Problem 72
Question
Use a graphing utility to solve the problem. If \(f(x)=|x|,\) graph \(f(x)\) and \(f(0.3 x)\) in the same viewing window. What is the relationship between the two graphs?
Step-by-Step Solution
Verified Answer
The graph of \(f(0.3x) = |0.3x|\) is a horizontal stretch of the graph of the original function \(f(x) = |x|\). This happens because the 'x' values are multiplied by a factor of 0.3, which stretch out the function horizontally.
1Step 1: Understand the nature of the Absolute function
The function \(f(x) = |x|\), also known as the absolute value function, always creates V-shaped graphs, indicating that the output value is always non-negative, regardless of the input value.
2Step 2: Graph the absolute function \(f(x)\)
Using a graphing utility tool, such as Desmos or GeoGebra, plot the function \(f(x) = |x|\). Notice the 'V' shape of the graph, where the vertex is at the origin (0,0).
3Step 3: Graph the compressed function \(f(0.3x)\)
Now, graph the function \(f(0.3x) = |0.3x|\). This is a horizontal compression of the original graph by a factor of 0.3 since the 'x' values are multiplied by 0.3.
4Step 4: Compare and analyze both graphs
Observe both graphs in the same viewing window. The graph of \(f(0.3x)\) is wider than the graph of \(f(x)\) because multiplying x by a factor less than 1 horizontally stretches the graph.
Key Concepts
Absolute Value FunctionGraphical TransformationsGraphing Utility Tools
Absolute Value Function
The absolute value function is represented by the equation f(x) = |x| and is pivotal in understanding how distances and magnitudes work in mathematics. When you take the absolute value of a number, you're essentially looking at how far the number is from zero on the number line, regardless of direction.
For example, the absolute value of 3 is 3, and the absolute value of -3 is also 3. This reflects in the graph of an absolute value function, which always takes on a 'V' shape. The 'V' shape occurs because the function produces only non-negative outputs. As a result, any negative input is reflected upwards, creating a mirror image across the x-axis.
In graphing the absolute value function, it's important to note the vertex, which is the point at the bottom of the 'V'. For the basic function f(x) = |x|, the vertex is at the origin (0,0), and the graph forms equal angles as it moves away from the vertex on either side.
For example, the absolute value of 3 is 3, and the absolute value of -3 is also 3. This reflects in the graph of an absolute value function, which always takes on a 'V' shape. The 'V' shape occurs because the function produces only non-negative outputs. As a result, any negative input is reflected upwards, creating a mirror image across the x-axis.
In graphing the absolute value function, it's important to note the vertex, which is the point at the bottom of the 'V'. For the basic function f(x) = |x|, the vertex is at the origin (0,0), and the graph forms equal angles as it moves away from the vertex on either side.
Graphical Transformations
Graphical transformations involve altering the basic graph of a function in various ways to produce a new graph. These transformations can be classified under translations, reflections, stretching, and compressions.
For instance, if we have f(x) = |x| and we compare it to f(0.3x) = |0.3x|, we notice that the input x is multiplied by a factor of 0.3. This change represents a horizontal stretch by the reciprocal of 0.3, which is approximately 3.33. Consequently, for every one unit that the graph of f(x) = |x| moves horizontally, the graph of f(0.3x) will move 3.33 units.
This stretching makes the 'V' shape of the graph of f(0.3x) look wider compared to that of f(x). Understanding how these transformations affect the graph can be particularly useful in visualizing functions without having to plot numerous points.
For instance, if we have f(x) = |x| and we compare it to f(0.3x) = |0.3x|, we notice that the input x is multiplied by a factor of 0.3. This change represents a horizontal stretch by the reciprocal of 0.3, which is approximately 3.33. Consequently, for every one unit that the graph of f(x) = |x| moves horizontally, the graph of f(0.3x) will move 3.33 units.
This stretching makes the 'V' shape of the graph of f(0.3x) look wider compared to that of f(x). Understanding how these transformations affect the graph can be particularly useful in visualizing functions without having to plot numerous points.
Graphing Utility Tools
Graphing utility tools like Desmos, GeoGebra, and graphing calculators, have revolutionized the way we understand and visualize mathematical functions. These tools allow you to enter any function and almost instantly view its graph, which is particularly handy when learning about transformations.
When using these tools, you can plot multiple functions on the same axes to directly compare their shapes and positions. This can clearly illustrate the effects of transformations like the one seen between f(x) = |x| and f(0.3x). You can also use sliders to dynamically vary parameters and instantly see how the graph changes in response, reinforcing your understanding of these concepts.
Perhaps the most significant benefit is the ability to experiment with functions and transformations, which can lead to a deeper comprehension and stimulate curiosity. In educational settings, these utilities can bridge the gap between abstract theory and visual, concrete understanding.
When using these tools, you can plot multiple functions on the same axes to directly compare their shapes and positions. This can clearly illustrate the effects of transformations like the one seen between f(x) = |x| and f(0.3x). You can also use sliders to dynamically vary parameters and instantly see how the graph changes in response, reinforcing your understanding of these concepts.
Perhaps the most significant benefit is the ability to experiment with functions and transformations, which can lead to a deeper comprehension and stimulate curiosity. In educational settings, these utilities can bridge the gap between abstract theory and visual, concrete understanding.
Other exercises in this chapter
Problem 72
Sketch a graph of the quadratic function, indicating the vertex, the axis of symmetry, and any \(x\)-intercepts. $$h(t)=-5 t+3-t^{2}$$
View solution Problem 72
In Exercises \(67-86,\) find expressions for \((f \circ g)(x)\) and \((g \circ f)(x)\) Give the domains of \(f \circ g\) and \(g \circ f\). $$f(x)=-2 x+1 ; g(x)
View solution Problem 72
Find all solutions of the quadratic equation. Relate the solutions of the equation to the zeros of an appropriate quadratic function. $$-6 t^{2}+2 t-\frac{1}{3}
View solution Problem 73
Sketch a graph of the quadratic function, indicating the vertex, the axis of symmetry, and any \(x\)-intercepts. $$F(s)=-2 s^{2}+3 s+1$$
View solution