In order to graph a parabola, you need at least the vertex and the x-intercepts. The y-intercept can also be helpful. Once you plot those points, sketch a parabola through them. Do not connect the dots.
Given:
#y=2x^2-x-1# is a quadratic equation in standard form:
#y=ax^2+bx+c#,
where:
#a=2#, #b=-1#, and #c=-1#
Axis of symmetry: vertical line that divides the parabola into two equal halves. It is also the #x#-value of the vertex. The formula for the axis of symmetry for a quadratic formula in standard form is:
#x=(-b)/(2a)#
Plug in the known values.
#x=(-(-1))/(2*2)#
Simplify.
#x=1/4##color(white)(.)# or#color(white)(.)# #x=0.25#
Vertex: maximum or minimum point of a parabola. If #a>0#, the vertex is the minimum point and the parabola opens upward. It is reversed if #a<0#. To find the #y#-value of the vertex, substitute #1/4# for #x# and solve for #y#.
#y=2x^2-x-1#
Plug in #1/4# for #x#.
#y=2(1/4)^2-1/4-1#
Simplify.
#y=2/16-1/4-1#
Simplify #2/16# to #1/8#.
Convert #1/4# and #1# to equivalent fractions with the denominator #8# by multiplying each number by a fraction equal to #1# that will give each number the denominator #16#. For example, #3/3=1#. Recall that a whole number has a denominator of #1#: #n=n/1#.
#y=1/8-1/4xxcolor(red)(2/2)-1/1xxcolor(blue)(8/8#
Simplify.
#y=1/8-2/8-8/8#
#y=(1-2-8)/8#
#y=-9/8##color(white)(.)# or#color(white)(.)##y=-1 1/8##color(white)(.)# or#color(white)(.)# #y=-1.125#
Vertex: #(1/4,-9/8)##color(white)(.)#or#color(white)(.)##(0.25,-1.125)#
Y-intercept: value of #y# when #x=0#
Substitute #0# for #x# and solve for #y#.
#y=2(x)^2-0-1#
#y=-1#
y-intercept: #(0,-1)#
X-intercepts: values of #x# when #y=0#.
Substitute #0# for #y#.
#0=2x^2-x-1#
Factor #2x^2-x-1#.
#0=2x^2-x-1#
Split the middle term into #x# and #-2x#.
#0=2x^2+x-2x-1#
Factor out the common terms in the first two and last two terms.
#0=x(2x+1)-1(2x-1)#
#0=2x+1#
#0=x-1#
Solutions for #x#.
#0=2x+1#
#-1/2=x#
#0=x-1#
#1=x#
#x=-1/2,1#
x-intercepts:
#x=(-1/2,0),##(1,0)#
Summary:
Axis of symmetry: #x=1/4=0.25#
Vertex: #(1/4,9/8)#=#(0.25,-1.125)#
Y-intercept: #(0,-1)#
X-intercepts: #(-1/2,0),##(1,0)#
graph{y=2x^2-x-1 [-10, 10, -5, 5]}
