Describe and correct the error in graphing and comparing f(x) = x² and g(x) = -0.5x².
Understand the Problem
The question presents two functions, f(x) = x² and g(x) = -0.5x², along with their graphs. It asks to identify and correct the error in graphing and comparing these two functions. The provided text makes a statement about the relationship between the graphs, which we need to evaluate for correctness.
Answer
The graph of $g(x)$ is a vertical compression by a factor of 0.5 and a reflection in the x-axis of the graph of $f(x)$.
Answer for screen readers
The error is stating that the graph of $g(x)$ is a vertical stretch of the graph of $f(x)$. The graph of $g(x)$ is a vertical compression by a factor of 0.5 and a reflection in the x-axis of the graph of $f(x)$.
Steps to Solve
- Identify the Vertex and Axis of Symmetry
Both $f(x) = x^2$ and $g(x) = -0.5x^2$ are parabolas with a vertex at the origin (0, 0). The axis of symmetry for both is the y-axis (x = 0). So, the statement "The graphs have the same vertex and the same axis of symmetry" is correct.
- Analyze the Transformation
The function $g(x) = -0.5x^2$ is a transformation of $f(x) = x^2$. The -0.5 affects the graph in two ways: -The negative sign indicates a reflection in the x-axis. -The 0.5 indicates a vertical compression (or shrink) by a factor of 0.5, since $0 < 0.5 < 1$. The original statement incorrectly identifies this as a vertical stretch
- Correct the Error
The error lies in stating that the graph of 'g' is a vertical stretch. It is actually a vertical compression (or shrink) by a factor of 0.5 and a reflection in the x-axis of the graph of 'f'.
The error is stating that the graph of $g(x)$ is a vertical stretch of the graph of $f(x)$. The graph of $g(x)$ is a vertical compression by a factor of 0.5 and a reflection in the x-axis of the graph of $f(x)$.
More Information
Vertical stretch/compression occurs when a function is multiplied by a factor. If the value is greater than 1, it's a vertical stretch. If the value is between 0 and 1, it's vertical compression. A negative sign indicates reflection across the x-axis.
Tips
A common mistake is confusing vertical stretches and compressions. A factor greater than 1 stretches the graph, while a factor between 0 and 1 compresses it. Also, forgetting the effect of the negative sign, which reflects the graph across the x-axis, is another common mistake.
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