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Chapter 5: Problem 25

Graph the function and specify the domain, range, intercept(s), and asymptote. $$y=-2^{x}+1$$

Short Answer

Expert verified
Domain: all real numbers; Range: \(y < 1\); y-intercept: (0, 0); Asymptote: \(y = 1\).

Step by step solution

01

Identify the Function

The function given is \(y = -2^x + 1\). It is an exponential function with a negative base that is reflected over the x-axis and shifted upward by 1 unit.
02

Determine the Domain

The domain of an exponential function is all real numbers, \(x \in \mathbb{R}\), because there are no restrictions on the values that \(x\) can take.
03

Determine the Range

For the function \(y = -2^x + 1\), the transformation affects the range. Since the function is reflected and shifted up, the range is \(y < 1\).
04

Find the Intercept(s)

To find the y-intercept, set \(x = 0\): \[y = -2^0 + 1 = -1 + 1 = 0\]Thus, the y-intercept is \((0, 0)\). There is no x-intercept for this function because the function never crosses the x-axis beyond the y-intercept point.
05

Identify Asymptote

The horizontal asymptote is the horizontal line that the function approaches as \(x\to \infty\). Here, this line is \(y = 1\) due to the vertical shift.

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Key Concepts

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

Function Graphing
Graphing the function \(y = -2^x + 1\) involves understanding its transformations. The base of the function \(2^x\) indicates exponential growth for positive values but the negative sign in front indicates a reflection over the x-axis. This flips the curve upside down. Additionally, the constant \(+1\) indicates a vertical shift upwards by 1 unit. By plotting key points and understanding these transformations, you can sketch a graph representing the exponential decline from the positive side along the x-axis.When you start plotting, remember:
  • The graph curves steeply downwards.
  • The basic shape of exponential growth becomes decline due to the negative sign.
  • The curve will never touch or cross a horizontal line, where \(y = 1\).
Grasping these concepts will help you visually represent the function accurately.
Domain and Range
The domain of an exponential function like \(y = -2^x + 1\) is the set of all real numbers. This means no matter the value of \(x\), you will get a valid result when plugging it into the function.The range, however, is affected by the vertical transformation. Typically, for a basic exponential function \(2^x\), the range is \(y > 0\). Here, due to the transformation of reflection and vertical shift, the range changes. The function outputs values less than 1, thus, the range becomes \(y < 1\). This means the graph will stay below the horizontal asymptote, which is at \(y = 1\).
Intercepts
Intercepts are points where the graph crosses the axes. For \(y = -2^x + 1\), the intercepts can be calculated to identify where it intersects with the y-axis and x-axis.- **Y-intercept:** To find the y-intercept, set \(x = 0\): \(y = -2^0 + 1 = 0\) Thus, the y-intercept is at the point \((0, 0)\).- **X-intercept:** Usually, you would solve \(y = 0\) to find the x-intercept. But here, after calculation, you find that the function never crosses the x-axis once you get past the y-intercept. Hence there are no additional x-intercepts.Understanding intercepts helps in quickly identifying important points on the graph.
Asymptotes
An asymptote is a line the graph approaches but never touches. In the context of \(y = -2^x + 1\), we deal with a horizontal asymptote. For transformed exponential functions, this horizontal line often defines the boundary for the range.For this function, due to the vertical shift, the horizontal asymptote is \(y = 1\). This means that as \(x\) becomes increasingly positive, the value of \(y\) will get closer and closer to 1, but will never actually reach or exceed it.Understanding asymptotes is crucial as they define the behavior of the graph in extreme conditions, providing insights into the limits of the function.

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Most popular questions from this chapter

Solve the inequalities. Where appropriate, give an exact answer as well as a decimal approximation. $$6\left(5-1.6^{x}\right) \geq 13$$

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