The vertex form for the equation of a parabola with vertical axis of
symmetry is
y = a(x - h)² + k
Be sure to memorize that equation, and the following facts about it:
Its vertex is the point (h,k)
Its axis of symmetry is the vertical line that passes through the vertex,
which has the equation x = h
It passes through the two points (h+1,k+a) and (h-1,k+a)
If a is positive, the parabola opens upward and its vertex is a minimum point.
If a is negative, the parabola opens downward and its vertex is a maximum point.
The y-intercept is found by substituting 0 for x and solving for y, It
will be the point (0,"that y-value")
The x-intercepts, (if any), are found by substituting 0 for y and solving
for x. They will be the points (r1, 0) and (r2, 0)
The minimum value is the number k if the vertex is a minimum point.
The maximum value is the number k if the vertex is a maximum point.
Its standard form is y = ax² + bx + c which is gotten from the vertex
form by multiplying it out, collecting like terms and placing it in
descending order.
Its factored form is y = a(x - r1)(x - r2) where
the r's stand for the x-values of the x-intercepts, if any. Not all
equations of parabolas have x-intercepts or a factored form.
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Your equation:
y = 
(x - 2)² - 3
is already in vertex form.
We compare it to
y = a(x - h)² + k
We see that a = , h = 2, and k = -3
So using the above facts:
Its vertex is the point (h,k) = (2,-3)
We plot that vertex point:
Its axis of symmetry is the vertical line that passes through the vertex,
which has the equation x = h or x = 2
So through that vertex we draw the axis of symmetry.
It passes through the two points (h+1,k+a) and (h-1,k+a).
h+1 = 2+1 = 3, k+a = -3+ = -2.25, the point (2,-2.25)
h-1 = 2-1 = 1, the point (1,-2.25)
We plot those two points
Since a is positive, the parabola opens upward and its vertex is a minimum point.
The y-intercept is found by substituting 0 for x and solving for y.
y = (x - 2)² - 3
y = (0 - 2)² - 3
y = (-2)² - 3
y = (4) - 3
y = 3 - 3
y = 0
So the y-intercept is the point (0,0), the origin. We plot that
and also another matching point on the other side of the axis of
symmetry, which is (4,0), We plot those:
Now we can draw in the parabola:
As it turns out in this particular problem, we have already found the
x intercepts, (0,0) and (4,0). In other problems we would have to
find them by setting y = 0. But we didn't need to here.
The minimum value is the number k or -3 because the vertex is a minimum point.
Its factored form is gotten from the vertex form by multiplying it out,
collecting like terms and placing it in descending order and factoring:
y = (x - 2)² - 3
Multiply through by 4
4y = 3(x - 2)² - 12
4y = 3(x - 2)(x - 2) - 12
4y = 3(x² - 4x + 4) - 12
4y = 3x² - 12x + 12 - 12
4y = 3x² - 12x
4y = 3x(x - 4)
y = x(x - 4)
if you think of the x factor as (x - 0) you can see it is
factored form y = a(x - r1)(x - r2) :
y = (x - 0)(x - 4)
but you can leave it
y = x(x - 4)
and that's the factored form.
The standard form is found by multiplying either of the forms out and collecting terms:
If we multiply the factored form out:
y = x(x - 4)
y = (x² - 4x)
y = x² - 3x
That is the standard form y = ax² + bx + c if you think of it as
y = x² - 3x + 0
But you can just leave it as
y = x² - 3x
and that will be considered the standard form
Edwin
