How do find the vertex and axis of symmetry, and intercepts for a quadratic equation #y= x^2 + 6x + 5#?

1 Answer
Mar 30, 2018

Vertex : #color(blue)((-3, -4)#

Axis of Symmetry is at : #color(blue)(x=(-3)#

x-intercepts: #color(blue)((-1,0) and (-5,0)#

y-intercept: #color(blue)((0,5)#

Explanation:

Given:

#color(red)(y = f(x) = x^2+6x+5#

The Vertex Form of a quadratic function is given by:

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

#color(green)(x=h# is the axis of symmetry.

Use completing the square method to convert #color(red)(f(x)# into Vertex Form.

#color(red)(y = f(x) = x^2+6x+5#

Standard Form #rArr ax^2+bx+c=0#

Consider the quadratic #x^2+6x+5=0#

#color(blue)(a=1; b=6 and c=5#

Step 1 - Move the constant value to the right-hand side.

Subtract 5 from both sides.

#x^2+6x+5-5 = 0-5#

#x^2+6x+cancel 5-cancel5 = 0-5#

#x^2+6x=-5#

Step 2 - Add a value to both sides.

What value to add?

Add the square of #b/2#

Hence,

#x^2+6x+[(6/2)^2]=-5+[(6/2)^2]#

#x^2+6x+9=-5+9#

#x^2+6x+9=4#

Step 3 - Write as Perfect Square.

#(x+3)^2=4#

Subtract #4# from both sides to get the vertex form.

#(x+3)^2-4=cancel 4-cancel 4#

#f(x)=(x+3)^2 - 4#

Now, we have the vertex form.

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

Hence, Vertex is at #color(blue)((-3,-4)#

Axis of Symmetry is at #color(red)(x=h#

Note that #h=-3#

#rArr color(blue)(x= -3#

Step 4 - Write the x, y intercepts.

Consider

#(x+3)^2=4#

To find the solutions, take square root on both sides.

#sqrt((x+3)^2)= +-sqrt(4)#

#rArr x+3=+-2#

There are two solutions.

#x+3 = 2#

#rArr x=2-3 = -1#

Hence, #x=-1# is one solution.

Next,

#x+3=-2#

#x=-2-3=-5#

Hence, #x=-5# is the other solution.

Hence, we have two x-intercepts: #(-1,0) and (-5,0)#

To find the y-intercept:

Let #x=0#

We have,

#f(x)=(x+3)^2 - 4#

#f(0)=(0+3)^2-4#

#rArr 3^2-4 = 9-4 = 5#

Hence, y-intercept is at #y=5#

#rArr color(blue)((0,5)#

Analyze the image of the graph below:

enter image source here