Function Model Construction and Application - AP Precalculus

1. What is the equation of the horizontal line at y=4?

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

2. What is the inverse function of $f(x) = \frac{1}{x}$?

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

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

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

4. What is the general form of a cubic function?

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

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

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

6. Find the x-intercept of $f(x) = 2x - 4$.

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

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

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

8. Identify the parent function of $f(x) = x^2$.

   Quadratic function. Basic parabola opening upward.

9. State the equation of a circle in standard form.

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

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

    Amplitude is 3. Coefficient determines maximum displacement from center.

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

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

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

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

13. State the domain of the square root function.

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

14. What is the effect of $f(x) = x^2 + 5$ on the graph?

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

15. What is the range of the logarithmic function $f(x) = \text{log}(x)$?

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

16. What is the effect of $f(x) = -x^2$ on the graph?

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

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

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

18. State the range of the cosine function.

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

19. What is the effect of $f(x) = \text{sin}(x) - 1$?

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

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

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

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

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

22. Find the range of the function $f(x) = 3x + 7$.

    All real numbers. Linear functions have unlimited output values.

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

    Rational function. Ratio of polynomials defines rational functions.

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

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

25. Find the zeros of $f(x) = x^2 - 4$.

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

26. What is the general form of a linear function?

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

27. What is the standard form of a quadratic function?

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

28. Identify the vertex form of a quadratic function.

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

29. Find the slope of the line $2x - 3y = 6$.

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

30. Which function represents exponential growth?

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

31. What is the period of $f(x) = \text{sin}(2x)$?

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

32. Find the inverse of $f(x) = 2x + 3$.

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

33. What is the domain of $f(x) = \text{log}(x)$?

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

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

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

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