EOC MOC Test

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What is a polynomial function?

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A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The general form is: $$f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$$ where $$a_n, a_{n-1}, ..., a_0$$ are constants and $$n$$ is a non-negative integer.

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What does it mean for a function to be decreasing on an interval?

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A function is decreasing on an interval if, for any two points $$x_1$$ and $$x_2$$ in that interval, where $$x_1 < x_2$$, the function value satisfies $$f(x_1) > f(x_2)$$.

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How do you find the intervals where a polynomial function is increasing or decreasing?

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To find the intervals where a polynomial function is increasing or decreasing, first find the derivative of the function. Then, determine the critical points by setting the derivative equal to zero and solving for $$x$$. Test the intervals between these critical points to see where the derivative is positive (increasing) or negative (decreasing).

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What is an inequality in mathematics?

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An inequality is a mathematical statement that compares two expressions and shows that one is greater than, less than, greater than or equal to, or less than or equal to the other. For example, $$2x - y > 6$$ is an inequality.

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What is the solution set of an equation?

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The solution set of an equation is the set of all values of the variable that make the equation true. For example, the solution set of $$x + 2 = 5$$ is $$\{3\}$$.

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What is a transformation in geometry?

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A transformation in geometry is an operation that moves or changes a shape in some way, such as translation (shifting), rotation (turning), reflection (flipping), or dilation (resizing).

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What is a vertical shift in transformations?

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A vertical shift is a transformation that moves a graph up or down on the coordinate plane. For example, the function $$f(x) + k$$ shifts the graph of $$f(x)$$ vertically by $$k$$ units.

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What is a rational exponent?

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A rational exponent is an exponent that is a fraction. For example, $$x^{\frac{m}{n}}$$ represents the nth root of $$x^m$$, which can also be written as $$\sqrt[n]{x^m}$$.

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How do you rewrite expressions with rational exponents?

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To rewrite expressions with rational exponents, use the property that $$x^{\frac{m}{n}} = \sqrt[n]{x^m}$$. For example, $$\left(\sqrt[3]{3x}\right)^3 = (3x)^{\frac{3}{3}} = 3x$$.

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What is the significance of critical points in calculus?

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Critical points are values of $$x$$ where the derivative of a function is zero or undefined. They are important for determining the local maxima and minima of the function.