Problem 12
Question
Use a graphing utility to graph \(f(x)=2 / x\) and the function \(g\) in the same viewing window. Describe the relationship between the two graphs. $$g(x)=\frac{1}{2} f(x+2)$$
Step-by-Step Solution
Verified Answer
The function \(g(x)\) is derived from \(f(x)\) by shifting the graph 2 units to the left and vertically stretching its local maximums and minimums. This can be seen as the graph of \(g(x)\) being closer to the axis than \(f(x)\). Thus, \(f(x)\) and \(g(x)\) have the same basic shape, but \(g(x)\) appears compressed (closer to the x-axis) and shifted left compared to \(f(x)\).
1Step 1: Identify Functions
Identify function \(f(x)=2/x\), and from it, derive the function \(g(x) = 1/2* f(x+2) = 1/(x+2)\) by replacing \(x\) in \(f(x)\) with \((x+2)\) and halving the result.
2Step 2: Graph the Original Function
Graph the function \(f(x)=2/x\) using a graphing utility. Observing the graph, this function will present a hyperbola shape with center at the origin. Keep in mind to use a varying x-value to plot the graph correctly.
3Step 3: Graph the Derived Function
Graph the derived function \(g(x) = 1/(x+2)\) in the same window as \(f(x)\). The resulting graph should present a similar hyperbola shape, but with center shifted 2 units to the left (due to the \(x+2\)) and stretched vertically (due to the factor \(1/2\)).
4Step 4: Describe Relationship Between Graphs
The graph of \(g(x)\) is a horizontally translated and vertically scaled version of the graph of \(f(x)\). The translation is 2 units to the left, and the scale factor is 1/2, which makes the graph of \(g(x)\) closer to the x-axis compared to the graph of \(f(x)\).
Key Concepts
Function TransformationHyperbolaGraph Translation
Function Transformation
The concept of function transformation is integral in understanding how altering a function changes its graph. Transformations can include translations (shifting the graph left, right, up, or down), stretches, compressions, and reflections. These modifications are applied to the basic form of a function, often revealing a deeper understanding of its behavior and properties.
For the function transformation from \(f(x) = \frac{2}{x}\) to \(g(x) = \frac{1}{2}f(x+2)\), a couple of transformations are at play. Firstly, recognizing \(f\) is transformed by replacing \(x\) with \(x+2\), this operation shifts the graph horizontally by 2 units to the left. Additionally, multiplying by \(\frac{1}{2}\) results in a vertical compression, making the graph flatten closer to the x-axis. Understanding these transformations helps in graphing functions accurately without actually plotting every point.
For the function transformation from \(f(x) = \frac{2}{x}\) to \(g(x) = \frac{1}{2}f(x+2)\), a couple of transformations are at play. Firstly, recognizing \(f\) is transformed by replacing \(x\) with \(x+2\), this operation shifts the graph horizontally by 2 units to the left. Additionally, multiplying by \(\frac{1}{2}\) results in a vertical compression, making the graph flatten closer to the x-axis. Understanding these transformations helps in graphing functions accurately without actually plotting every point.
Hyperbola
A hyperbola is a type of curve on a graph that can be formed by intersecting a double cone with a plane. It consists of two separate branches which appear as mirror images of each other and is defined by its asymptotes, which are lines the branches approach but never actually intersect.
For the given function, \(f(x) = \frac{2}{x}\), it forms a hyperbola with the x and y-axes serving as asymptotes. This is because as \(x\) approaches 0, the function values ( or the y-values) grow infinitely large and negative. Similarly, as \(x\) tends toward infinity or negative infinity, the function values approach 0, never quite touching the x-axis.
For the given function, \(f(x) = \frac{2}{x}\), it forms a hyperbola with the x and y-axes serving as asymptotes. This is because as \(x\) approaches 0, the function values ( or the y-values) grow infinitely large and negative. Similarly, as \(x\) tends toward infinity or negative infinity, the function values approach 0, never quite touching the x-axis.
- A hyperbola has a simple symmetrical shape, centered around the origin.
- The center of our specific hyperbola, \(f(x)\), is at the origin (0,0).
Graph Translation
Graph translation is a fundamental transformation involving moving the entire graph of a function without altering its shape or orientation. This operation is achieved through modifying the input or output values. For instance, adding a constant inside the function changes the x-values, shifting the graph left or right, depending on the sign.
In the function \(g(x) = \frac{1}{2}f(x+2)\), the term \(x+2\) suggests a horizontal shift. Specifically, it results in a leftward shift of 2 units. Such translations help in predicting where a graph will be relocated on the coordinate plane without recalculating every point.
In the function \(g(x) = \frac{1}{2}f(x+2)\), the term \(x+2\) suggests a horizontal shift. Specifically, it results in a leftward shift of 2 units. Such translations help in predicting where a graph will be relocated on the coordinate plane without recalculating every point.
- Horizontal translations are determined by constants added or subtracted within the function argument (inside the parentheses).
- Vertical translations are determined by constants added or subtracted to the function itself (outside the parentheses).
Other exercises in this chapter
Problem 11
Sketch the graph of the function and compare it with the graph of \(y=x^{2}\) \(y=(x+3)^{2}\)
View solution Problem 12
(a) use a graphing utility to create a scatter plot of the data, (b) determine whether the data could be better modeled by a linear model or a quadratic model,
View solution Problem 12
Confirm that the function has the indicated zeros. $$f(x)=2 x^{5}-2 x ; 0,1,-1-i, i$$
View solution Problem 12
Use long division to divide and use the result to factor the dividend completely. $$\left(2 x^{3}-3 x^{2}-50 x+75\right) \div(2 x-3)$$
View solution