Let’s learn some vocabulary for
quadratic functions. First, vertex. What is a vertex of a quadratic function? Well,
the word vertex means peak. So the vertex of a quadratic function is the peak point
on a parabola. Which means for a parabola opening down, the vertex is the highest
point. Now for a parabola opening up, the vertex is the lowest point. So in this
case, we have a parabola opening up, so the vertex is the lowest point, right
here. Sitting at x equals 1, y equals negative 9. So the vertex is
at 1 and negative 9. Next, axis of symmetry. The axis of symmetry of a
quadratic function, is a vertical line that divides the parabola into two equal
halves, like this. And the axis of symmetry must always pass through the
vertex. The symmetry behavior can be best understood when we fold the parabola in
half along the axis of symmetry. Then the right half of the parabola will completely
overlap the left half of the parabola. Can you see that? Let me show you. Here we
have a parabola, just like this one, and the axis of symmetry runs down the middle
of a the parabola. Now, I’m going to fold the graph along the axis of symmetry.
Can you see that? The two halves of the parabola completely overlap each other.
Yeah? Cool. Now let’s determine the equation of the axis of symmetry. For a
vertical line that passes through x equals 1, the equation of the line is
simply x equals 1. Because for every point on this vertical line, the x
coordinate is always 1. So in this case, the equation of the axis of symmetry is x
equals 1. Next, y intercept. The y intercept is the point where the parabola
intersects the y axis. So in this case, that’s right here and the y intercept is
negative 8. Next, x intercepts. The x intercepts are the points where the
parabola intersects the x axis. So in this case, the parabola intersects the x axis
at two points. One at x equals negative 2 another one at x equals 4.
Next, domain. By definition, domain is the set of x values for which a function is
defined. So as we can see, a parabola is defined for all x values. Let me show you.
This point, the x value is positive 1. This point x value is positive 2. This
point x value is positive 3. This point x value is positive 4, and this
point x value is positive 5. So a parabola is defined for all positive integers for
x. Now if you take any points in between, now see, these points the x values are
positive decimal numbers, right? Yeah. So a parabola can take on any positive values
for x, whether they are integers or whether they are decimal numbers. Parabola
can take on any positive numbers for x. Now, guess what? Parabola can also take on
any negative values for x. For example, this point x value is negative 1.
This point x value is negative 2. This point x value is negative 3. You
see, negative integers. Now, for any point in between them, for any points between
the negative integers, this point the x values are all negative decimal numbers.
So in conclusion, parabola can take on any positive values for x. Parabola can also
take on any negative values for x and parabola can even take on zero for x.
That would be this point, for the y intercept, the x value is zero, right?
So therefore, the domain for a quadratic function is all real numbers. And we write
a mathematical notation like this, x is all real numbers. This symbol means “is”
and this symbol here, represents, see this symbol here is an R with an extra bar,
represents “all real numbers.” In fact, the domain for any quadratic function is
always x is all real numbers. Next, range. By definition, range is the
set of y values for which a function is defined. So here, the parabola only goes
as low as right here, where the y value is negative 9. However, you will go as
high it likes and therefore we say the range . . . Range talks about y right? So
the range is y must be negative 9 and anything above. So the range is y is
greater than or equal to negative 9. Does this parabola have a minimum value or
a maximum value? Well first of all, in math, whenever we talk about the value of
a function, we are always referring to the y value. Remember that. Value of a
function always refers to the y value. Now, since the parabola can go as high as
it likes, there’s no maximum value. However, parabola can only go as low as
right here, where the y value is negative 9. Therefore, we say this parabola
has a minimum value of negative 9.