3.6 Quadratic Functions

QUADRATIC FUNCTIONS WOOOO PARTYYY

Let's start off with the quadratic formula.

or

because y is a function of x

A function f is a quadratic function if

The graph of a quadratic function is a parabola.

• is a parabola with vertex at the origin (0,0)
• has a vertical axis
• opens upward if a>0 and downward if a<0

the quadratic function

can be obtained by vertical and/or horizontal shifts of the graph:

Some things observed from a quadratic equation:

• c is the y-intercept
• set y=0 and solve to get x-intercepts
• a>0 open upwards
• a<0 open downwards
• a tells to what extreme the parabola is vertically stretched or compressed. the larger the number, the more extremely it is vertically stretched
• b=0 if axis of symmetry is the y-axis
• b is a vertical line; it serves as the axis of symmetry, splitting the parabola in two at the vertex
• to find the vertex of a parabola solve for the x-coordinate and then plug in the value to solve for the y coordinate

A quadratic formula can also be expressed in the form

this is known as the standard equation of a parabola with vertical axis

to reach this form of an equation from the quadratic formula you complete the square

ex:

given

factor out 3

complete the square for

is the equivelant equation in form

when an equation is in this form its has a vertex (h,k) and depending on a being > or < than 0, (h,k) will be the minimum value or the maximum value

Finding an equation of a parabola with a given vertex ex:

find an equation of a parabola that has

vertex(2,3)

vertical axis

passes through the point (5,1)

we will try to put the parabola in form

vertex (3,2) gives us h=2 and k=3

plug in the coordinate (5,1) because we know it is on the parabola

solve to get that

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