Clase resumen de Parábolas - Profe Mauro Quintana

Understanding Parabolas and Their Graphs

Introduction to the Function

• The function presented is f(x) = 3x^2 - 7x + 2 , which represents a parabola. The speaker emphasizes the importance of understanding how to graph this function.

Graphing Basics

• The speaker explains that when finding intercepts on the axes, substituting zero into the opposite variable gives the corresponding value. For example, setting x = 0 yields y = c .

Types of Intercepts

• A parabola can intersect the x-axis in three ways:

• Two intersections (cuts twice)

• One intersection (tangential)

• No intersection (does not cut)

Discriminant and Its Role

• The discriminant, calculated as B^2 - 4AC , determines how many times a parabola intersects the x-axis:

• Positive discriminant: two intersections

• Zero discriminant: one intersection

• Negative discriminant: no intersections

Symmetry and Vertex Calculation

• The axis of symmetry divides the parabola into two equal halves and can be found using x_1 + x_2 / 2 or -B / (2A) .

Finding Key Points of the Parabola

Calculating Vertex Coordinates

• To find the vertex's coordinates, use:

• X-coordinate: x = -B / (2A)

• Y-coordinate can be derived by substituting back into the function.

Analyzing Given Function for Graph Representation

• By substituting values into the function, it’s determined that if f(0)=2 , then it touches positive y-values. This helps eliminate certain graphical options.

Discriminant Calculation Example

• For this specific function, calculating:

• Discriminant as B^2 - 4AC = 49 - (4 * 3 * 2), results in a positive number indicating two cuts on the x-axis.

Conclusion on Axis of Symmetry

• Finally, determining whether axis of symmetry is negative or positive based on calculations leads to identifying correct graphical representation among given options.