key term - 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 are characterized by their U-shaped graphs called parabolas, which can open upwards or downwards depending on the sign of 'a'. The vertex of the parabola represents either the maximum or minimum point of the function, while the axis of symmetry divides the parabola into two mirror-image halves.

5 Must Know Facts For Your Next Test

1. The graph of a quadratic function is always a parabola that can either open upward (if 'a' > 0) or downward (if 'a' < 0).
2. The vertex can be found using the formula $$x = -\frac{b}{2a}$$ and this x-value can be substituted back into the function to find the y-coordinate.
3. Quadratic functions can be expressed in different forms: standard form ($$f(x) = ax^2 + bx + c$$), vertex form ($$f(x) = a(x - h)^2 + k$$), and factored form ($$f(x) = a(x - r_1)(x - r_2)$$).
4. The solutions to a quadratic equation (roots) can be found using factoring, completing the square, or the quadratic formula $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$.
5. The vertex form is particularly useful for graphing because it makes it easy to identify the vertex and shift the parabola accordingly.

Review Questions

• How do transformations such as translations and reflections affect the graph of a quadratic function?
  • Transformations like translations and reflections change the position and orientation of the parabola on the graph. For instance, adding a constant to 'c' in the standard form translates the graph vertically, while changing the sign of 'a' reflects it across the x-axis. These transformations allow for shifts and flips without altering the basic U-shape of the graph, providing insights into how changes in parameters impact its appearance.
• Discuss how to determine whether a quadratic function has maximum or minimum values based on its coefficients.
  • To determine if a quadratic function has a maximum or minimum value, examine the coefficient 'a' in its standard form $$f(x) = ax^2 + bx + c$$. If 'a' is positive, the parabola opens upwards, meaning it has a minimum value located at its vertex. Conversely, if 'a' is negative, the parabola opens downwards and possesses a maximum value at its vertex. This understanding is crucial for optimizing scenarios modeled by quadratic functions.
• Evaluate how changes in parameters 'a', 'b', and 'c' affect key features of quadratic functions like vertex location and width of the parabola.
  • Changes in parameter 'a' affect both the width and direction of the parabola; larger absolute values of 'a' result in a narrower graph, while smaller absolute values make it wider. The parameter 'b' impacts the x-coordinate of the vertex through its relationship with 'a', shifting it left or right along the x-axis. Lastly, 'c' determines the vertical position by shifting the entire parabola up or down. Understanding these relationships helps predict how each parameter influences key features of quadratic functions.

Related terms

Vertex:

The highest or lowest point on a parabola, which represents the maximum or minimum value of a quadratic function.

Axis of Symmetry:

A vertical line that divides the parabola into two symmetric halves, passing through the vertex.

Discriminant:

The expression $$b^2 - 4ac$$ used to determine the nature of the roots of a quadratic equation.