How to graph a parabola #y = (x + 5)^2 - 3#?

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The Vertex Form of a quadratic function is :

#color(blue)(y=f(x)=a(x-h)^2+k#, where #color(green)(( h, k )# is the Vertex of the Parabola.

Quadratic Function is given in Vertex Form: #color(red)(y = (x + 5)^2 - 3#

#color(brown)(h=-5 and k = -3#

Vertex is at #color(green)((h,k)#: #color(blue)((-5, -3)#

Plot the Vertex and the #color(red)((x,y)# values from the data table.

To find the x-intercepts:

#f(x)=(x+5)^2-3#

Let #f(x)=0#

# :. (x+5)^2-3 = 0#

Add #color(red)(3# to both sides:

#(x+5)^2-3+color(red)(3)= 0+color(red)(3#

#(x+5)^2-cancel 3+color(red)(cancel 3)= 0+color(red)(3#

#(x+5)^2 = 3#

Take Square root on both sides to simplify:

#sqrt((x+5)^2) = sqrt(3)#

#(x+5) = +- sqrt(3)#

Subtract #color(red)(5)# from both sides:

#(x+5)-color(red)(5) = +- sqrt(3)-color(red)(5)#

#(x+cancel 5)-color(red)(cancel 5) = +- sqrt(3)-color(red)(5)#

#x=+-sqrt(3)-5#

Hence, #color(blue)(x=[sqrt(3)-5]# is one solution and #color(blue)(x=[-sqrt(3)-5]# is the other.

Using a calculator,

#color(blue)(x~~ -3.26795)# is one solution.

#color(blue)(x~~ -6.73205)# is another solution.

Hence, x-intercepts are: #x~~ -3.3#, #x~~ -6.7#

Verify this solution by using graphs:

#color(green)("Graph 1"#

Graph of #color(blue)(y=x^2#

This is the Parent Graph.

Use this graph to understand the behavior of the given quadratic function.

#color(green)("Graph 2"#

Graph of #color(blue)(y = (x + 5)^2 - 3#

Study the graphs of both the Parent function and the given function.

Next, verify the x-intercepts:

Hope it helps.