Problem 65
Question
a. Use a graphing utility to graph \(f(x)=x^{2}+1\)
b. Graph \(f(x)=x^{2}+1,\) and \(g(x)=f\left(\frac{1}{2} x\right),\) and
\(h(x)=f\left(\frac{1}{4} x\right)\) in the same viewing rectangle.
c. Describe the relationship among the graphs of \(f, g,\) and \(h,\) with
emphasis on different values of \(x\) for points on all three graphs that give
the same \(y\) -coordinate.
d. Generalize by describing the relationship between the graph of \(f\) and the
graph of \(g,\) where \(g(x)=f(c x)\) for \(0
Step-by-Step Solution
Verified Answer
By graphing and analyzing, we see that as \(c\) decreases from 1 to less than 1, the graph of \(g(x)\) gets wider or stretches horizontally away from the y-axis. This implies that modifying the input with a factor \(c\) where \(0
1Step 1: Graph the Function \(f(x)=x^{2}+1\)
Use a graphing utility to plot the function \(f(x)=x^{2}+1\). This will provide a baseline for the additional functions to be graphed.
2Step 2: Graph the Functions on the Same Viewing Rectangle
In the same viewing rectangle, use the graphing utility to plot the functions \(f(x)=x^{2}+1\), \(g(x)=f\left(\frac{1}{2} x\right)\), and \(h(x)=f\left(\frac{1}{4} x\right)\). This will allow a clear visual comparison of the three functions.
3Step 3: Describe the Relationship
Analyze the plot and provide an explanation of the relationship between the graphs. Be sure to focus on different values of \(x\) that share the same \(y\) -coordinate across all three graphs.
4Step 4: Generalize the Relationship
Use your observations to generalize and describe the relationship between the graph of \(f\) and the graph of \(g\), where \(g(x)=f(c x)\) for \(0
5Step 5: Sketch the Graphs Using a Different Function
Choose another function and sketch the graphs of \(f(c x)\) for \(c=1,\) and \(c=\frac{1}{2},\) and \(c=\frac{1}{4}\). Verify if this new set of graphs aligns with the generalized relationship established in Step 4.
Key Concepts
Function TransformationGraphical AnalysisScaling FunctionsParabolas
Function Transformation
Function transformation is an important concept in mathematics that helps to understand how graphs of functions change with various operations. For instance, the function \(f(x) = x^2 + 1\) describes a parabola opening upwards, shifted one unit above the x-axis.
A common transformation involves altering the input variable. When we examine equations like \(g(x) = f\left(\frac{1}{2}x\right)\) or \(h(x) = f\left(\frac{1}{4}x\right)\), we are scaling the input variable. This operation affects the width of the parabola.
By scaling the x-variable by constants \(\frac{1}{2}\) and \(\frac{1}{4}\), we compress the graph horizontally. As a result, these transformations demonstrate how manipulating the input (x-value) can affect the shape and orientation of the graph.
A common transformation involves altering the input variable. When we examine equations like \(g(x) = f\left(\frac{1}{2}x\right)\) or \(h(x) = f\left(\frac{1}{4}x\right)\), we are scaling the input variable. This operation affects the width of the parabola.
By scaling the x-variable by constants \(\frac{1}{2}\) and \(\frac{1}{4}\), we compress the graph horizontally. As a result, these transformations demonstrate how manipulating the input (x-value) can affect the shape and orientation of the graph.
Graphical Analysis
Graphical analysis involves closely studying the behavior and relationships of graphs. In this exercise, analyzing how the graph of \(f(x) = x^2 + 1\) compares to those of \(g(x)\) and \(h(x)\) reveals a clear pattern.
The function \(f(x) = x^2 + 1\) serves as our original or baseline graph. When we introduce transformations such as \(g(x) = f\left(\frac{1}{2}x\right)\), the graph squishes horizontally by a factor of 2. Analyzing these changes allows us to compare specific points:
These x-coordinate shifts determine the amount of "stretch" or "compression" that occurs for any value of \(y\) on overlapping x-values.
The function \(f(x) = x^2 + 1\) serves as our original or baseline graph. When we introduce transformations such as \(g(x) = f\left(\frac{1}{2}x\right)\), the graph squishes horizontally by a factor of 2. Analyzing these changes allows us to compare specific points:
- For \(f(x)\), a point (x, y) will have a new counterpart in \(g(x)\) at location \(\left(2x, y\right)\).
- Similarly, in \(h(x) = f\left(\frac{1}{4}x\right)\), each point moves to \(\left(4x, y\right)\).
These x-coordinate shifts determine the amount of "stretch" or "compression" that occurs for any value of \(y\) on overlapping x-values.
Scaling Functions
Scaling functions generally involves multiplying the x-variable by a constant factor. This manipulation changes the graph's appearance, particularly its width.
When considering functions like \(f(x)\) and its transformations \(g(x) = f(c x)\) where \(0 < c < 1\), scaling impacts how wide or narrow the parabola becomes.
If \(c\) is a fraction, it causes the graph to stretch horizontally, appearing wider. Conversely, if \(c\) was greater than 1 (not covered in this exercise), it would cause the graph to compress. By observing these changes, we understand that scaling functions provides control over graph dimensions, making it crucial for graphical representation and analysis.
When considering functions like \(f(x)\) and its transformations \(g(x) = f(c x)\) where \(0 < c < 1\), scaling impacts how wide or narrow the parabola becomes.
If \(c\) is a fraction, it causes the graph to stretch horizontally, appearing wider. Conversely, if \(c\) was greater than 1 (not covered in this exercise), it would cause the graph to compress. By observing these changes, we understand that scaling functions provides control over graph dimensions, making it crucial for graphical representation and analysis.
Parabolas
A parabola is a symmetric curve that represents quadratic functions like \(f(x) = x^2 + 1\). Its recognizable U-shape is crucial in both mathematics and engineering.
Parabolas are defined by a quadratic equation that determines their orientation, position, and shape. For instance, \(f(x) = x^2 + 1\) is a simple parabola centered at the y-axis, shifted upwards by one unit.
When transformations are applied, such as those observed in this exercise, the basic shape of a parabola is retained, but its width and position may change. The primary attributes of parabolas include:
These key features help identify how transformations affect the parabola's appearance, providing a handy tool for analysis.
Parabolas are defined by a quadratic equation that determines their orientation, position, and shape. For instance, \(f(x) = x^2 + 1\) is a simple parabola centered at the y-axis, shifted upwards by one unit.
When transformations are applied, such as those observed in this exercise, the basic shape of a parabola is retained, but its width and position may change. The primary attributes of parabolas include:
- Vertex: The highest or lowest point, depending on its orientation (open upwards or downwards).
- Axis of Symmetry: A vertical line that divides the parabola into mirror-image halves.
- Direction: Indicates whether the parabola opens upwards (minimum) or downwards (maximum).
These key features help identify how transformations affect the parabola's appearance, providing a handy tool for analysis.
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