Identify the slope of the line through $(2,1)$ and $(6,9)$.

     $m=2$. Apply slope formula: $\frac{9-1}{6-2}=\frac{8}{4}=2$.

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