PSAT Math Flashcards: Graphing Functions

What this deck covers

This deck focuses on Graphing Functions, giving you a quick way to review the definitions, rules, and examples that matter most for PSAT Math.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Flashcard 37: Identify the vertical asymptote of f(x)=rac{3}{x+5}.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Answer: m=rac{y_2-y_1}{x_2-x_1}. This formula gives the rate of change between two points.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Flashcard 72: Identify the domain of f(x)=rac{1}{x-2}.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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