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Questions and Answers
What are the coordinates of the vertex of a quadratic function with the equation $f(x) = 2x^2 - 7x + 3$?
In the quadratic function $f(x) = 3x^2 - 5x + 2$, what is the axis of symmetry?
What is the end behavior of a polynomial function if its leading coefficient is negative?
How do different roots or intercepts affect the behavior of a polynomial within its domain?
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If a polynomial has zeros at -4 and 5, what is the equation of the polynomial?
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What is the formula to find a polynomial function if its zeros are -2 and 7?
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What is the equation of the axis of symmetry for a quadratic function?
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If a quadratic function has the general form \(5x^2 + 3x - 2\), what is the x-coordinate of its vertex?
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When finding the y-coordinate of the vertex of a quadratic function, what formula is used?
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What is an end behavior of a polynomial function?
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When writing a formula for a polynomial function given its zeros, what form is commonly used?
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If you know the behavior at zeros of a polynomial function, what can you determine?
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Study Notes
Polynomial functions are mathematical expressions consisting of variables and coefficients raised to integer powers, combined using addition, subtraction, multiplication, division, and non-negative integer exponents. These functions play a central role in mathematics and its applications in various fields like physics, engineering, and economics. This article will delve into some aspects of polynomials, specifically the vertex and axis of symmetry of a quadratic function, finding a quadratic function given vertex and another point, end behaviors and behaviors at zeros of a polynomial function, writing formulas for polynomial functions given zeros, and finding formulas for polynomial functions given information about the function.
Vertex and Axis of Symmetry of a Quadratic Function
A parabola is symmetric with respect to the vertical line through its vertex. For a quadratic function, this means that the axis of symmetry is vertical and passes through the vertex. To find the equation of the axis of symmetry, we can use the general form of a quadratic function: (ax^2 + bx + c). Since it's symmetric with respect to the y-axis, the x-coordinate of the vertex is (-b / 2a). Therefore, if we set (x = -b / 2a), we get the y-coordinate of the vertex, which is ((-b)^2 / 4a) when (a \neq 0).
For example, consider the quadratic function (f(x) = 2x^2 + 7x + 1). In this case, (a = 2), (b = 7), and (c = 1). So, the x-coordinate of the vertex is (-\frac{b}{2a} = -\frac{7}{2 \times 2} = -\frac{7}{4}). Plugging this back into the equation gives the y-coordinate of the vertex, which is ((-\frac{7}{4})^2 \times 2 = \frac{49}{8}). Thus, the vertex of the graph lies at ((\frac{-7}{4}, \frac{49}{8})).
Finding a Quadratic Function Given Vertex and Another Point
To find a quadratic function given a vertex and another point ((-b/2a,c)), we first know the value of (a). By plugging the x-coordinate of the other point into the quadratic formula, we obtain a second point ((x_1,y_1)). Then, by solving the system of equations formed by the two points, we find the values of (b) and (c). Afterward, we can substitute these values back into the standard form of a quadratic function to have our desired quadratic function.
Consider the following example: Suppose we want to find a quadratic function whose vertex is ((-3,-1)) and another point is ((2,-2)). We already know that (a = 1) from the vertex. Let (p(2) = -2), so we need to solve the system of linear equations: (2(1) + b(-3) + c = -2) and (2^2 = 4). Solving this system, we find that (b = -2) and (c = -4). Substituting these values into the standard form of a quadratic function, we get (f(x) = x^2 - 2x - 4).
End Behaviors and Behaviors at Zeros of a Polynomial Function
The end behavior of a polynomial function depends on the leading coefficient term ((a)). If (a > 0), the overall trend is upwards and open; if (a < 0), the overall trend is downwards and closed. Additionally, different roots or intercepts may cause local extrema or inflection points within the interval where the domain exists.
As for the behavior at zeros of a polynomial function, it depends on the nature of the factor(s) in the polynomial. A positive factor will cause the curve to intersect the x-axis, while a negative factor will cause the curve to not intersect the x-axis.
Writing Formula for Polynomial Function Given Zeros
To write a polynomial function given its zeros, we need to find the coefficients of the polynomial. If the polynomial is a quadratic function, the equation can be written as (ax^2 + bx + c = 0). We can use the quadratic formula to find the value of (x) given the zeros of the polynomial: (x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}).
For example, if the zeros of a quadratic function are (2) and (3), we have (x^2 + 3x + 2 = 0). Plugging these values into the quadratic formula gives (\frac{-3 \pm \sqrt{9 - 4(2)(2)}}{2(1)} = \frac{-3 \pm \sqrt{-4}}{2} = x = \frac{-3}{2} \pm \sqrt{\frac{5}{4}}). Thus, the equation of the polynomial function is (f(x) = x^2 + 3x + 2 = 0).
Finding Formula for Polynomial Function Given Information About the Function
To find a polynomial function given certain conditions or information about the function, such as zeros, intercepts, and coefficient, we need to analyze the given information and determine the coefficients accordingly. For example, if we are given the vertex ((-3,-1)), (y)-intercept ((0,-2)), and root (1), we can set up the system of equations: (2(-3)^2 + (-1)(0) + c = -2), ((-3)^2 = 9), and (-1(1) + c = 0). Solving this system, we obtain the equation of the polynomial function: (f(x) = 2x^2 - 1).
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Description
Explore different aspects of polynomial functions including finding the vertex and axis of symmetry of a quadratic function, determining a quadratic function given a vertex and another point, analyzing end behaviors and behaviors at zeros of polynomial function, writing formulas for polynomial functions based on zeros, and finding formulas given specific information about the function.
