A quadratic function is a polynomial function of degree two, which can be expressed in the standard form as $$f(x) = ax^2 + bx + c$$, where $$a$$, $$b$$, and $$c$$ are constants and $$a \neq 0$$. This type of function graphs as a parabola, which opens either upward or downward depending on the sign of the leading coefficient $$a$$. Understanding quadratic functions is crucial for grasping concepts like composition and inverses, analyzing polynomial graphs, and modeling real-world scenarios.
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The vertex form of a quadratic function is given by $$f(x) = a(x-h)^2 + k$$, where (h, k) is the vertex of the parabola.
The axis of symmetry for a quadratic function can be found using the formula $$x = -\frac{b}{2a}$$.
Quadratic functions can have zero, one, or two real roots depending on the value of the discriminant, which is computed as $$D = b^2 - 4ac$$.
Quadratic equations can be solved using various methods including factoring, completing the square, or applying the quadratic formula: $$x = \frac{-b \pm \sqrt{D}}{2a}$$.
Quadratic functions are commonly used to model scenarios like projectile motion, area problems, and economic profit maximization.
Review Questions
How do you find the vertex and axis of symmetry for a given quadratic function?
To find the vertex of a quadratic function in standard form $$f(x) = ax^2 + bx + c$$, you can use the formula for the axis of symmetry, which is given by $$x = -\frac{b}{2a}$$. Substituting this x-value back into the function will give you the y-coordinate of the vertex. Therefore, the vertex coordinates are (x, f(x)), and this point plays a critical role in graphing the parabola.
Explain how to use the discriminant to determine the number and type of roots of a quadratic equation.
The discriminant is calculated using the formula $$D = b^2 - 4ac$$ from the quadratic equation $$ax^2 + bx + c = 0$$. If D > 0, there are two distinct real roots; if D = 0, there is exactly one real root (the parabola touches the x-axis); if D < 0, there are no real roots (the parabola does not intersect the x-axis). This understanding helps in predicting the behavior of the graph and its intersections with the x-axis.
Analyze how quadratic functions can be used to model real-world situations, providing an example.
Quadratic functions are frequently used to model situations where there is a maximum or minimum value involved. For example, in physics, projectile motion can be modeled with a quadratic function where the height of an object over time forms a parabolic shape. By defining height as a function of time with respect to initial velocity and acceleration due to gravity, we can predict how high an object will go before it starts to fall back down. This practical application highlights how understanding quadratic functions aids in solving real-life problems.