Question

How do you graph the parabola `3x^2+18x+21` using vertex, intercepts and additional points?

Answer 1

See method and graph below.

Explanation:

`"Let "y=3x^2+18x+21`

We will manipulate this to find points on the graph, and thus we can sketch the parabola.

Start off with the y-intercept, and make clear everything that you are doing (anything formatted you should write down).

`"Let " x=0`
`y=21`

So we know there is a y-intercept of 21. Now, for the x-intercepts

`"Let " y=0`
`0=3x^2+18x+21`
`0=x^2+6x+7`

A quick check of the discriminant tells us that this does not factorise (the discriminant is 8 which is not a square number). So to solve this, we might as well use the Quadratic formula.

`x=(-b+-sqrt(b^2-4ac))/(2a)`
`=(-6+-sqrt(6^2-4xx1xx7))/(2)`
`x=-3+-sqrt2`

We don't need to find the values of these as a decimal, since when we sketch our graph, we will give the x-intercepts as exact values. The decimals are (if you're interested) round about -4.41 and -1.59.

This is enough to sketch the graph, although the question asks us to find the vertex (turning point), so we should probably find that too. You can complete the square or differentiate to find the turning point.

Completing the square

`y=3x^2+18x+21`
`y=3(x^2+6x+7)`
`y=3([x+3]^2-9+7)`
`y=3([x+3]^2-2)`
`y=3[x+3]^2-6`

So the turning point is `(-3, -6)`

Differentiation

`y=3x^2+18x+21`
`dy/dx=6x+18`
`"Let " dy/dx=0`
`6x+18=0`
`x+3=0`
`x=-3`

`"Let " x=-3`
`y=3(-3)^2+18(-3)+21`
`y=-6`

`:. t.p.=(-3,-6)`

---

If the question didn't specify, and I had real roots to my quadratic, I wouldn't bother to find the turning point. The x-coordinate of the turning point will always lie halfway between the roots.

---

Now we have enough information to sketch the curve. Make sure to mark all of your intercepts exactly (even though this graph leaves it as a decimal). Aim for a smooth curve.

When sketching, you don't need to have the y-axis go up in the same step as the x-axis. Just make sure you label your points. (If you need me to go into more detail on this, I will do, just ask).

graph{y=3x^2+18x+21 [-31.85, 33.1, -7.4, 25.07]}