Math Problem Statement
Problem 2: Use the table of f (x) to construct tables for g(x) and h(x). x -4 -2 0 2 4 f (x) 1 3 6 8 9 a. g(x) = f (x − 2) + 3 b. h(x) = −2f (x + 1) Problem 3: Write an equation that transforms the graph of f (x) = x3 in the desired ways. Do not simplify. a. Shifted right 2 units, downward 4 units. b. Reflected about the x-axis. c. Shifted left 6 units, reflected about the y-axis. d. Reflected about the y-axis, shifted upward 2 units
Solution
Problem 2:
We are given a table for , and we need to construct tables for and .
Table:
\hline x & f(x) \\ \hline -4 & 1 \\ -2 & 3 \\ 0 & 6 \\ 2 & 8 \\ 4 & 9 \\ \hline \end{array}$$ #### a. Construct $$g(x) = f(x - 2) + 3$$: To find $$g(x)$$: 1. Replace $$x$$ with $$x - 2$$ in $$f(x)$$, so $$g(x) = f(x - 2)$$. This shifts $$f(x)$$ to the right by 2 units. 2. Add 3 to the resulting values of $$f(x - 2)$$. Steps: - Shift $$x - 2$$, so we use values of $$x$$ from $$f(x)$$ corresponding to $$x - 2$$: For $$x = -4$$, look up $$f(-6)$$. For $$x = -2$$, look up $$f(-4$$. pandas as pd # Original f(x) table data_f = {'x': [-4, -2, 0, 2, 4], 'f(x)': [1, 3, 6, 8, 9]} df_f = pd.DataFrame(data_f) # Function g(x) = f(x - 2) + 3 # Shift x by -2, find corresponding values, and add 3 df_g = df_f.copy() df_g['x'] = df_g['x'] + 2 df_g['g(x)'] = df_f['f(x)'] + 3 # Function h(x) = -2f(x + 1) # Shift x by +1, find corresponding values, and multiply by -2 df_h = df_f.copy() df_h['x'] = df_h['x'] - 1 df_h['h(x)'] = -2 * df_f['f(x)'] df_g_h = df_g[['x', 'g(x)']].merge(df_h[['x', 'h(x)']], on='x', how='outer') df_g_hHere are the tables for $$g(x) = f(x - 2) + 3$$ and $$h(x) = -2f(x + 1)$$. #### $$g(x) = f(x - 2) + 3$$: $$\begin{array}{|c|c|} \hline x & g(x) \\ \hline -2 & 4 \\ 0 & 6 \\ 2 & 9 \\ 4 & 11 \\ 6 & 12 \\ \hline \end{array}$$ #### $$h(x) = -2f(x + 1)$$: $$\begin{array}{|c|c|} \hline x & h(x) \\ \hline -5 & -2 \\ -3 & -6 \\ -1 & -12 \\ 1 & -16 \\ 3 & -18 \\ \hline \end{array}$$ --- ### Problem 3: Write the equations for the transformations of $$f(x) = x^3$$. #### a. Shifted right 2 units and downward 4 units: The transformation equation is: $$f(x) = (x - 2)^3 - 4$$ #### b. Reflected about the $$x$$-axis: The transformation equation is: $$f(x) = -x^3$$ #### c. Shifted left 6 units and reflected about the $$y$$-axis: The transformation equation is: $$f(x) = (-x + 6)^3$$ #### d. Reflected about the $$y$$-axis and shifted upward 2 units: The transformation equation is: $$f(x) = (-x)^3 + 2$$ Let me know if you have further questions or need detailed explanations! 😊 --- ### Questions to Explore: 1. How do transformations in $$g(x) = f(x - c) + d$$ affect the graph of $$f(x)$$? 2. What happens when a function is reflected about the $$y$$-axis compared to the $$x$$-axis? 3. How do we construct tables for functions involving multiple transformations like $$h(x)$$? 4. How does the cubic graph $$x^3$$ behave under reflections and shifts? 5. Can you verify the computed table values for $$g(x)$$ and $$h(x)$$ manually? ### Tip: When shifting or reflecting graphs, always remember to adjust the signs and order of operations carefully—this avoids common errors!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Function transformation
Table construction
Graph translation
Reflection over axes
Formulas
g(x) = f(x - 2) + 3
h(x) = -2f(x + 1)
f(x) = (x - c)^3 + d (for translation)
f(x) = -x^n (for reflection over x-axis)
f(x) = (-x)^n (for reflection over y-axis)
Theorems
Function translation rules
Reflection properties of functions
Suitable Grade Level
Grades 10-12