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Chapter 8: Problem 1

Graph each function. Label the asymptote of each graph. $$ y=-5^{x} $$

Short Answer

Expert verified
The graph of the function \(y = -5^{x}\) is a decreasing curve that reflects the graph of \(y = 5^{x}\) about the x-axis. The horizontal asymptote of the graph is \(y = 0\).

Step by step solution

01

Identify the base and exponent

The base of the exponential function is -5 and the exponent is \(x\). The negative sign indicates a reflection in the x-axis.
02

Identify the asymptote

For the function \(y=-5^{x}\), the asymptote is the x-axis, which is \(y=0\).
03

Plot the points and sketch the graph

First, plot a few points for some values of \(x\). For example, when \(x=0\), \(y=-5^{0}=-1\). When \(x=1\), \(y=-5^{1}=-5\). When \(x=-1\), \(y=-5^{-1}=-0.2\). Connect these points and sketch the rest of the graph. It should start from the top on the left (approaching but never reaching the x-axis), cross the y-axis at \(-1\), and then keep decreasing towards negative infinity.
04

Label the asymptote

The horizontal asymptote, \(y=0\), should be labeled on the graph.

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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 an essential skill in mathematics. It helps us understand and visualize the behavior of different kinds of mathematical equations. In the case of exponential functions like \(y = -5^{x}\), the process involves:
  • Identifying the base and the exponent
  • Determining any transformations such as reflections
  • Plotting points on a coordinate plane
  • Sketching the curve based on these points
The function \(y = -5^{x}\) is an exponential function with a base \(-5\), and the transformation caused by the negative sign affects how we sketch the graph.
Asymptotes
Asymptotes are lines that a graph approaches but never actually touches. They act as boundaries for the behavior of a function as it heads towards very large or very small inputs. For the exponential function \(y = -5^{x}\), a horizontal asymptote exists at \(y = 0\). This means:
  • No matter how large or small the value of \(x\) gets, \(y\) will approach \(0\) but never touch it.
  • This behavior is crucial for understanding the long-term trends of the graph.
When plotting \(y = -5^{x}\), you'll see the curve getting closer to the \(x\)-axis (which is actually \(y = 0\)) from the negative side, but it will never actually intersect that line.
Reflections
Reflections in graphing refer to flipping a graph over a particular line. For \(y = -5^{x}\), the negative sign in front of the base of the function results in a reflection over the \(x\)-axis. This alteration leads to a curve that:
  • Originally it might grow upwards; due to the reflection, it moves downward.
  • The negative output values suggest that the graph takes a direction opposite to what it would normally follow without the negative sign.
Visualizing this helps in sketching and understanding the actual graph of the function. You can better predict where the graph is headed by recognizing these reflections.
Sketching Graphs
Sketching graphs involves using all the elements we have identified to draw a visual representation of the function. For \(y = -5^{x}\), consider:
  • Starting by plotting calculated points such as \(x = 0, 1, -1\)
  • Noting how the graph starts far above and dips below each subsequent value of \(x\)
  • Connecting the dots with a smooth curve, keeping in mind the approach towards the asymptote \(y = 0\)
Remember that sketching is about visualizing how all the pieces come together. With \(y = -5^{x}\), it's crucial to illustrate how the graph plummets further negative, near the origin, then continues to decrease endlessly.

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