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Chapter 5: Problem 15

In Exercises \(15-20\) , sketch the graph of the function. $$y=2^{x}$$

Short Answer

Expert verified
The graph of the function \(y=2^{x}\) crosses the y-axis at (0,1), increases exponentially as it goes to the right and approaches the x-axis as it goes to the left.

Step by step solution

01

Plot the y-intercept

The y-intercept is the point where the graph crosses the y-axis. For any exponential function, this will occur when x equals 0. For this function \(y=2^{x}\), \(y=2^0=1\). So the y-intercept of the function is (0, 1).
02

Identify the direction of the graph as x increases

As \(x\) increases in \(y=2^{x}\), \(y\), or the output of the function, increases exponentially. That is, for every unit increase in \(x\), \(y\) doubles. This means the graph will be increasing as it goes to the right.
03

Identify the direction of the graph as x decreases

As \(x\) decreases in \(y=2^{x}\), \(y\), or the output of the function, decreases and approaches 0 but never reaches it. This means the graph will be getting closer and closer to the x-axis (y = 0) as it goes to the left but will never touch or cross it.
04

Draw the sketch of the graph

Having identified the y-intercept and the direction of the graph as x increases or decreases, draw a smooth curve to represent the function. The curve should cross the y-axis at (0,1) and grow exponentially as it goes to the right and approach y = 0 as it goes to the left.

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Key Concepts

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

Graphing Techniques
Graphing exponential functions can initially seem challenging, but with the right techniques, it becomes a lot easier. An exponential function like \( y = 2^x \) is characterized by its consistent rapid growth as you move from left to right on the graph.
Here are some graphing techniques you can use:
  • Start by identifying key points, especially the y-intercept, which is crucial for getting the graph's shape right.
  • Consider how the function behaves as \( x \) values become very large or very small. This helps in understanding the general direction the graph takes.
  • Use a smooth curve to connect your plotted points. Exponential graphs are continuous and never have breaks or sharp turns.
Once you've identified your points and directions, sketch the curve on your graph grid, making sure it gets steeper as it moves to the right.
Y-intercept
The y-intercept is a fundamental component of graphing any function. For the exponential function \( y = 2^x \), the y-intercept shows us exactly where the graph starts when \( x = 0 \).
To determine the y-intercept:
  • Set \( x = 0 \) in the function, which gives \( y = 2^0 \).
  • Calculate \( 2^0 \), which equals 1. Therefore, the y-intercept is the point \( (0, 1) \).
This point gives you a fixed starting position for graphing the function, and it's an essential anchor for drawing the curve accurately. Without the y-intercept, it can be challenging to properly initiate the graph on the graph paper.
Exponential Growth
Understanding exponential growth is key to mastering the concept of exponential functions like \( y = 2^x \). Unlike linear growth, where increases are consistent, exponential growth involves the growth rate itself increasing.
Key characteristics of exponential growth include:
  • For every unit increase in \( x \), the value of \( y \) doubles. This rapid increase is why exponential graphs rise so steeply compared to linear graphs.
  • As \( x \) becomes negative, \( y \) values diminish but never reach zero. This creates a horizontal asymptote at the x-axis, indicating the graph's approach but never touching the axis.
When you see a problem involving exponential growth, expect the graph to start relatively flat for negative \( x \) values and then rise sharply as \( x \) becomes positive.

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Most popular questions from this chapter

(a) Show that $$\int_{0}^{1} \frac{4}{1+x^{2}} d x=\pi$$ (b) Approximate the number \(\pi\) by using the integration capabilities of a graphing utility.

(a) Show that \(\left(2^{3}\right)^{2} \neq 2^{(3)}\) . (b Are \(f(x)=\left(x^{x}\right)^{x}\) and \(g(x)=x^{\left(x^{\prime}\right)}\) the same function? Why or why not? (c) Find \(f^{\prime}(x)\) and \(g^{\prime}(x)\)

Calculus History InL'Hopital's 1696 calculus textbook, he illustrated his rule using the limit of the function $$f(x)=\frac{\sqrt{2 a^{3} x-x^{4}}-a \sqrt[3]{a^{2} x}}{a-\sqrt[4]{a x^{3}}}$$ as \(x\) approaches \(a, a>0 .\) Find this limit.

Finding an Equation of a Tangent Line In Exercises \(61-64,\) find an equation of the tangent line to the graph of the function at the given point. $$y=\log _{3} x, \quad(27,3)$$

In Exercises 55 and 56, a differential equation, a point, and a slope field are given. (a) Sketch two approximate solutions of the differential equation on the slope field, one of which passes through the given point. (To print an enlarged copy of the graph, go to MathGraphs.com.) (b) Use integration and the given point to find the particular solution of the differential equation and use a graphing utility to graph the solution. Compare the result with the sketch in part (a) that passes through the given point. $$\frac{d y}{d x}=\frac{2}{9+x^{2}}, \quad(0,2)$$
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