%5E2%2F10-2x.jpeg "(x-5)^2/10-2x")
Simplifying and Analyzing the Expression (x-5)^2/10 - 2x
This article will explore the expression (x-5)^2/10 - 2x, analyzing its properties, simplifying it, and examining its graph.
Expanding and Simplifying
The first step is to expand the expression and simplify it:
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Expand the square: (x-5)^2 = (x-5)(x-5) = x^2 - 10x + 25
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Substitute back into the original expression: (x^2 - 10x + 25)/10 - 2x
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Distribute the 1/10: (x^2/10) - x + 2.5 - 2x
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Combine like terms: (x^2/10) - 3x + 2.5
The simplified expression is (x^2/10) - 3x + 2.5.
Analyzing the Expression
The expression is a quadratic function because the highest power of x is 2. This means the graph of the function will be a parabola.
Key Features
- Leading Coefficient: The leading coefficient is 1/10, which is positive. This indicates that the parabola will open upwards.
- Vertex: The vertex of the parabola represents the minimum value of the function. We can find the x-coordinate of the vertex using the formula: x = -b / 2a, where a = 1/10 and b = -3. Therefore, x = 15. We can substitute this value back into the expression to find the y-coordinate of the vertex.
- Y-intercept: The y-intercept is the point where the graph intersects the y-axis. We can find it by setting x = 0 in the simplified expression. This gives us a y-intercept of 2.5.
- Roots: The roots are the points where the graph intersects the x-axis. We can find them by setting the expression equal to zero and solving for x. This will involve solving a quadratic equation.
Graphing the Function
By plotting the vertex, the y-intercept, and finding the roots (if applicable), we can sketch the graph of the function. The shape of the parabola, opening upwards and its position based on the vertex and intercepts, will illustrate the behavior of the function.
Conclusion
By simplifying the expression and analyzing its key features, we gain a deeper understanding of the function (x-5)^2/10 - 2x. This knowledge allows us to predict its behavior, visualize its graph, and determine its important values, such as its minimum and roots.