Quadratic function

Definition

A quadratic function is a polynomial function of degree two, typically expressed in the standard form $$f(x) = ax^2 + bx + c$$, where $a$, $b$, and $c$ are constants, and $a \neq 0$. This type of function graphs as a parabola, which can either open upwards or downwards depending on the sign of $a$. Understanding the properties of quadratic functions helps in analyzing their behavior, graphing them effectively, and recognizing their applications across various mathematical concepts.

5 Must Know Facts For Your Next Test

1. A quadratic function has a parabolic graph that can either open upward when $a > 0$ or downward when $a < 0$.
2. 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.
3. The axis of symmetry for a quadratic function can be found using the formula $$x = -\frac{b}{2a}$$.
4. The roots or x-intercepts of a quadratic function can be found using factoring, completing the square, or applying the quadratic formula: $$x = \frac{-b \pm \sqrt{D}}{2a}$$.
5. The discriminant not only indicates how many real roots a quadratic has but also reveals if they are distinct, repeated, or complex.

Review Questions

• How can you determine the vertex and axis of symmetry for a given quadratic function?
  • To find the vertex of a quadratic function in standard form $$f(x) = ax^2 + bx + c$$, use the formula for the x-coordinate: $$x = -\frac{b}{2a}$$. Once you have this x-value, substitute it back into the function to get the corresponding y-coordinate. The axis of symmetry is then given by the line $$x = -\frac{b}{2a}$$, which runs vertically through the vertex and divides the parabola into two equal halves.
• Explain how to use the discriminant to determine the nature of roots for a quadratic equation.
  • The discriminant is calculated using the formula $$D = b^2 - 4ac$$ from a quadratic equation in standard form. If $D > 0$, there are two distinct real roots. If $D = 0$, there is exactly one real root (a repeated root). If $D < 0$, there are no real roots, indicating that both roots are complex. This understanding helps in predicting how many solutions exist without solving for them directly.
• Evaluate the impact of changing coefficients in a quadratic function on its graph and roots.
  • Changing the coefficients $a$, $b$, and $c$ in a quadratic function significantly alters its graph and roots. For instance, increasing $a$ makes the parabola narrower while decreasing it makes it wider. The value of $b$ affects the position of the vertex along the x-axis and thus shifts the parabola left or right. Changing $c$ moves the parabola up or down without affecting its shape. Each modification impacts not only where the graph lies but also how many x-intercepts it has based on the discriminant's value.

Related terms

Vertex:

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

Axis of Symmetry: A vertical line that divides the parabola into two symmetrical halves, passing through the vertex.

Discriminant: The value calculated from the coefficients of a quadratic equation given by $$D = b^2 - 4ac$$, which determines the nature and number of roots.