How do you graph the parabola #f(x)=x ^2 + 3x-10# using vertex, intercepts and additional points?

2 Answers
Jan 11, 2018

After calculating the vertex and intercepts and plotting them, calculate additional points, #(x,f(x))#, near these crucial spots of our parabola.

Explanation:

To find the vertex, we use the formula #x = -b/(2a)#, but as you may have guessed, this will only give us the #x# coordinate of the vertex. In order to get the #y# coordinate, we would need to put our #x# value back into #f(x)#.

Standard form is
#ax^2 + bx - 10#

In our case:
#a = 1#
#b = 3#

This means the #x# coordinate of our vertex will be

#-(3)/(2(1))# or simply #-3/2#

Plugging the #x# coordinate back into #f(x)#, we get the #y# coordinate.

#(-3/2)^2+3(-3/2)-10#

#= 9/4-9/2-10#

#= -49/4#

Our vertex is therefore #(-3/2,-49/4)#.

Now, we can calculate the #x#-intercept and the #y#-intercept, which, alongside our vertex, will both help us determine what other nearby points we may wish to calculate.

The #x#-intercept is where #y=0#. This case may be solved by factoring.

#x^2 + 3x -10 = 0#

#(x + 5)(x - 2) = 0#

#x + 5 = 0 or x - 2 = 0#

#x = - 5 or x = 2#

So, our #x#-intercepts are #(-5,0)# and #(2,0)#.

The #y#-intercept is where #x=0#. Simply plug this value into #f(x)#.

#(0)^2+3(0)-10#

#= -10#

So, our #y#-intercept is #(0,-10)#.

Now, we have four relevant points that we can plot. These are:

  • The vertex, #(-3/2,-49/4)#
  • The #x#-intercepts, #(-5,0)# and #(2,0)#
  • And the #y#-intercept, #(0,-10)#

After plotting these, we likely wish to plot some additional points in between to more accurately graph it. We would choose #x# values and plug them into #f(x)#.

These would be points within our range of crucial coordinates, so anywhere between our lowest #x# coordinate, #(-5,0)#, and our greatest #x# coordinate, #(2,0)#.

You may also plot more points outside of this range to give a more complete picture.

Jan 11, 2018

ME (Harold Walden)

Explanation:

I have handwritten my explanation, I hope it is legible :)Again, the source is me :)

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