Describe and correct the error in graphing and comparing f(x) = x² and g(x) = -0.5x².
Understand the Problem
The question asks us to describe and correct the error in graphing and comparing the functions f(x) = x² and g(x) = -0.5x². We need to analyze the provided graph and identify any discrepancies between the graphed functions and their actual mathematical representation.
Answer
The graph of $g$ is a vertical *compression* by a factor of 0.5 and a reflection in the x-axis of the graph of $f$, not a vertical stretch.
Answer for screen readers
The error in the description is that the graph of $g$ is described as a vertical stretch by a factor of 0.5. The correct description is: The graphs have the same vertex and axis of symmetry. The graph of $g$ is a vertical compression by a factor of 0.5 and a reflection in the x-axis of the graph of $f$.
Steps to Solve
- Analyze the given functions
The functions are $f(x) = x^2$ and $g(x) = -0.5x^2$. We can see that $g(x)$ is a transformation of $f(x)$. The transformation includes a vertical compression by a factor of 0.5 (because the coefficient is 0.5) and a reflection across the x-axis (because of the negative sign).
- Examine the properties of $f(x)$
$f(x) = x^2$ is a parabola opening upwards with its vertex at the origin (0, 0). It's a standard quadratic function.
- Examine the properties of $g(x)$
$g(x) = -0.5x^2$ is a parabola opening downwards (due to the negative sign) with its vertex at the origin (0, 0). It is vertically compressed compared to $f(x)$. For any given x-value, the y-value of $g(x)$ will be -0.5 times the y-value of $f(x)$.
- Identify the error in the description
The error in the original statement is that the graph of $g$ is described as a vertical stretch by a factor of 0.5. It should be described as a vertical compression by a factor of 0.5. A stretch would imply the coefficient is greater than 1, but here it is 0.5 which is less than 1.
- Correct the error
The correct description is: The graphs have the same vertex and axis of symmetry. The graph of $g$ is a vertical compression by a factor of 0.5 and a reflection in the x-axis of the graph of $f$.
The error in the description is that the graph of $g$ is described as a vertical stretch by a factor of 0.5. The correct description is: The graphs have the same vertex and axis of symmetry. The graph of $g$ is a vertical compression by a factor of 0.5 and a reflection in the x-axis of the graph of $f$.
More Information
Vertical stretch/compression factors are determined by the coefficient of the squared term. If the absolute value of the coefficient is greater than 1, it's a stretch. If it's between 0 and 1, it's a compression. A negative sign indicates a reflection across the x-axis.
Tips
A common mistake is confusing vertical stretches and compressions. Remembering that a coefficient with an absolute value less than 1 results in a compression and an absolute value greater than 1 results in a stretch helps to avoid this error.
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