Algebra Know-It-ALL

400 Graphs of Quadratic Functions

Are you confused?

By now, you might ask, “How can we can tell whether the parabola for a particular quadratic function opens

upward or downward?” This is easy to figure out. If the coefficient of x^2 in the polynomial is larger than 0

(that is, if a > 0 in the general form of the function), then the parabola opens upward. If a < 0, then the parab-

ola opens downward. If a= 0, then we don’t have a quadratic function at all, and its graph is not a parabola.

Finding the absolute minimum or maximum

When we know the x-intercepts of a quadratic function with two real zeros, it’s easy to find

thex-value of the absolute minimum or maximum of its graph. It’s the arithmetic mean, or

average, of the zeros r and s.

If a > 0 in the polynomial, then the parabola opens upward, and we have an absolute

minimum somewhere. Let’s call its x-value xmin. Then

xmin= (r+s)/2

If a < 0, then the parabola opens downward, and it has an absolute maximum. If we call its

x-value xmax, then again

xmax= (r+s)/2

To find the y-value of the absolute minimum or maximum, we can plug the x-value into the

function once we’ve found it. A little arithmetic will give us the y-value. Then we can plot the

point on a coordinate grid.

When we have plotted the two x-intercepts along with the extremum, we can draw a fair

approximation of the parabola representing the quadratic function.

Here’s a challenge!

Consider the following quadratic function:

y=x^2 − 3 x+ 2

Determine whether the parabola for this function opens upward or downward. Then find the two real

zeros, r and s. After that, find the x-value of the extremum. Then determine its y-value. Finally, plot the

zeros and the absolute minimum or maximum, and draw an approximate curve through these three points

that represents the graph of the function.

Solution

The coefficient of x^2 is positive, so we know that the parabola opens upward. We’re lucky here because the

polynomial equation factors into

(x− 1)(x− 2) = 0