Find a function, g left-parenthesis x right-parenthesis, that transforms f left-parenthesis x right-parenthesis equals Root Index 3 Start Root x End Root by shifting f left-parenthesis x right-parenthesis right 3 units.
To shift the function f(x) = ∛x right 3 units, we can add 3 to the input of the function.
Therefore, g(x) = f(x + 3) = ∛(x + 3) is the function that transforms f(x) by shifting it right 3 units.
options
a) g left-parenthesis x right-parenthesis equals Root Index 3 Start Root x plus 3 End Root- image with description: g left-parenthesis x right-parenthesis equals Root Index 3 Start Root x plus 3 End Root - - no response given
b) g left-parenthesis x right-parenthesis equals Root Index 3 Start Root x minus 3 End Root
- image with description: g left-parenthesis x right-parenthesis equals Root Index 3 Start Root x minus 3 End Root - - no response given
c) g left-parenthesis x right-parenthesis equals Root Index 3 Start Root x End Root plus 3
- image with description: g left-parenthesis x right-parenthesis equals Root Index 3 Start Root x End Root plus 3 - - incorrect
d) g left-parenthesis x right-parenthesis equals Root Index 3 Start Root x End Root minus 3
The correct answer is option D:
g(x) = ∛x - 3
Describe how the function g left-parenthesis x right-parenthesis equals one-half Start Absolute Value x minus 4 End Absolute Value transforms the function f left-parenthesis x right-parenthesis equals Start Absolute Value x minus 4 End Absolute Value.
The function g(x) = 1/2 |x - 4| is obtained by scaling the function f(x) = |x - 4| by a factor of 1/2.
Recall that the graph of f(x) = |x - 4| is a V-shaped graph centered at x = 4, with the vertex at (4, 0). This means that as x approaches positive infinity or negative infinity, the value of f(x) increases without bound, taking positive values. On the other hand, for x values less than 4, f(x) takes negative values, while for x values greater than 4, f(x) takes positive values.
When we introduce the factor of 1/2 in g(x) = 1/2 |x - 4|, the graph of g(x) is compressed horizontally. This means that the graph is narrower compared to the graph of f(x). Additionally, the scaling factor also decreases the amplitude of g(x) compared to f(x). Specifically, the maximum distance between the graph of g(x) and the x-axis is half of the maximum distance of the graph of f(x) and the x-axis.
Hence, when we compare the graphs of f(x) and g(x), the graph of g(x) is narrower, and the distances between the graph and the x-axis are halved compared to the graph of f(x).
options
a) The graph of g left-parenthesis x right-parenthesis is the parent function f left-parenthesis x right-parenthesis vertically compressed by a factor of 2.
The graph of Image with alt text: g left-parenthesis x right-parenthesis is the parent function Image with alt text: f left-parenthesis x right-parenthesis vertically compressed by a factor of 2.
b) The graph of g left-parenthesis x right-parenthesis is the parent function f left-parenthesis x right-parenthesis vertically stretched by a factor of 2.
The graph of Image with alt text: g left-parenthesis x right-parenthesis is the parent function Image with alt text: f left-parenthesis x right-parenthesis vertically stretched by a factor of 2.
c) The graph of g left-parenthesis x right-parenthesis is the parent function f left-parenthesis x right-parenthesis horizontally stretched by a factor of 2.
The graph of Image with alt text: g left-parenthesis x right-parenthesis is the parent function Image with alt text: f left-parenthesis x right-parenthesis horizontally stretched by a factor of 2.
d) The graph of g left-parenthesis x right-parenthesis is the parent function f left-parenthesis x right-parenthesis horizontally compressed by a factor of 2.
The correct answer is option a:
The graph of g(x) is the parent function f(x) vertically compressed by a factor of 2.