Question #81645

2 Answers
Mar 1, 2018

graph{y=x^2-x-6 [-9.96, 10.04, -7.64, 2.36]}

This is the graph of #y=x^2-x-6#

Explanation:

#y=(x+2)(x-3)#

#y=x^2-x-6#

We can plot a parabola by substituting x=-5 to +5, which will give
24, 14, 6, 0, -4, -6, -6, -4, 0, 14, 24.

By graphing the points, we can link the points to get a parabola.

Note:
#y=x^2-x-6#
#y=(x-0.5)^2-0.25-6#
#y=(x-0.5)^2-6.25#

Hence, the vertex of the parabola is at #(0.5,-6.25)#

Mar 1, 2018

Refer to the explanation.

Explanation:

Graph:

#y=(x+2)(x-3)#

In order to graph a parabola, useful points are the vertex and x-intercepts (if it crosses the x-axis), and other points determined by plugging in different values for #x# and solving for #y#.

X-intercepts: values of #x# when #y=0#

The given equation is in the factored form of a quadratic equation. We can substitute #0# for #y# and solve for #x#, which will give us the two x-intercepts.

#(x+2)=0#

#x=-2#

#(x-3)=0#

#x=3#

#color(red)("x-intercepts":# #(-2,0), (3,0)#

Now we need to convert the equation to standard form by FOILing the binomials.

http://www.mathcaptain.com/algebra/foil-method.html

#y=x^2-x-6# is a quadratic equation in standard form:

#y=ax^2+bx+c#,

where:

#a=1#, #b=-1#, #c=-6#

For a quadratic equation in standard form, the #x#-value of the vertex is the same as the axis of symmetry.

Axis of Symmetry: vertical line that divides the parabola into equal halves

Formula for axis of symmetry for a quadratic equation in standard form:

#x=(-b)/(2a)#

#x=(-(-1))/(2*1)#

#x=1/2=0.5#

Vertex: minimum or maximum point on the parabola

Substitute #0.5# for #x# in the equation and solve for #y#.

#y=(0.5)^2-0.5-6#

#y=0.25-0.5-6=-6.25#

#color(blue)("vertex":# #(0.5,-6.25)#

Other points

First point:

Substitute #-3# for #x# and solve for #y#.

#y=(-3)^2-(-3)-6#

#y=9+3-6=6#

#color(green)("first point:"# #(-3,6)#

Second point:

Substitute #4# for #x# and solve for #y#.

#y=4^2-4-6#

#y=16-4-6=6#

#color(purple)"second point:"# #(4,6)#

Plot the vertex, x-intercepts, and other points. Sketch a parabola through the points. Do not connect the dots.

graph{y=x^2-x-6 [-10, 10, -5, 5]}