Transformations of Functions AP Precalculus PDF - Past Paper

# Study Sheet: Transformations of Functions (AP Precalculus) **Key Topics Covered:** ## 1. Transformations of Functions * Vertical Translation: $g(x) = f(x) + k$ * Moves graph up or down by k units. * Horizontal Translation: $g(x) = f(x+h)$ * Moves graph left ($h > 0$) or right...

# Study Sheet: Transformations of Functions (AP Precalculus) **Key Topics Covered:** ## 1. Transformations of Functions * Vertical Translation: $g(x) = f(x) + k$ * Moves graph up or down by k units. * Horizontal Translation: $g(x) = f(x+h)$ * Moves graph left ($h > 0$) or right ($h < 0$). * Vertical Dilation: $g(x) = af(x)$ * Stretches/compresses vertically by a factor of $|a|$. Reflects over x-axis if $a < 0$. * Horizontal Dilation: $g(x) = f(bx)$ * Stretches/compresses horizontally by a factor of $\frac{1}{|b|}$. Reflects over y-axis if $b < 0$. ## 2. Function Model Selection and Assumption Articulation * Selecting appropriate models (linear, quadratic, etc.) for real-world scenarios. * Interpreting regression equations, correlation coefficients $(r)$, and residuals. ## 3. Function Model Construction and Application * Constructing models using transformations. * Applying models to solve contextual problems (e.g. temperature vs. fish distance, bacterial growth). ## 4. Examples of Parent Functions and Their Transformations Constant, Identity, Quadratic, Cubic, Rational, Square Root, Cube Root, Absolute Value functions. # 60-Point Test on Transformations of Functions **Section 1: Multiple Choice (20 points)** 1. Which transformation shifts the graph of $f(x)$ three units to the right? A. $f(x + 3)$ B. $f(x - 3)$ C. $f(x) + 3$ D. $f(x) - 3$ 2. What is the effect of $g(x) = -f(x)$ on the graph of $f(x)$? A. Reflection over the x-axis B. Reflection over the y-axis C. Vertical stretch by a factor of 2 D. Horizontal compression by a factor of 2 3. If $f(x) = x^2$, what is the equation of the function after a vertical dilation by a factor of 3 and a horizontal translation 2 units left? A. $g(x) = 3(x + 2)^2$ B. $g(x) = 3(x - 2)^2$ C. $g(x) = (3x + 2)^2$ D. $g(x) = (3x - 2)^2$ 4. The domain of $f(x) = \sqrt{x}$ is transformed to $g(x) = \sqrt{x} - 4$. What is the new domain of $g(x)$? A. $[0, \infty)$ B. $[4, \infty)$ C. $[-4, 0]$ D. $(-\infty, 4]$ 5. If $h(x) = 2f(3x) - 1$, which sequence of transformations describes $h(x)$ from $f(x)$? A. Horizontal compression by $\frac{1}{3}$, vertical stretch by 2, vertical shift down by 1 B. Horizontal stretch by 3, vertical compression by $\frac{1}{2}$, vertical shift up by 1 C. Horizontal compression by $\frac{1}{3}$, vertical compression by $\frac{1}{2}$, vertical shift down by 1 D. Horizontal stretch by 3, vertical stretch by 2, vertical shift up by 1 **Section 2: Free Response (40 points)** **Question 1 (10 points):** Let $f(x) = x^2$ a) Write the equation of $g(x)$ if it is obtained by shifting $f(x)$ 4 units to the right and 3 units up. b) Sketch the graphs of $f(x)$ and $g(x)$ on the same axes. Clearly label key points. c) State the domain and range of $g(x)$. **Question 2 (10 points):** The table below shows values of a function $p(x)$. Let $h(x) = 3p(2x) - 1$ | x | -2 | 0 | 2 | 4 | 6 | | --- | --- | -- | -- | -- | -- | | p(x) | 1 | -1 | 0 | 3 | 7 | a) Find $h(-1)$. b) Find $h(2)$. c) Explain why $h(h(0))$ cannot be determined without additional information. **Question 3 (10 points):** A nuclear power plant releases water into a lake at varying temperatures. The distance $d$ (in meters) from the outflow pipe to the nearest fish is modeled by $d(T) = 2T + 10$, where $T$ is the water temperature in degrees Celsius. a) Predict the distance when $T = 29^\circ C$. b) Calculate the residual if the actual distance at $T = 29^\circ C$ is 70 meters. Interpret the residual. c) Is this model appropriate for predicting distances at $T = 50^\circ C$? Justify your answer. **Question 4 (10 points):** The number of bacteria $N(T)$ in refrigerated food is given by $N(T) = 30T^2 - 30T + 120$, where $T$ is the temperature in degrees Celsius $(-2 < T < 14)$. a) At what temperature is the number of bacteria minimized? b) How many bacteria are present at $T = 2.7^\circ C$? c) Is this model valid for $T = -16^\circ C$? Why or why not? # Answers **Multiple Choice:** 1. B 2. A 3. A 4. B 5. A **Free Response:** **Question 1:** a) $g(x) = (x - 4)^2 + 3$ b) Graphs should show $f(x) = x^2$ as a parabola opening upwards with vertex at $(0, 0)$, and $g(x)$ as a parabola shifted 4 units right and 3 units up with vertex at (4,3). c) Domain: $(-\infty, \infty)$; Range: $[3, \infty)$. **Question 2:** a) $h(-1) = 3p(-2) - 1 = 3(1) - 1 = 2$ b) $h(2) = 3p(4) - 1 = 3(3) - 1 = 8$ c) $h(0) = 3p(0) - 1 = 3(-1) - 1 = -4$. Since $h(-4)$ requires $p(-8)$, which is undefined in the table, $h(h(0))$ cannot be determined. **Question 3:** a) $d(29) = 2(29) + 10 = 68$ meters b) Residual = Actual - Predicted = 70 - 68 = 2. This means the actual distance is 2 meters farther than predicted c) No, because extrapolating beyond the data range risks inaccuracies. **Question 4:** a) Minimum occurs at $T = \frac{-b}{2a} = \frac{-(-30)}{2(30)} = 0.5^\circ C$ b) $N(2.7) = 30(2.7)^2 - 30(2.7) + 120 = 137.7$ bacteria c) No, because $T = -16^\circ C$ is outside the model's domain $(-2 < T < 14)$.