Quadratic functions

Definition

A quadratic function is a polynomial function of degree two, generally expressed in the form $$f(x) = ax^2 + bx + c$$, where $a$, $b$, and $c$ are constants, and $a \neq 0$. This type of function produces a parabolic graph that opens either upwards or downwards depending on the sign of the coefficient $a$. The characteristics of the graph, including its vertex, axis of symmetry, and intercepts, can be influenced by transformations such as shifts and reflections.

5 Must Know Facts For Your Next Test

1. Quadratic functions can be transformed by adding or subtracting constants to shift them vertically or horizontally.
2. A positive value of $a$ in the quadratic function causes the parabola to open upwards, while a negative value causes it to open downwards.
3. The vertex form of a quadratic function is given by $$f(x) = a(x - h)^2 + k$$, where $(h,k)$ represents the vertex of the parabola.
4. Completing the square is a method used to derive the vertex form of a quadratic function from its standard form.
5. The discriminant, given by $$D = b^2 - 4ac$$, determines the number and type of roots of the quadratic equation.

Review Questions

• How do transformations affect the graph of a quadratic function?
  • Transformations can shift the graph of a quadratic function horizontally or vertically by adding or subtracting values to/from $x$ and $f(x)$ respectively. For instance, if we have $$f(x) = ax^2 + bx + c$$ and we change it to $$f(x) = a(x-h)^2 + k$$, the graph moves horizontally to $h$ and vertically to $k$. Reflections can occur by changing the sign of $a$, flipping the parabola over the x-axis.
• What role does the vertex play in understanding the behavior of quadratic functions?
  • The vertex is crucial for understanding quadratic functions as it represents either the maximum or minimum point of the parabola. Knowing the vertex helps in sketching the graph since it identifies where the function reaches its peak or trough. Additionally, it allows us to determine critical points where changes in direction occur, aiding in further analysis of its behavior.
• Evaluate how different values of coefficients in a quadratic function affect its transformations and overall shape.
  • Different coefficients in a quadratic function lead to various transformations and shapes. The coefficient $a$ affects whether the parabola opens upwards or downwards and its width; larger absolute values lead to narrower parabolas. The coefficients $b$ and $c$ influence horizontal positioning and vertical shifts respectively. By analyzing these coefficients collectively, one can predict how the graph will be transformed in relation to its standard position.

Related terms

Vertex:

The vertex is the highest or lowest point of a parabola, depending on whether it opens upwards or downwards.

Axis of Symmetry: The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves, passing through the vertex.

Transformations: Transformations involve changes to the graph of a function, including shifts, stretches, compressions, and reflections.