Problem 77 In the following exercises, grap... [FREE SOLUTION]

Chapter 10: Problem 77

In the following exercises, graph each function in the same coordinate system. $$ f(x)=4^{x}, g(x)=4^{x-1} $$

Short Answer

Expert verified

Graph both functions, noting that $g(x) = 4^{x-1}$ is a horizontal shift of $f(x) = 4^x$ to the right by 1 unit.

Step by step solution

Understand the Functions

Identify the functions that need to be graphed: $f(x) = 4^x$ and $g(x) = 4^{x-1}$. Notice that the second function is a horizontal shift of the first function.

Determine Key Points for f(x)

Calculate key points for $f(x) = 4^x$ by substituting different values of $x$. For example:$f(0) = 4^0 = 1$$f(1) = 4^1 = 4$$f(2) = 4^2 = 16$Mark these points: (0, 1), (1, 4), (2, 16).

Determine Key Points for g(x)

Calculate key points for $g(x) = 4^{x-1}$ by substituting different values of $x$. For example:$g(0) = 4^{-1} = \frac{1}{4}$$g(1) = 4^{0} = 1$$g(2) = 4^{1} = 4$Mark these points: (0, 0.25), (1, 1), (2, 4).

Plot the Points

On a graph, plot the points determined in Steps 2 and 3. For $f(x)$: (0, 1), (1, 4), (2, 16); and for $g(x)$: (0, 0.25), (1, 1), (2, 4).

Draw the Graphs

Draw smooth curves through the points plotted for $f(x) = 4^x$ and $g(x) = 4^{x-1}$ to complete the graphs. Ensure both curves are exponential growth functions.

Compare the Graphs

Observe that $g(x) = 4^{x-1}$ is a horizontal shift of $f(x) = 4^x$ to the right by 1 unit. Both functions should show exponential growth but start at different initial points.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Growth

Exponential growth refers to an increase that occurs at a consistently growing rate. In mathematical terms, an exponential growth function can be described by the equation $f(x) = a \cdot b^x$, where $a > 0$ and $b > 1$. As we increase the value of $x$, the function value $f(x)$ grows rapidly. For example, in the function $f(x) = 4^x$:

When $x = 0$, $f(0) = 1$.
When $x = 1$, $f(1) = 4$.

This rapid growth means we quickly move to much larger values. Key features of exponential growth include:

A rapid increase in value.
A consistent rate of growth reflected by the base value.
An initial value often derived from the constant term $a$ when $x = 0$.

Function Transformation

To transform a function means to change its position or shape on a graph. Common transformations include translations (shifts), reflections, stretching, and compressions. Considering the functions $f(x) = 4^x$ and $g(x) = 4^{x-1}$, we observe a horizontal shift: $g(x)$ moves $f(x)$ one unit to the right. Mathematically, the new function $g(x) = f(x - 1)$.Transformation types:

Vertical Shift: $f(x) + c$.
Horizontal Shift: $f(x + c)$.
Reflection: $-f(x)$ or $f(-x)$.
Stretching/Compressing: Applying a coefficient to $x$ or $f(x)$.

In our example, the horizontal shift demonstrates how the graph is moved without altering the function's shape.

Horizontal Shift

A horizontal shift moves the entire graph of a function left or right. For a function $f(x)$, replacing $x$ with $x - c$ results in a shift of $c$ units to the right. Conversely, $f(x + c)$ shifts the graph $c$ units to the left.Exploring $f(x) = 4^x$ and $g(x) = 4^{x-1}$ illustrates this concept:

The graph of $f(x)$ shifts 1 unit to the right to form $g(x)$.
The shift doesn鈥檛 change the exponential growth nature of the graph.
Key points on $f(x)$ move to new positions on $g(x)$: $(0, 1)$ becomes $(1, 1)$, and $(1, 4)$ becomes $(2, 4)$.

Horizontal shifts are straightforward yet essential for understanding how functions translate along the $x$-axis.

Key Points on a Graph

Identifying key points on a graph helps us understand function behavior. For exponential functions, calculating points for various $x$-values aids in sketching the graph accurately.For $f(x) = 4^x$:

$f(0) = 1$ at (0, 1).
$f(1) = 4$ at (1, 4).
$f(2) = 16$ at (2, 16).

For $g(x) = 4^{x-1}$:

$g(0) = \frac{1}{4}$ at (0, 0.25).
$g(1) = 1$ at (1, 1).
$g(2) = 4$ at (2, 4).

Graphing these points enables us to visualize their exponential growth clearly. The points serve as anchors to draw precise curves, reflecting each function's nature and transformations.

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91影视

In the following exercises, graph each function in the same coordinate system. $$ f(x)=4^{x}, g(x)=4^{x-1} $$

Short Answer

Step by step solution

Understand the Functions

Determine Key Points for f(x)

Determine Key Points for g(x)

Plot the Points

Draw the Graphs

Compare the Graphs

Key Concepts

Exponential Growth

Function Transformation

Horizontal Shift

Key Points on a Graph

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