Problem Analysis:
The task asks us to determine which of the given functions has a graph that is symmetric with respect to the origin. Symmetry with respect to the origin means that if a point \((x, y)\) lies on the graph of the function, then the point \((-x, -y)\) must also lie on the graph. Mathematically, this property is expressed as:
\[
f(-x) = -f(x)
\]
for all \(x\) in the domain of \(f\). Functions that satisfy this property are called
odd functions.
We are provided with four graphs labeled (a), (b), (c), and (d). Our goal is to identify which graph exhibits symmetry with respect to the origin.
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Step-by-Step Solution:
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Graph (a):
- The graph appears to be a sinusoidal wave (similar to the sine function).
- Sinusoidal waves like \(y = \sin(x)\) are odd functions because \(\sin(-x) = -\sin(x)\).
- Visually, the graph is symmetric with respect to the origin: for every point \((x, y)\), there is a corresponding point \((-x, -y)\).
-
Conclusion for (a): This graph is symmetric with respect to the origin.
####
Graph (b):
- The graph resembles a cubic function (e.g., \(y = x^3\)).
- Cubic functions like \(y = x^3\) are odd functions because \((-x)^3 = -x^3\).
- Visually, the graph is symmetric with respect to the origin: for every point \((x, y)\), there is a corresponding point \((-x, -y)\).
-
Conclusion for (b): This graph is symmetric with respect to the origin.
####
Graph (c):
- The graph resembles an exponential decay function (e.g., \(y = e^{-x}\)).
- Exponential functions like \(y = e^{-x}\) are not odd functions because \(e^{-(-x)} = e^x \neq -e^{-x}\).
- Visually, the graph does not exhibit symmetry with respect to the origin: for a point \((x, y)\), the point \((-x, -y)\) does not lie on the graph.
-
Conclusion for (c): This graph is
not symmetric with respect to the origin.
####
Graph (d):
- The graph resembles a hyperbola (e.g., \(y = \frac{1}{x}\)).
- Hyperbolas like \(y = \frac{1}{x}\) are odd functions because \(\frac{1}{-x} = -\frac{1}{x}\).
- Visually, the graph is symmetric with respect to the origin: for every point \((x, y)\), there is a corresponding point \((-x, -y)\).
-
Conclusion for (d): This graph is symmetric with respect to the origin.
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Final Answer:
From the analysis, the graphs that are symmetric with respect to the origin are (a), (b), and (d). However, since the question asks for
one correct answer, we select the most straightforward or typical example of an odd function.
The most common and easily recognizable odd function is the cubic function (graph (b)).
Thus, the final answer is:
\[
\boxed{b}
\]
Parent Tip: Review the logic above to help your child master the concept of end behavior worksheet.
