Textbook Question
Begin by graphing the cube root function, f(x) = ∛x. Then use transformations of this graph to graph the given function. g(x) = (1/2)∛(x-2)
674
views
3. Functions
Transformations
1:23 minutesProblem 11
Textbook Question
Use the graph of y = f(x) to graph each function g.
g(x) = ½ f(x)
Verified step by step guidance1
Step 1: Observe the graph of y = f(x). It is a horizontal line segment from (1, -3) to (4, -3). This means that the function f(x) has a constant value of -3 for all x in the interval [1, 4].
Step 2: The new function g(x) = ½ f(x) involves scaling the values of f(x) by a factor of ½. Since f(x) = -3, g(x) = ½ * (-3) = -3/2 for all x in the interval [1, 4].
Step 3: To graph g(x), keep the x-values the same (from 1 to 4), but adjust the y-values to reflect the scaled value of -3/2. This means the new graph will be a horizontal line segment at y = -3/2.
Step 4: Plot the points (1, -3/2) and (4, -3/2) on the graph. These points represent the endpoints of the horizontal line segment for g(x).
Step 5: Draw a horizontal line segment connecting the points (1, -3/2) and (4, -3/2). This is the graph of g(x) = ½ f(x).
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Function Transformation
Function transformation refers to the process of altering the graph of a function through operations such as scaling, translating, or reflecting. In this case, the function g(x) = ½ f(x) represents a vertical compression of the function f(x) by a factor of ½, meaning that all y-values of f(x) are halved, resulting in a graph that is closer to the x-axis.
Recommended video:
4:22
Domain & Range of Transformed Functions
Graphing Linear Functions
Graphing linear functions involves plotting points that satisfy the function's equation and connecting them to form a straight line. For the function f(x) shown in the graph, which is a horizontal line at y = -3 between x = 1 and x = 4, understanding how to plot these points is essential for accurately representing the transformed function g(x).
Recommended video:
5:26Graphs of Logarithmic Functions
Horizontal and Vertical Scaling
Horizontal and vertical scaling are techniques used to stretch or compress the graph of a function. Vertical scaling, as seen in g(x) = ½ f(x), compresses the graph vertically, affecting the y-values while keeping the x-values unchanged. This concept is crucial for predicting how the graph of g will appear compared to f.
Recommended video:
5:28Horizontal Parabolas
Related Videos
Related Practice