Question

How do you graph the parabola `y = 0.3x^2 + 1.8x - 2` using vertex, intercepts and additional points?

Answer 1

First, convert your equation to vertex form...

Explanation:

...which I can never remember, I always have to look it up. It is:

`y = a(x-h)^2+ k`

...and if you can do this, you can read off the vertex coordinates directly - they are `(h,k)`

I start by dividing both sides of the initial equation by 0.3, so that we have the `x^2` multiplied by 1, not 0.3:

`y/0.3 = x^2 + 6x -2/0.3`

Next, on the right side, we execute the infamous "complete the square" maneuver. If we add (and then subtract) 9, we have:

`y/0.3 = x^2 + 6x + 9 - 9 - 2/0.3`

...and now the first 3 terms are a perfect square, so we can rewrite as:

`y/0.3 = (x + 3)^2 - 9 - 2/0.3`

`y/0.3 = (x + 3)^2 - (9(0.3))/0.3 - 2/0.3`

`y/0.3 = (x+3)^2 - (2.7 + 2)/0.3`

`y/0.3 = (x+3)^2 - 4.7/0.3`

...and now we can multiply both sides by 0.3:

`y = 0.3(x+3)^2 - 4.7`

...and if the vertex form of the equation is:

`y = a(x-h)^2+ k`, then

your vertex coordinates are (-3,-4.7)

Now, your intercepts. For the y intercepts, just set x = 0, then calculate y:

`y = 0.3(0+3)^2 - 4.7 = 0.3 * 3^2 - 4.7 = 0.3 * 9 - 4.7 = 2.7 - 4.7 = -2`

So the curve intercepts the y axis at point (0,-2).

For x intercepts, set y = 0, and solve for x. You can use the original form of the equation for this, since it's a quadratic equation of form
`ax^2 + bx + c = 0` where `a` = 0.3, `b` = 1.8, and `c` = -2

Plug this into

`x = (-b +- sqrt(b^2 - 4ac))/2a`

and this gives you the x intercepts, which are the roots of the quadratic formula above. For `a` = 0.3, `b` = 1.8, and `c` = -2, these are -6.96 and 0.958

Your problem statement asks about the use of additional points to help plot the graph. Pick any other value of `x` you want, and plug that into the equation. For example, let `x = 1`:

`y = 0.3(4^2) - 4.7 = 0.3(16) - 4.7 = 4.8 - 4.7 = 0.1`

So now you can plot points:
(-3,-4.7) - the vertex
(-6.96,0) and (0.958) - the x intercepts
(0,-2) - the y intercept, and:
(1,0.1) - an additional point.

Connect the dots with a nice, swoopy parabola:
graph{0.3x^2 + 1.8x -2 [-10, 10, -5, 5]}

GOOD LUCK!