AP Precalculus Flashcards: Function Model Construction And Application

What this deck covers

This deck focuses on Function Model Construction And Application, giving you a quick way to review the definitions, rules, and examples that matter most for AP Precalculus.

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 is the range of $f(x) = e^x$?

Answer: All positive real numbers. Exponential function always produces positive outputs.

Flashcard 2: What is the equation of the horizontal line at y=4?

Answer: $y = 4$. Horizontal line has constant y-value for all x.

Flashcard 3: What is the effect of $f(x) = \text{sin}(x) - 1$?

Answer: Vertical shift down by 1 unit. Subtracting constant moves entire graph downward.

Flashcard 4: Identify the vertex form of a quadratic function.

Answer: $f(x) = a(x-h)^2 + k$. Shows vertex at $(h,k)$ with $a$ controlling width and direction.

Flashcard 5: What is the inverse function of $f(x) = \frac{1}{x}$?

Answer: $f^{-1}(x) = \frac{1}{x}$. Function is its own inverse (self-inverse).

Flashcard 6: Identify the transformation: $f(x) = 2x^2$.

Answer: Vertical stretch by a factor of 2. Coefficient 2 multiplies all y-values, stretching vertically.

Flashcard 7: What is the general form of a cubic function?

Answer: $f(x) = ax^3 + bx^2 + cx + d$. Highest degree term $ax^3$ with lower degree terms.

Flashcard 8: Identify the transformation: $f(x) = 2x^2$.

Answer: Vertical stretch by a factor of 2. Coefficient 2 multiplies all y-values, stretching vertically.

Flashcard 9: State the domain of the square root function.

Answer: All non-negative real numbers. Cannot take square root of negative numbers in real domain.

Flashcard 10: What is the y-intercept of $f(x) = -5x + 10$?

Answer: $y = 10$. Evaluate $f(0)$ to find where graph crosses y-axis.

Flashcard 11: What is the equation of the horizontal line at y=4?

Answer: $y = 4$. Horizontal line has constant y-value for all x.

Flashcard 12: What is the y-intercept of the function $f(x) = 3x^2 + 2x + 1$?

Answer: $y = 1$. Evaluate $f(0) = 3(0)^2 + 2(0) + 1 = 1$.

Flashcard 13: Find the x-intercept of $f(x) = 2x - 4$.

Answer: $x = 2$. Set $f(x) = 0$ and solve: $2x - 4 = 0$.

Flashcard 14: Find the inverse of $f(x) = 2x + 3$.

Answer: $f^{-1}(x) = \frac{x-3}{2}$. Switch variables and solve: $x = 2y + 3$ gives $y = \frac{x-3}{2}$.

Flashcard 15: What is the y-intercept of the function $f(x) = 3x^2 + 2x + 1$?

Answer: $y = 1$. Evaluate $f(0) = 3(0)^2 + 2(0) + 1 = 1$.

Flashcard 16: What is the effect of $f(x) = \frac{1}{2}x$ on the graph?

Answer: Vertical compression by a factor of $\frac{1}{2}$. Coefficient $\frac{1}{2}$ reduces all y-values by half.

Flashcard 17: Identify the parent function of $f(x) = x^2$.

Answer: Quadratic function. Basic parabola opening upward.

Flashcard 18: What is the general form of a cubic function?

Answer: $f(x) = ax^3 + bx^2 + cx + d$. Highest degree term $ax^3$ with lower degree terms.

Flashcard 19: Identify the parent function of $f(x) = x^2$.

Answer: Quadratic function. Basic parabola opening upward.

Flashcard 20: State the equation of a circle in standard form.

Answer: $(x-h)^2 + (y-k)^2 = r^2$. Center at $(h,k)$ with radius $r$.

Flashcard 21: Find the amplitude of $f(x) = 3\text{sin}(x)$.

Answer: Amplitude is 3. Coefficient determines maximum displacement from center.

Flashcard 22: Identify the transformation: $f(x) = (x-3)^2$.

Answer: Horizontal shift right by 3 units. Subtracting inside parentheses shifts graph right.

Flashcard 23: What is the effect of $f(x) = \frac{1}{2}x$ on the graph?

Answer: Vertical compression by a factor of $\frac{1}{2}$. Coefficient $\frac{1}{2}$ reduces all y-values by half.

Flashcard 24: State the domain of the square root function.

Answer: All non-negative real numbers. Cannot take square root of negative numbers in real domain.

Flashcard 25: What is the effect of $f(x) = x^2 + 5$ on the graph?

Answer: Vertical shift up by 5 units. Adding constant moves entire graph upward.

Flashcard 26: What is the range of the logarithmic function $f(x) = \text{log}(x)$?

Answer: All real numbers. Logarithm can output any real number value.

Flashcard 27: What is the standard form of a quadratic function?

Answer: $f(x) = ax^2 + bx + c$. Standard form with leading coefficient $a$, linear term $bx$, and constant $c$.

Flashcard 28: What is the effect of $f(x) = -x^2$ on the graph?

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

Flashcard 29: Identify the function type: $f(x) = \frac{1}{x}$.

Answer: Rational function. Ratio of polynomials defines rational functions.

Flashcard 30: Identify the axis of symmetry for $f(x) = x^2 - 4x + 3$.

Answer: $x = 2$. Use formula $x = -\frac{b}{2a} = -\frac{-4}{2(1)} = 2$.

Flashcard 31: Identify the axis of symmetry for $f(x) = x^2 - 4x + 3$.

Answer: $x = 2$. Use formula $x = -\frac{b}{2a} = -\frac{-4}{2(1)} = 2$.

Flashcard 32: State the range of the cosine function.

Answer: From -1 to 1 inclusive. Cosine oscillates between minimum -1 and maximum 1.

Flashcard 33: What is the effect of $f(x) = \text{sin}(x) - 1$?

Answer: Vertical shift down by 1 unit. Subtracting constant moves entire graph downward.

Flashcard 34: Identify the transformation: $f(x) = \frac{1}{x+2}$.

Answer: Horizontal shift left by 2 units. Adding inside parentheses shifts graph left.

Flashcard 35: What is the range of $f(x) = e^x$?

Answer: All positive real numbers. Exponential function always produces positive outputs.

Flashcard 36: What is the range of the logarithmic function $f(x) = \text{log}(x)$?

Answer: All real numbers. Logarithm can output any real number value.

Flashcard 37: Find the range of the function $f(x) = 3x + 7$.

Answer: All real numbers. Linear functions have unlimited output values.

Flashcard 38: Identify the function type: $f(x) = \frac{1}{x}$.

Answer: Rational function. Ratio of polynomials defines rational functions.

Flashcard 39: Find the domain of $f(x) = \frac{1}{x-2}$.

Answer: All real numbers except $x = 2$. Denominator cannot equal zero, so $x \neq 2$.

Flashcard 40: State the equation of a circle in standard form.

Answer: $(x-h)^2 + (y-k)^2 = r^2$. Center at $(h,k)$ with radius $r$.

Flashcard 41: Find the zeros of $f(x) = x^2 - 4$.

Answer: $x = 2, x = -2$. Set $f(x) = 0$: $x^2 - 4 = 0$, so $x^2 = 4$.

Flashcard 42: Find the x-intercept of $f(x) = 2x - 4$.

Answer: $x = 2$. Set $f(x) = 0$ and solve: $2x - 4 = 0$.

Flashcard 43: What is the general form of a linear function?

Answer: $f(x) = mx + b$. Standard slope-intercept form where $m$ is slope and $b$ is y-intercept.

Flashcard 44: Find the amplitude of $f(x) = 3\text{sin}(x)$.

Answer: Amplitude is 3. Coefficient determines maximum displacement from center.

Flashcard 45: What is the standard form of a quadratic function?

Answer: $f(x) = ax^2 + bx + c$. Standard form with leading coefficient $a$, linear term $bx$, and constant $c$.

Flashcard 46: What is the effect of $f(x) = x^2 + 5$ on the graph?

Answer: Vertical shift up by 5 units. Adding constant moves entire graph upward.

Flashcard 47: Find the slope of the line $2x - 3y = 6$.

Answer: Slope $m = \frac{2}{3}$. Rearrange to $y = \frac{2}{3}x - 2$ form.

Flashcard 48: What is the effect of $f(x) = -x^2$ on the graph?

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

Flashcard 49: Find the range of the function $f(x) = 3x + 7$.

Answer: All real numbers. Linear functions have unlimited output values.

Flashcard 50: Identify the vertex form of a quadratic function.

Answer: $f(x) = a(x-h)^2 + k$. Shows vertex at $(h,k)$ with $a$ controlling width and direction.

Flashcard 51: Find the vertex of $f(x) = (x-1)^2 + 3$.

Answer: Vertex at $(1, 3)$. Vertex form directly shows vertex coordinates $(h,k)$.

Flashcard 52: Find the slope of the line $2x - 3y = 6$.

Answer: Slope $m = \frac{2}{3}$. Rearrange to $y = \frac{2}{3}x - 2$ form.

Flashcard 53: What is the domain of $f(x) = \text{log}(x)$?

Answer: All positive real numbers. Logarithm undefined for zero and negative inputs.

Flashcard 54: What is the inverse function of $f(x) = \frac{1}{x}$?

Answer: $f^{-1}(x) = \frac{1}{x}$. Function is its own inverse (self-inverse).

Flashcard 55: Which function represents exponential growth?

Answer: $f(x) = a \times b^x$ where $b > 1$. Base $b > 1$ ensures growth; $a$ is initial value.

Flashcard 56: What is the period of $f(x) = \text{sin}(2x)$?

Answer: $\frac{\text{Period}}{2}$. Coefficient of $x$ doubles frequency, halving period to $\pi$.

Flashcard 57: Find the inverse of $f(x) = 2x + 3$.

Answer: $f^{-1}(x) = \frac{x-3}{2}$. Switch variables and solve: $x = 2y + 3$ gives $y = \frac{x-3}{2}$.

Flashcard 58: Find the zeros of $f(x) = x^2 - 4$.

Answer: $x = 2, x = -2$. Set $f(x) = 0$: $x^2 - 4 = 0$, so $x^2 = 4$.

Flashcard 59: What is the domain of $f(x) = \text{log}(x)$?

Answer: All positive real numbers. Logarithm undefined for zero and negative inputs.

Flashcard 60: Find the vertex of $f(x) = (x-1)^2 + 3$.

Answer: Vertex at $(1, 3)$. Vertex form directly shows vertex coordinates $(h,k)$.