The graph of [tex]y=-0.2 x^2[/tex] is the graph of [tex]y=x^2[/tex].
Answer (2)
The graph of y = − 0.2 x 2 is obtained from the graph of y = x 2 by a vertical compression by a factor of 0.2 and a reflection about the x-axis.
Explanation
Understanding the Problem We are asked to describe how the graph of y = − 0.2 x 2 is obtained from the graph of y = x 2 . This involves understanding transformations of functions, specifically vertical stretches/compressions and reflections.
General Transformation The general transformation from y = x 2 to y = a x 2 involves a vertical stretch or compression by a factor of ∣ a ∣ . If 1"> ∣ a ∣ > 1 , it's a vertical stretch. If 0 < ∣ a ∣ < 1 , it's a vertical compression. If a < 0 , there's also a reflection about the x-axis.
Specific Transformation In our case, a = − 0.2 . Since ∣ − 0.2∣ = 0.2 and 0 < 0.2 < 1 , there is a vertical compression by a factor of 0.2. Also, since a = − 0.2 < 0 , there is a reflection about the x-axis.
Examples
Understanding transformations of functions is crucial in various fields. For example, in physics, the trajectory of a projectile under gravity can be modeled by a quadratic function. Changing the parameters of the function (like the coefficient of the x 2 term) can help us understand how the trajectory changes with different initial conditions or gravitational forces. Similarly, in engineering, understanding how transformations affect the shape of a curve is essential in designing structures and optimizing their performance. For instance, the shape of a bridge or an arch can be modeled using mathematical functions, and transformations can help engineers adjust the design to meet specific requirements.
The graph of y = − 0.2 x 2 is derived from y = x 2 through a vertical compression by a factor of 0.2 and a reflection across the x-axis. This means the new graph opens downwards and is flatter than the original. Understanding these transformations is key in analyzing quadratic functions.
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