Quadratic Functions

A quadratic function is a nonlinear function that can be written in standard from y = ax2 + bx + c The U-Shaped graph of a quadratic function is called a parabola The parent function of a quadratic function is y=x2 The highest or lowest point on a parabola is the vertex. Therefore, the minimum or maximum value of a quadratic function occurs at the vertex.

End Behavior of a Graph Given a quadratic function in the form 𝒇 𝒙 =𝒂 𝒙 𝟐 + bx + c Standard form 𝒇 𝒙 =𝒂 𝒙 −𝒉 𝟐 Vertex Form Quadratic function is said to open up if a > 0 and open down if a < 0.

The minimum value of a function is the least possible y-value for that function. If a > 0 then f has a minimum at the x -coordinate of the vertex, The maximum value of a function is the greatest possible y-value for that function. If a < 0 then has a maximum at x-coordinate of the vertex

Review Domain is all the possible x values of a function From left to right Range is all the possibly y values of a function From bottom to top

Unless a specific domain is given, the domain of a quadratic function is all real numbers. One way to find the range of a quadratic function is by looking at its graph.

Roots Vs Solutions Roots are solutions to a one variable equations. So roots and solutions are synonymous. Zeroes indicate you are looking at a graph and a 2 variable equation. They are the x-values where y = 0. Imaginary roots cannot be located on a rectangular on a graph

Two Solutions A quadratic equation has two solutions if the graph of its related function has two x-intercepts Would have 2 real roots and 2 zeroes

One Solution A quadratic equation has one solution if the graph of its related function has one x-intercept One zero and a double real root

No Real Solutions A quadratic equation has no real solution if the graph of its related function has no x-intercepts No zeroes but 2 non real roots

State the vertex, the solutions, the domain and the range of each

State the vertex, the solutions, the domain and the range of each