Question

How do you identify the important parts of `y= 2x^2+7x-21` to graph it?

Answer 1

y intercept = -21 ; shape is an upward horse shoe with the minimum at `(1 3/4, - 1 3/4)` Crosses the x axis at y=0 so factorise and equate to 0

Explanation:

equation standard form is `y =ax^2 + bx +c`

in this case:

`+x^2` is upwards horse shoe shape
`-x^2` is downward horse shoe shape
So this an upwards with a minimum y value but goes on increasing from there to infinity.

y intercept at `x=0` so by substitution `y=-21`
which is the value of the constant c.

The value of `x` at the minimum may be found by completing the square ( changing the way the equation looks without changing its intrinsic value). We need to end up with both the x^2 and the x part together inside the brackets. In this case we have:

write `y=2x^2 + 7x -21` as:

`y = 2(x^2 + 7/2)^2 - 21 -("correction bit of" (7/2)^2)`

the correction is to compensate for the constant we have introduced. We still have the -21 but we have introduced `(7/2)^2` due to `7/2 times 7/2` from `( x^2 + 7/2)(x^2+7/2)`

Now we look inside the brackets at the `7/2x` part
The minimum occurs at `x_(min) = -1/2 times 7/2 = - 1 3/4`

To find `y_(min)` we substitute the found value for `x_(min)` in the original equation.

Answer 2

ABR again could not correct my answer. The value of `x_(min)` is 1/ 3/4# Sorry about that!!!