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 $a \neq 0$. This function creates a parabolic graph, which can open either upwards or downwards, depending on the sign of the leading coefficient $a$. Quadratic functions are significant because they model various real-world scenarios, such as projectile motion and area problems.

5 Must Know Facts For Your Next Test

1. The graph of a quadratic function can intersect the x-axis at zero, one, or two points, which corresponds to the number of real roots.
2. The vertex of the parabola represents either the maximum or minimum point of the quadratic function, depending on whether it opens upwards or downwards.
3. The axis of symmetry of a quadratic function can be found using the formula $$x = -\frac{b}{2a}$$, dividing the parabola into two mirror-image halves.
4. Quadratic functions are used in various fields such as physics, economics, and engineering to model phenomena like projectile motion and profit maximization.
5. Factoring a quadratic function can sometimes provide quicker solutions to finding its roots, especially if it can be expressed as a product of two binomials.

Review Questions

• How does changing the value of the leading coefficient $a$ affect the graph of a quadratic function?
  • Changing the value of the leading coefficient $a$ in a quadratic function significantly impacts the graph's shape and orientation. If $a$ is positive, the parabola opens upwards; if negative, it opens downwards. Additionally, larger absolute values of $a$ make the parabola narrower, while smaller absolute values result in a wider shape. This effect on width and direction is crucial for understanding how quadratic functions behave graphically.
• In what ways can you derive the vertex form from standard form for a quadratic function, and why is this form useful?
  • To convert a quadratic function from standard form $$f(x) = ax^2 + bx + c$$ to vertex form $$f(x) = a(x - h)^2 + k$$, you can complete the square. This process involves manipulating the equation to isolate a perfect square trinomial. The vertex form is useful because it clearly shows the vertex $(h, k)$ of the parabola, making it easier to graph and understand its maximum or minimum point. It also helps in determining transformations applied to the basic parabola.
• Evaluate how the discriminant can be used to predict the nature of roots in a quadratic equation and discuss its implications.
  • The discriminant, given by $$D = b^2 - 4ac$$ for a quadratic equation $$ax^2 + bx + c = 0$$, provides critical information about the roots. If $D > 0$, there are two distinct real roots; if $D = 0$, there is one real root (the roots are repeated); and if $D < 0$, there are no real roots (the roots are complex). Understanding this allows us to predict not just how many solutions exist but also their behavior, which is essential in many applications such as optimization problems in real-world contexts.

Related terms

Parabola: The U-shaped graph that represents a quadratic function, characterized by its vertex and axis of symmetry.

Vertex Form: A way to express a quadratic function as $$f(x) = a(x - h)^2 + k$$, where $(h, k)$ is the vertex of the parabola.

Discriminant: A value calculated from the coefficients of a quadratic equation that determines the nature of the roots; given by $$D = b^2 - 4ac$$.