Quadratic functions

Definition

Quadratic functions are polynomial functions of degree two, typically expressed in the standard form $$f(x) = ax^2 + bx + c$$, where 'a', 'b', and 'c' are constants and 'a' is not equal to zero. They graph as parabolas that can open upwards or downwards depending on the sign of 'a'. Understanding their composition with other functions is crucial since it can reveal new behaviors and characteristics of combined functions.

5 Must Know Facts For Your Next Test

1. Quadratic functions can be represented in multiple forms: standard form, vertex form, and factored form, each providing different insights into the function's characteristics.
2. The discriminant of a quadratic function, calculated as $$D = b^2 - 4ac$$, determines the number and type of roots: two distinct real roots, one repeated real root, or two complex roots.
3. When composing a quadratic function with another function, the resulting function can also be quadratic or a higher-degree polynomial, depending on the nature of the other function involved.
4. Quadratic functions are widely used in various real-world applications, such as physics for projectile motion, economics for profit maximization problems, and engineering for designing parabolic structures.
5. Understanding transformations of quadratic functions, like shifts and stretches, is essential when considering their compositions with linear functions or other non-linear functions.

Review Questions

• How does the composition of a quadratic function with a linear function affect the overall behavior of the resulting function?
  • When a quadratic function is composed with a linear function, the outcome is generally a cubic function. This means that instead of just having a U-shaped curve, the resulting graph can display more complex behaviors such as inflection points or additional turns. The overall characteristics such as intercepts and end behavior also change based on the specific linear function involved in the composition.
• Evaluate how changes in coefficients of a quadratic function impact its graph when combined with other polynomial functions.
  • Altering the coefficients of a quadratic function directly influences its shape and position. For instance, increasing 'a' makes the parabola narrower, while decreasing it broadens the shape. When combined with another polynomial function, these changes can affect intersection points and overall growth rates of the resulting function. Analyzing these interactions helps understand how quadratic behavior interacts within more complex polynomial expressions.
• Synthesize how knowledge of quadratic functions aids in solving real-world problems involving projectile motion when integrated with trigonometric functions.
  • Quadratic functions are crucial for modeling projectile motion, as they represent paths described by parabolas. By integrating these functions with trigonometric functions to account for angles of launch and initial velocity, one can derive equations that predict maximum height and distance traveled. This synthesis allows for practical applications in fields like engineering and physics, where predicting trajectories accurately is essential for design and analysis.

Related terms

Parabola: The U-shaped graph of a quadratic function, which can either open upwards if 'a' is positive or downwards if 'a' is negative.

Vertex: The highest or lowest point of a parabola, depending on whether it opens upwards or downwards, located at the point $$(-\frac{b}{2a}, f(-\frac{b}{2a}))$$.

Axis of Symmetry: A vertical line that divides the parabola into two mirror-image halves, given by the equation $$x = -\frac{b}{2a}$$.