SAT 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 SAT 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: Identify the range of the function $f(x) = -2x^2 + 4$.

Answer: $y \leq 4$. Parabola opens downward with maximum value $y = 4$.

Flashcard 2: Which function represents a parabola opening upwards?

Answer: $y = ax^2 + bx + c$, $a > 0$. When $a > 0$, the parabola opens upward with minimum vertex.

Flashcard 3: Determine the y-intercept of the function $y = -3x + 7$.

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

Flashcard 4: What is the range of the function $y = \ln(x)$?

Answer: All real numbers. Natural logarithm outputs all real values for positive inputs.

Flashcard 5: State the domain for $f(x) = \frac{1}{x^2 - x - 6}$.

Answer: $x \neq 3, x \neq -2$. Factor: $(x-3)(x+2) \neq 0$, so exclude both roots.

Flashcard 6: Identify the range of the function $f(x) = |x|$.

Answer: $y \geq 0$. Absolute value function always produces non-negative outputs.

Flashcard 7: Identify the axis of symmetry for $y = ax^2 + bx + c$.

Answer: $x = -\frac{b}{2a}$. Vertical line through the vertex of any parabola.

Flashcard 8: State the equation for the asymptote of $y = 2^x - 3$.

Answer: $y = -3$. Exponential functions approach horizontal asymptotes as $x \to -\infty$.

Flashcard 9: State the transformation: $f(x) = x^2$ to $f(x) = 3x^2$.

Answer: Vertical stretch by 3. Coefficient greater than 1 stretches graph vertically.

Flashcard 10: State the intercepts of $y = x^2 - 4x + 4$.

Answer: $(2, 0)$. Perfect square trinomial $(x-2)^2$ has double root at $x = 2$.

Flashcard 11: What is the equation of a parabola opening upwards with vertex $(h, k)$?

Answer: $y = a(x - h)^2 + k$. Vertex form with $a > 0$ opens upward from vertex $(h, k)$.

Flashcard 12: What is the range of the function $y = \ln(x)$?

Answer: All real numbers. Natural logarithm outputs all real values for positive inputs.

Flashcard 13: Identify the vertex form of a quadratic function.

Answer: $y = a(x - h)^2 + k$. Shows vertex at $(h, k)$ with horizontal shifts and vertical shifts.

Flashcard 14: What is the general form of the exponential function?

Answer: $y = ab^x$. Base $b$ raised to variable power $x$ with coefficient $a$.

Flashcard 15: Find the x-intercept of the line: $y = 3x + 6$.

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

Flashcard 16: State the formula for the slope of a line through $(x_1, y_1)$ and $(x_2, y_2)$.

Answer: $m = \frac{y_2 - y_1}{x_2 - x_1}$. Rise over run formula between two points.

Flashcard 17: State the equation of a horizontal line.

Answer: $y = c$. Horizontal lines have zero slope and constant y-value.

Flashcard 18: State the domain for $f(x) = \frac{1}{x^2 - x - 6}$.

Answer: $x \neq 3, x \neq -2$. Factor: $(x-3)(x+2) \neq 0$, so exclude both roots.

Flashcard 19: Which function represents exponential growth?

Answer: $y = a \cdot b^x$, $b > 1$. Base $b > 1$ creates increasing exponential function.

Flashcard 20: Identify the vertex form of a quadratic function.

Answer: $y = a(x-h)^2 + k$. Standard vertex form with vertex at $(h,k)$ and vertical stretch $a$.

Flashcard 21: Find the vertex of $y = 2(x - 3)^2 + 4$.

Answer: $(3, 4)$. Vertex form directly shows vertex coordinates $(h, k)$.

Flashcard 22: What is the vertex of the quadratic function $y = x^2 - 6x + 9$?

Answer: $(3, 0)$. Complete the square: $(x-3)^2$ gives vertex at $(3, 0)$.

Flashcard 23: What is the shape of the graph of $y = \frac{1}{x}$?

Answer: Hyperbola. Rational function with two branches in opposite quadrants.

Flashcard 24: Find the x-intercept of the line: $y = 3x + 6$.

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

Flashcard 25: Identify the range of $y = 3x^2 - 5$.

Answer: $y \geq -5$. Parabola opens upward with minimum value at vertex $y = -5$.

Flashcard 26: Identify the transformation: $f(x) = (x - 4)^2 + 2$.

Answer: Right 4, Up 2. $(x - 4)$ shifts right 4, $+2$ shifts up 2.

Flashcard 27: Identify the axis of symmetry for $y = ax^2 + bx + c$.

Answer: $x = -\frac{b}{2a}$. Vertical line through the vertex of any parabola.

Flashcard 28: Identify the equation for a circle centered at the origin.

Answer: $x^2 + y^2 = r^2$. Circle equation with center $(0,0)$ and radius $r$.

Flashcard 29: State the domain of the function $f(x) = \sqrt{x-3}$.

Answer: $x \geq 3$. Square root requires non-negative input, so $x - 3 \geq 0$.

Flashcard 30: Identify the vertex form of a quadratic function.

Answer: $y = a(x-h)^2 + k$. Standard vertex form with vertex at $(h,k)$ and vertical stretch $a$.

Flashcard 31: State the equation of a horizontal line.

Answer: $y = c$. Horizontal lines have zero slope and constant y-value.

Flashcard 32: What is the domain of the function $f(x) = \frac{1}{x - 2}$?

Answer: $x \neq 2$. Denominator cannot equal zero, so $x - 2 \neq 0$.

Flashcard 33: State the intercepts of $y = x^2 - 4x + 4$.

Answer: $(2, 0)$. Perfect square trinomial $(x-2)^2$ has double root at $x = 2$.

Flashcard 34: What is the slope of a vertical line?

Answer: Undefined. Vertical lines have infinite slope, expressed as undefined.

Flashcard 35: What is the standard form of a quadratic function?

Answer: $y = ax^2 + bx + c$. General quadratic form with leading coefficient $a$ and constant term $c$.

Flashcard 36: Identify the transformation: $f(x) = x^2$ to $f(x) = (x + 5)^2$.

Answer: Left 5. $(x + 5)$ represents horizontal shift left by 5 units.

Flashcard 37: Identify the transformation: $f(x) = x^2$ to $f(x) = (x + 5)^2$.

Answer: Left 5. $(x + 5)$ represents horizontal shift left by 5 units.

Flashcard 38: What is the standard form of a quadratic function?

Answer: $y = ax^2 + bx + c$. General quadratic form with leading coefficient $a$ and constant term $c$.

Flashcard 39: Identify the transformation: $f(x) = (x - 4)^2 + 2$.

Answer: Right 4, Up 2. $(x - 4)$ shifts right 4, $+2$ shifts up 2.

Flashcard 40: Identify the range of the function $f(x) = -2x^2 + 4$.

Answer: $y \leq 4$. Parabola opens downward with maximum value $y = 4$.

Flashcard 41: What is the equation of a parabola opening upwards with vertex $(h, k)$?

Answer: $y = a(x - h)^2 + k$. Vertex form with $a > 0$ opens upward from vertex $(h, k)$.

Flashcard 42: Find the y-intercept of the function $f(x) = 4x^2 - 3x + 5$.

Answer: $y = 5$. Y-intercept is the constant term when $x = 0$.

Flashcard 43: What is the domain of the function $f(x) = \frac{1}{x - 2}$?

Answer: $x \neq 2$. Denominator cannot equal zero, so $x - 2 \neq 0$.

Flashcard 44: Which function represents exponential growth?

Answer: $y = a \cdot b^x$, $b > 1$. Base $b > 1$ creates increasing exponential function.

Flashcard 45: What is the result of $f(x) = x^3$ being reflected over the y-axis?

Answer: $f(x) = -x^3$. Reflection over y-axis changes $x$ to $-x$ in odd functions.

Flashcard 46: What is the general form of a linear function?

Answer: $y = mx + b$. Slope-intercept form where $m$ is slope and $b$ is y-intercept.

Flashcard 47: Find the x-intercepts of $y = x^2 - 9$.

Answer: $x = 3, x = -3$. Factor as $(x-3)(x+3) = 0$ to find roots.

Flashcard 48: What is the general form of the exponential function?

Answer: $y = ab^x$. Base $b$ raised to variable power $x$ with coefficient $a$.

Flashcard 49: What is the slope of the line $y = -5x + 2$?

Answer: $m = -5$. Coefficient of $x$ in linear form $y = mx + b$.

Flashcard 50: Find the y-intercept of the function $f(x) = 4x^2 - 3x + 5$.

Answer: $y = 5$. Y-intercept is the constant term when $x = 0$.

Flashcard 51: State the domain of the function $f(x) = \sqrt{x-3}$.

Answer: $x \geq 3$. Square root requires non-negative input, so $x - 3 \geq 0$.

Flashcard 52: What is the vertex of the quadratic function $y = x^2 - 6x + 9$?

Answer: $(3, 0)$. Complete the square: $(x-3)^2$ gives vertex at $(3, 0)$.

Flashcard 53: What is the general form of a linear equation?

Answer: $y = mx + b$. Standard linear form with slope $m$ and y-intercept $b$.

Flashcard 54: What is the general form of a linear equation?

Answer: $y = mx + b$. Standard linear form with slope $m$ and y-intercept $b$.

Flashcard 55: Find the x-intercept of the function $y = 2x + 3$.

Answer: $x = -\frac{3}{2}$. Set $y = 0$ and solve: $0 = 2x + 3$, so $x = -\frac{3}{2}$.

Flashcard 56: Identify the range of the function $f(x) = |x|$.

Answer: $y \geq 0$. Absolute value function always produces non-negative outputs.

Flashcard 57: Identify the range of $y = 3x^2 - 5$.

Answer: $y \geq -5$. Parabola opens upward with minimum value at vertex $y = -5$.

Flashcard 58: Find the domain of the function $f(x) = \sqrt{x - 4}$.

Answer: $x \geq 4$. Square root requires non-negative argument: $x - 4 \geq 0$.

Flashcard 59: What is the result of $f(x) = x^3$ being reflected over the y-axis?

Answer: $f(x) = -x^3$. Reflection over y-axis changes $x$ to $-x$ in odd functions.

Flashcard 60: What is the y-intercept of the line: $y = -4x + 7$?

Answer: $y = 7$. Y-intercept occurs when $x = 0$, giving $y = 7$.

Flashcard 61: What is the shape of the graph of $y = \frac{1}{x}$?

Answer: Hyperbola. Rational function with two branches in opposite quadrants.

Flashcard 62: What is the effect of $f(x) = x^2$ becoming $f(x) = -x^2$?

Answer: Reflection over x-axis. Negative coefficient flips parabola upside down.

Flashcard 63: State the formula for the slope of a line through $(x_1, y_1)$ and $(x_2, y_2)$.

Answer: $m = \frac{y_2 - y_1}{x_2 - x_1}$. Rise over run formula between two points.

Flashcard 64: State the transformation: $f(x) = x^2$ to $f(x) = 3x^2$.

Answer: Vertical stretch by 3. Coefficient greater than 1 stretches graph vertically.

Flashcard 65: Find the x-intercepts of $y = x^2 - 9$.

Answer: $x = 3, x = -3$. Factor as $(x-3)(x+3) = 0$ to find roots.

Flashcard 66: Find the domain of the function $f(x) = \sqrt{x - 4}$.

Answer: $x \geq 4$. Square root requires non-negative argument: $x - 4 \geq 0$.

Flashcard 67: Find the vertex of $y = 2(x - 3)^2 + 4$.

Answer: $(3, 4)$. Vertex form directly shows vertex coordinates $(h, k)$.

Flashcard 68: Identify the equation for a circle centered at the origin.

Answer: $x^2 + y^2 = r^2$. Circle equation with center $(0,0)$ and radius $r$.

Flashcard 69: Identify the vertex form of a quadratic function.

Answer: $y = a(x - h)^2 + k$. Shows vertex at $(h, k)$ with horizontal shifts and vertical shifts.

Flashcard 70: What is the slope of the line $y = -5x + 2$?

Answer: $m = -5$. Coefficient of $x$ in linear form $y = mx + b$.

Flashcard 71: Find the x-intercept of the function $y = 2x + 3$.

Answer: $x = -\frac{3}{2}$. Set $y = 0$ and solve: $0 = 2x + 3$, so $x = -\frac{3}{2}$.

Flashcard 72: What is the effect of $f(x) = x^2$ becoming $f(x) = -x^2$?

Answer: Reflection over x-axis. Negative coefficient flips parabola upside down.

Flashcard 73: What is the general form of a linear function?

Answer: $y = mx + b$. Slope-intercept form where $m$ is slope and $b$ is y-intercept.

Flashcard 74: What is the slope of a vertical line?

Answer: Undefined. Vertical lines have infinite slope, expressed as undefined.

Flashcard 75: State the equation for the asymptote of $y = 2^x - 3$.

Answer: $y = -3$. Exponential functions approach horizontal asymptotes as $x \to -\infty$.

Flashcard 76: Determine the y-intercept of the function $y = -3x + 7$.

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

Flashcard 77: Which function represents a parabola opening upwards?

Answer: $y = ax^2 + bx + c$, $a > 0$. When $a > 0$, the parabola opens upward with minimum vertex.

Flashcard 78: What is the y-intercept of the line: $y = -4x + 7$?

Answer: $y = 7$. Y-intercept occurs when $x = 0$, giving $y = 7$.