How do you graph the parabola f(x)=x ^2 + 3x-10f(x)=x2+3x10 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))(x,f(x)), near these crucial spots of our parabola.

Explanation:

To find the vertex, we use the formula x = -b/(2a)x=b2a, but as you may have guessed, this will only give us the xx coordinate of the vertex. In order to get the yy coordinate, we would need to put our xx value back into f(x)f(x).

Standard form is
ax^2 + bx - 10ax2+bx10

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

This means the xx coordinate of our vertex will be

-(3)/(2(1))32(1) or simply -3/232

Plugging the xx coordinate back into f(x)f(x), we get the yy coordinate.

(-3/2)^2+3(-3/2)-10(32)2+3(32)10

= 9/4-9/2-10=949210

= -49/4=494

Our vertex is therefore (-3/2,-49/4)(32,494).

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

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

x^2 + 3x -10 = 0x2+3x10=0

(x + 5)(x - 2) = 0(x+5)(x2)=0

x + 5 = 0 or x - 2 = 0x+5=0orx2=0

x = - 5 or x = 2x=5orx=2

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

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

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

= -10=10

So, our yy-intercept is (0,-10)(0,10).

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

  • The vertex, (-3/2,-49/4)(32,494)
  • The xx-intercepts, (-5,0)(5,0) and (2,0)(2,0)
  • And the yy-intercept, (0,-10)(0,10)

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

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

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

Jan 11, 2018

ME (Harold Walden)ME (Harold Walden)

Explanation:

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

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