Quadratic function

Definition

A quadratic function is a type of polynomial function of degree two, typically expressed in the form $$f(x) = ax^2 + bx + c$$, where $$a$$, $$b$$, and $$c$$ are constants and $$a eq 0$$. This function creates a parabolic graph that opens either upward or downward, depending on the sign of the leading coefficient $$a$$. Quadratic functions have unique features like vertex, axis of symmetry, and roots that provide valuable information about their behavior and characteristics.

5 Must Know Facts For Your Next Test

1. Quadratic functions can have zero, one, or two real roots depending on the value of the discriminant.
2. The vertex form of a quadratic function is given by $$f(x) = a(x - h)^2 + k$$, where (h, k) represents the coordinates of the vertex.
3. The graph of a quadratic function is always a parabola, which can open upwards if $$a > 0$$ or downwards if $$a < 0$$.
4. The maximum or minimum value of a quadratic function occurs at its vertex and is determined by the direction in which the parabola opens.
5. The roots of a quadratic function can be found using the quadratic formula: $$x = \frac{-b \pm \sqrt{D}}{2a}$$, where $$D$$ is the discriminant.

Review Questions

• How does changing the value of the coefficient 'a' in a quadratic function affect its graph?
  • Changing the value of 'a' in a quadratic function affects both the direction in which the parabola opens and its width. If 'a' is positive, the parabola opens upward, creating a minimum point at the vertex. Conversely, if 'a' is negative, it opens downward, resulting in a maximum point at the vertex. Additionally, larger absolute values of 'a' make the parabola narrower, while smaller absolute values make it wider.
• What role does the discriminant play in determining the roots of a quadratic function?
  • The discriminant, given by $$D = b^2 - 4ac$$ in a quadratic equation $$ax^2 + bx + c = 0$$, indicates how many real roots the function has. If $$D > 0$$, there are two distinct real roots; if $$D = 0$$, there is exactly one real root (the vertex); and if $$D < 0$$, there are no real roots but two complex roots. This makes understanding the discriminant crucial for predicting how a quadratic function will intersect the x-axis.
• Evaluate how understanding quadratic functions can be applied in real-world scenarios or problem-solving situations.
  • Understanding quadratic functions is vital for modeling various real-world situations such as projectile motion, maximizing area in optimization problems, and analyzing profit and loss in business contexts. For example, when studying projectile motion, the height of an object can be modeled with a quadratic function where time is an independent variable. By analyzing the properties of this function, including its vertex and roots, one can determine key aspects such as maximum height and time when it reaches the ground. Thus, mastery of quadratic functions equips individuals with tools for effective analysis and problem-solving across numerous fields.

Related terms

Vertex: The vertex is the highest or lowest point on the graph of a quadratic function, depending on whether it opens upward or downward.

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

Discriminant: The discriminant is a component of the quadratic formula given by $$D = b^2 - 4ac$$; it determines the nature and number of roots of a quadratic equation.