How do you use the important points to sketch the graph of #y=x^2+14x+29#?

Given:

#color(red)(y = x^2+14x+29)#

Factoring Method is unsuitable for this problem situation.

To find the Solutions, we use the Quadratic Formula:

#color(green)((-b+- sqrt(b^2 - 4*a*c))/(2*a))#

We can see that

#color(blue)(a = 1; b = 14 and c= 29)#

#x = [-14+-sqrt(14^2-4*1*29)]/(2*1)#

#x = [-14+-sqrt(80)]/(2)#

#x = [-14+-sqrt(16*5)]/(2)#

#x = [-14+-4*sqrt(5)]/(2)#

#x = [-7+-2sqrt(5)]#

Therefore,

#x = [-7+-4.472136]#

#x = [-7+4.472136], [-7-4.472136]#

#x =-2.52786, -11.4721#

Hence,

x-Intercepts are: #-2.52786, -11.4721#

To get the y-intercept we substitute #x=0# in

#color(red)(y = x^2+14x+29)#

#:.# our y-intercept is at #(0, 29)#

Next, we will work on our Vertex

We can use the formula #color(blue)[[-b/(2*a)]# to find our Vertex

#rArr -14/(2.1)#

#rArr -7#

This is the x-coordinate value of our vertex

To find the y-coordinate value of our vertex, we substitute this value of #x# in #color(red)(y = x^2+14x+29)#

We get,

#y = (-7)^2 + 14*(-7) + 29#

Therefore,

#y = 49 -98 + 29#

#y = -20#

This is our y-coordinate value of our vertex.

Hence, our vertex is #color(green)[(-7, -20)#

Now, we have all the important points with us for the graph.

Analyze the graph below to understand better:

Hope you find the solution useful.