Graphing
Quadratic Functions: The
Leading Coefficient
/ The Vertex (page
2 of 4)

Sections: Introduction,
The meaning of the leading coefficient / The vertex, Examples

The general form of a quadratic
is "y = ax^{2} + bx + c".
For graphing, the leading coefficient "a"
indicates how "fat" or how "skinny" the parabola will
be.

For | a | > 1 (such
as a = 3 or a = –4), the parabola
will be "skinny", because it grows more quickly (three times
as fast or four times as fast, respectively, in the case of our sample
values
of a).

For | a | < 1 (such
as a = ^{1}/_{3} or a = ^{–1}/_{4 }),
the parabola will be "fat", because it grows more slowly (one-third
as fast or one-fourth as fast, respectively, in the examples). Also, if a is negative, then the parabola is upside-down.

You can see these
trends when you look at how the curve y = ax^{2} moves as "a"
changes:

This can be useful information:
If, for instance, you have an equation where a is negative, but you're somehow coming up with plot points that make it
look like the quadratic is right-side-up, then you will know that you
need to go back and check your work, because something is wrong.

Parabolas always have a
lowest point (or a highest point, if the parabola is upside-down). This
point, where the parabola changes direction, is called the "vertex".

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If the quadratic is written
in the form y = a(x – h)^{2} + k, then
the vertex is the point (h, k). This makes
sense, if you think about it. The squared part is always positive (for
a right-side-up parabola), unless it's zero. So you'll always have that
fixed value k,
and then you'll always be adding something to it to make y bigger, unless of course the squared part is zero. So the smallest y can possibly be is y = k,
and this smallest value will happen when the squared part, x – h, equals
zero. And the squared part is zero when x – h = 0, or
when x = h. The same
reasoning works, with k being the largest value and the squared part always subtracting from it,
for upside-down parabolas.

(Note: The "a"
in the vertex form "y = a(x – h)^{2} + k"
of the quadratic is the same as the "a"
in the common form of the quadratic equation, "y = ax^{2} + bx + c".)

Since the vertex is a useful
point, and since you can "read off" the coordinates for the
vertex from the vertex form of the quadratic, you can see where the vertex
form of the quadratic can be helpful, especially if the vertex isn't one
of your T-chart values. However, quadratics are not usually written in
vertex form. You can complete
the square to convert ax^{2} + bx + c to vertex form, but, for finding the vertex, it's simpler to just use a formula. (The vertex formula
is derived from the completing-the-square process, just as is the Quadratic
Formula. In each case, memorization is probably simpler than completing
the square.)

For a given quadratic y = ax^{2} + bx + c,
the vertex (h, k) is found
by computing h = ^{–b}/_{2a},
and then evaluating y at h to find k.
If you've already learned the Quadratic
Formula, you may
find it easy to memorize the formula for k,
since it is related to both the formula for h and the discriminant in the Quadratic Formula: k = (4ac – b^{2})
/ 4a.

Find the vertex
of y = 3x^{2} + x – 2 and graph the parabola.

To find the vertex, I
look at the coefficients a, b,
and c.
The formula for the vertex gives me:

h = ^{–b}/_{2a} = ^{–(1)}/_{2(3)} = ^{–1}/_{6}

Then I can find k by evaluating y at h = ^{–1}/_{6}:

k = 3( ^{–1}/_{6 })^{2} + ( ^{–1}/_{6 }) – 2

= ^{3}/_{36} – ^{1}/_{6} – 2

= ^{1}/_{12} – ^{2}/_{12}
– ^{24}/_{12}

= ^{–25}/_{12}

So now I know that the
vertex is at ( ^{–1}/_{6} , ^{–25}/_{12} ). Using
the formula was helpful, because this point is not one that I was likely
to get on my T-chart.

I need additional
points for my graph:

Now I can do my
graph, and I will label the vertex:

When you write down the
vertex in your homework, write down the exact coordinates: "( ^{–1}/_{6} , ^{–25}/_{12} )".
But for graphing purposes, the decimal approximation of "(–0.2,
–2.1)" may be
more helpful, since it's easier to locate on the axes.

The only other consideration
regarding the vertex is the "axis of symmetry". If you look
at a parabola, you'll notice that you could draw a vertical line right
up through the middle which would split the parabola into two mirrored
halves. This vertical line, right through the vertex, is called the axis
of symmetry. If you're asked for the axis, write down the line "x = h", where h is
just the x-coordinate
of the vertex. So in the example above, then the axis would be the vertical
line x = h = ^{–1}/_{6}.

Helpful note: If your quadratic's x-intercepts
happen to be nice neat numbers (so they're relatively easy to work with),
a shortcut for finding the axis of symmetry is to note that this vertical
line is always exactly between the two x-intercepts.
So you can just average the two intercepts to get the location of the
axis of symmetry and the x-coordinate
of the vertex. However, if you have messy x-intercepts
(as in the example above) or if the quadratic doesn't actually cross the x-axis
(as you'll see on the next page), then you'll need to use the formula
to find the vertex.

Stapel, Elizabeth.
"Graphing Quadratic Functions: The Leading Coefficient / The Vertex." Purplemath. Available from https://www.purplemath.com/modules/grphquad2.htm.
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