Chapter 2 Quiz

less than

<

quadratic function

an equation that can be written in the form f(x)=a(x-h)^2+k/y=ax^2+bx+c

greater than

>

-f(x)=-(x^2)=-x^2 ; reflected over the x-axis

Reflections in the x-axis

axis of symmetry

a line that divides a parabola into mirror images and passes through the vertex ; x=h

directrix

a line that is perpendicular to the axis of symmetry that is the same distance to the parabola as the parabola is to the focus

focus

a point in the interior of the parabola that lies on the axis of symmetry

vertex form

f(x)=a(x-h)^2+k where (h,k) is the vertex

standard form of a quadratic equation

f(x)=ax^2+bx+c ; use this form to find y-intercept (0,c)

x=1/4p y^2 ; a=1/4p ; p=1/4a ; when the parabola is sideways add up p to x-coordinate of vertex to find focus ; subtract p from x-coordinate of vertex to find directrix ; axis of symmetry is y=k when parabola is sideways

horizontal axis of symmetry (y=0)

f(ax)=(ax)^2 ; horizontal stretch (away from y-axis) when 0<a<1 / horizontal shrink (toward y-axis) when a>1

horizontal stretches and shrinks

f(x-h)=(x-h)^2 ; shifts left when h<0 / shifts right when h>0

horizontal translations

f(-x)=x^2

reflection in the y-axis

parabola

the graph of a quadratic equation

vertex of a parabola

the lowest point on a parabola that opens up or the highest point on a parabola that opens down

y=1/4p x^2 ; subtract p from y-coordinate of focus to get directrix

vertical axis of symmetry (x=0)

a•f(x)=ax^2 ; vertical stretch (away from x-axis) when a>1/ vertical shrink (towards x-axis) when 0<a<1

vertical stretches and shrinks

f(x)+k=x^2+k ; shifts down when k<0 / shifts up when k>0

vertical translations

intercept form

when the graph of a quadratic function has at least one x-intercept, the function can be written in intercept form ; f(x)=a(x-p)(x-q) ; when p & q are the x-intercepts (also called roots or zeros)

Properties of the graph of f(x)=ax^2+bx+c

•the parabola opens up when a>0 and open down when a<0 •the axis of symmetry is x=-b/2a and the vertex is (-b/2a,f(-b/2a))<---evaluate the function at the x-coordinate of the vertex •the y-intercept is c

maximum value

•the y-coordinate of the vertex when a<0 ; y=k •maximum value: f(-b/2a) •domain: all real numbers •range: y≤f(-b/2a) ; smaller than the minimum •increasing to the left of x=-b/2a •decreasing to the right of x=-b/2a

minimum value

•the y-coordinates of the vertex when a<0 ; y=k •minimum value: f(-b/2a) •domain: all real numbers •range: y≥f(-b/2a) ; larger than the minimum •decreasing to the left of x=-b/2a •increasing to the right of x=-b/2a