Question

Without drawing the graph of the equation, determine how many points the parabola has in common with the x-axis and whether its vertex lies above, on, or below the x-axis.

y = 3x2 – 12x + 12
How do I do this?

Answer 1

if you factor out that expression, you'll see that the factor \(\large (x-2) \) has mulitplicity of 2 which means the graph will only touch the x-axis only at x=2.

Answer 2

To determine how many points the parabola has in common find the discriminant.
To find the discriminant use
\[ b^2 - 4ac \]
Now just plugin are values
\[ -12^2 - 4(3)(12) = 0\]
So our discriminant equals 0. Now that our discriminant is equal to 0 we can determine how man point the parabola has in common.

If discriminant < 0 then there will be 0 points
If discriminant > 0 then there will be 2 points
If discriminant = 0 then there will be 1 point

Now to fine the whether its vertex lies above, on, or below the x-axis we find the x coordinate of the vertex by using \[ x = -\frac{b}{2a}\]
Now just plugin your number for the formula above.
\[x = -\frac{-12}{2(3)} \]
\[x = -\frac{-12}{2(3)} \]
\[x =\frac{12}{6} \]
\[x =2\]
Now we know the x cord of our vertex so we can find y which will tell us if the vertex lies above, on, or below the x-axis.

Just plugin 2 for x in the equation y = 3x2 – 12x + 12
\[ y = 3(2)^{2} – 12(2) + 12 \]
\[ y = 12 – 24 + 12 \]
\[ y = 0 \]
Since y = 0 the vertex lies on the x-axis

Answer 3

That means it's : 1 point in common; vertex on x-axis
Right?