Quadratic equations form the cornerstone of algebra, characterized by their parabolic graphs, axis of symmetry, and intercepts. Understanding their properties, such as the discriminant's role in determining real or complex solutions, is crucial. Techniques like factoring, completing the square, and the Quadratic Formula are pivotal for solving these equations, while graphical transformations aid in visualizing their behavior.
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Quadratic equations are algebraic expressions in the form y = ax^2 + bx + c, where 'a' is nonzero, and graph as parabolas
Axis of Symmetry
The axis of symmetry is a vertical line that divides the parabola into two congruent halves and can be found using the formula x = -b/(2a)
Vertex
The vertex is the highest or lowest point on the parabola, lies on the axis of symmetry, and can be a maximum or minimum point of the function
Intercepts
The y-intercept is where the parabola crosses the y-axis, and the x-intercepts are the points where the parabola intersects the x-axis, representing the roots of the quadratic equation
The direction in which the parabola opens is determined by the sign of 'a': positive 'a' results in an upward opening parabola, while negative 'a' leads to a downward opening parabola
Factoring is a method used to solve quadratic equations by expressing them in the form y = a(x - p)(x - q), where 'p' and 'q' are the roots
The Zero Product Property states that if a product equals zero, then at least one of the factors must be zero, and it is applied to the factored equation to find the roots
The Quadratic Formula, x = (-b ± √(b^2 - 4ac)) / (2a), is a universal method for finding the roots of any quadratic equation, including those that are not factorable or have irrational or complex solutions
The Location Principle states that if the y-values of a function change sign between two consecutive x-values, a root lies between them, and it is useful for approximating roots when they cannot be calculated exactly
Adjusting 'a'
Changing the coefficient 'a' alters the parabola's width, resulting in a narrower or wider graph
Adjusting 'h'
Modifying 'h' translates the graph horizontally, shifting it left or right
Adjusting 'k'
Changing 'k' shifts the graph vertically, moving it up or down
The Square Root Property and Completing the Square are techniques used to solve quadratic equations by finding the square root of a perfect square trinomial or transforming the equation into a perfect square trinomial, respectively
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Quadratic equation standard form
y = ax^2 + bx + c, where a ≠ 0.
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Axis of symmetry formula
x = -b/(2a), vertical line dividing parabola.
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Finding y-intercept in quadratics
Evaluate at x = 0, gives parabola's y-axis intersection.
