vertex, minimum or maximum

Question

Consider the graph of . Find the intercepts. Find the vertex and state whether it is a minimum or a maximum. Determine the intervals where  is increasing and decreasing and the intervals where it is positive and negative.

Answer 1

f(x) = 2x² -10x+8

X intercept

first we will write equation in factor form

y=2(x² -5x+4)

=2(x-4)(x-1)

To find x intercept substitute y=0

0=2(x-4)(x-1)

x=4 ,x= 1

so x-intercept is (4,0),(1,0)

Y intercept

to find y intercept substitute x=0

y=2(0)² -10(0)+8 =8

so y intercept is (0,8)

vertex

f(x) = 2x² -10x+8

a=2,b=-10,c=8

so

x co ordinate of vertex is

x=-b/2a =-(-10)/2(2)=2.5

to find y coordinate substitute value of x in equation

y = 2x² -10x+8

=2(2.5)² - 10(2.5)+8

=2(6.25)-25+8

=-4.5

so vertex is(2.5,-4.5)

Vertex is at minimum as parabola opens upward (as a is positive)

The function decreases on(-∞ , 2.5 ) and increases on (2.5 , +∞)

Positive : ( -∞,1)∪ (4,+∞)

Negative :(1,4)