Quadratic FunctionsSections 5.1-5.3

What is a “quadratic” function?a quadratic function is one that can be

written in the following form:, where

(called “standard form”)

3 parts to a quadratic function:- : “quadratic” term (MUST be present)- : “linear” term- : “constant” term or just the

“constant”

Graph of a quadratic function…the graph of a quadratic function is called a

“parabola” (“u”-shaped or “bowl” shaped graph)

can either open up or down:- if , the parabola opens up- if , the parabola opens down

Graph of a quadratic function…vertex”: either the minimum (when opens up)

or maximum (when opens down) point on the parabola

parabolas are symmetric- have an “axis of symmetry” (represented by the equation (vertical line))

the vertex is the only point of the parabola that intersects the axis of symmetry (x-coordinate/# can be found by using the following formula:

“y-intercept” of the parabola:

Graph of a quadratic function…

What must you be able to do with quadratic functions right now?1) Graph a quadratic function (either in

standard or vertex form)2) Convert a quadratic function to either

standard or vertex form given the other form3) Write a quadratic function (in either

form) given information

Procedure for graphing quadratic functions (in standard form)…1) Determine whether the parabola will “open up” or

“open down”

2) Find the x-coordinate of the vertex by using the formula

3) Find the y-coordinate of the vertex by substituting the x-value into the function and solving for y

4) Plot the vertex

Procedure for graphing quadratic functions (in standard form)…5) Select two other values of x and solve for

their y-values; plot these points (if possible, choose 0 as one of your values!)

6) Find the “reflection (corresponding) points” of those two points and plot these points as well

7) Draw your graph through your points…done!

No parab-lem!

Let’s graph some parabolas!Graph the following quadratic functions in standard form:

A)

B)

C)

D)

There is another form…in addition to standard form, quadratic equations can also

be written in “vertex form”:

: coordinates of vertexaxis of symmetry:

if a minus sign inside parentheses, is positiveif a plus sign inside parentheses, is negativeif a plus sign at the end, is positiveif a minus sign at the end, is negative

Procedure for graphing quadratic functions (in vertex form)…use same procedure as before (start at step

#4)

Let’s graph some more parabolas!Graph the following quadratic functions in

vertex form:

A)

B)

C)

D)

Family Reunion!the standard form and vertex forms of a quadratic function

are related! (can convert from one to another)Procedure for converting standard form into vertex form:

1) Find the coordinates of the vertex2) Place your -value and the coordinates of the vertex

into their places within the vertex form

Procedure for converting vertex form into standard form:1) Multiply the squared expression by using the FOIL

Method2) Distribute the -value if necessary, then combine like

terms

Let’s try some…Convert the following quadratic functions

from standard form into vertex form:A)

B)

C)

D)

Let’s try some…Convert the following quadratic functions

from vertex form into standard form:A)

B)

C)

D)

Writing Quadratic Functions…Quadratic functions can be written given certain

pieces of information (vertex, axis of symmetry, another point on the parabola, etc…)

when given the vertex and another point on the parabola:

- place coordinates of vertex into their correct places for vertex form (adjust signs if necessary)

- use the other point given to solve for the-value- write the function with the -value in vertex

form

Writing Quadratic Functions…When given a table of values:

- press STAT, then choose option 1 EDIT- enter the data (each row is a column on

the calculator)- after you finished entering the data,

press STAT, move the cursor to CALC, then choose option 5 QUADREG (Quadratic Regression Line)

- write the quadratic function using the information given

Let’s try some…Given the following table of values,

determine the quadratic function associated with them:

A)

B)

Translations…one can use vertex form to write “translation

equations” of quadratic functions

translation- when a graph is moved around the coordinate plane (does not affect either size of shape of graph)

- three types of translations: horizontal, vertical, and diagonal

translations start from “parent functions” (the function representing the graph that is to be moved)

Translations…Rules for writing translation equations: - if you move left, the sign inside the parentheses is positive - if you move right, the sign inside the parentheses is negative - if you move up, the sign at the end of the function is positive - if you move down, the sign at the end of the function is negative

Let’s try some…Given the following translation equations,

state the type of translation, the parent function, and the shift(s):

A)

B)

C)

Let’s try some…Given the following information, write the

translation equation:A) parent function: ; right 8

B) parent function: ; up 5

C) parent function; ; left 4, down 7