Parabolic Function: Graph of y=ax^n

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SUMMARY

The graph of the function y=ax^n can be derived from the graph of y=x^n by adjusting the coefficient 'a'. When 'a' is positive, increasing its value results in a narrower graph, while decreasing 'a' leads to a wider graph. For instance, the comparison between y=x^2 and y=5x^2 illustrates this effect, with the latter appearing "skinnier." When 'a' is negative, the graph is reflected about the x-axis, as seen in the example of y=-x^2, which retains the shape of y=x^2 but points downward.

PREREQUISITES
  • Understanding of polynomial functions
  • Familiarity with graph transformations
  • Knowledge of the effects of coefficients on function shapes
  • Basic algebra skills
NEXT STEPS
  • Explore the impact of different coefficients on cubic functions, such as y=ax^3
  • Investigate graph reflections and their mathematical implications
  • Learn about the vertex form of parabolas and how it relates to transformations
  • Study the concept of asymptotes in relation to polynomial functions
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Students in mathematics, educators teaching algebra, and anyone interested in understanding the graphical behavior of polynomial functions.

batballbat
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how can the graph of the function y=ax^n be obtained from the graph of y=x^n if a is positive?negative
 
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If you graph two functions, y=x^2 and y=5x^2 you will notice the first graph appears "fatter" and the second graph is "skinnier".
As you increase the coefficient a the more narrow the graph becomes.
As you decrease the coefficient a the wider the graph becomes.
I would experiment with this concept until you notice a mathematical pattern by which you can predict the graph's appearance at any a.
 
When a is negative, the graph is reflected about the x-axis. For example, the parabola y=-x^2 has the same shape as y=x^2, but is pointing down instead of up.
 

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