/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 9 Sketch the graph of the function... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 9

Sketch the graph of the function by making a table of values. Use a calculator if necessary. $$f(x)=2^{x}$$

Short Answer

Expert verified
The graph of \( f(x) = 2^x \) is an increasing exponential curve.

Step by step solution

01

Understand the Function

The given function is an exponential function of the form \( f(x) = 2^x \). Exponential functions have a constant base raised to the power of a variable exponent.
02

Choose Values for x

Select a range of \( x \) values to evaluate the function. For this exercise, use \( x = -2, -1, 0, 1, 2, 3 \). These values will help us observe how the function behaves as \( x \) changes.
03

Create a Table of Values

Calculate \( f(x) \) for each chosen \( x \) value:- If \( x = -2 \), then \( f(-2) = 2^{-2} = \frac{1}{4} = 0.25 \).- If \( x = -1 \), then \( f(-1) = 2^{-1} = \frac{1}{2} = 0.5 \).- If \( x = 0 \), then \( f(0) = 2^0 = 1 \).- If \( x = 1 \), then \( f(1) = 2^1 = 2 \).- If \( x = 2 \), then \( f(2) = 2^2 = 4 \).- If \( x = 3 \), then \( f(3) = 2^3 = 8 \).
04

Plot the Points on a Graph

Using the table of values, plot the corresponding points on a coordinate grid: \((-2, 0.25), (-1, 0.5), (0, 1), (1, 2), (2, 4), (3, 8)\).
05

Draw the Graph

Connect the plotted points smoothly to illustrate the continuous curve of the exponential function. The graph should show a rapid increase as \( x \) becomes positive and approach zero as \( x \) decreases.

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Key Concepts

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

Graphing Functions
Graphing functions is a fundamental skill in understanding mathematical concepts. It involves plotting points on a graph to visualize the relationship between variables, such as the input \( x \) and the output \( f(x) \) for a function. When graphing \( f(x) = 2^x \), you start by creating a table of values, which are pairs of \( x \) and \( f(x) \).
  • Select a set of \( x \) values, such as -2, -1, 0, 1, 2, and 3.
  • Compute \( f(x) \) for each \( x \). For instance, when \( x = 1 \), \( f(1) = 2^1 = 2 \).
  • Plot these points on a graph to create a visual representation.
  • Use these points to draw a smooth curve.
A graph is a useful way to illustrate how values change and relate to one another, making functions easier to understand and apply. By observing the curve, one can easily see the behavior of the function, such as rapid increases and decreases.
Exponential Growth
Exponential growth is a fascinating pattern observed in exponential functions where values increase at an increasing rate. The function \( f(x) = 2^x \) is a classic example of this growth.
Exponential growth differs from linear growth because it is multiplicative rather than additive. Here, the growth rate itself magnifies over time, resulting in a steep curve on a graph.
  • At each step, the output \( f(x) \) is the base (2 in this case) raised to the power of \( x \).
  • As \( x \) becomes larger, \( f(x) \) grows rapidly.
Such functions model many real-world scenarios, like population growth, where quantities double over consistent intervals. This pattern is why understanding and being able to graph exponential functions is crucial in various scientific and economic contexts.
Coordinate Systems
Coordinate systems provide a framework for plotting points and graphs on a grid. In two dimensions, this involves the \( x \)-axis (horizontal) and \( y \)-axis (vertical), intersecting at what鈥檚 known as the origin (0,0).
To plot the function \( f(x) = 2^x \):
  • Identify the point where each pair of values \( (x, f(x)) \) meets on the grid.
  • Each \( x \)-value has a corresponding \( y \)-value, giving a precise location on the plane.
  • For instance, when \( x = -1 \), plot the point at \( (x, 0.5) \).
The intersection of the axes lets us divide the plane into four quadrants, allowing positive and negative plotting of coordinates. This system is crucial for accurately representing the relationships in mathematical functions, providing a clear visual map of their behavior.

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Most popular questions from this chapter

Estimating a Solution Without actually solving the equation, find two whole numbers between which the solution of \(9^{x}=20\) must lie. Do the same for \(9^{x}=100 .\) Explain how you reached your conclusions.

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Find the inverse function of \(f\). $$f(x)=\log _{2}(x-1)$$

Find the domain of the function. $$f(x)=\log _{5}(8-2 x)$$

Use a graphing device to find all solutions of the equation, rounded to two decimal places. $$\log x=x^{2}-2$$
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