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Lesson 149: Graphing Quadratic Equations (Parabolas)
Until now, we've been working with graphing linear equations. In this lesson we'll learn how to handle equations that have a squared term in them.
Recall that a quadratic equation is an equation of the form y = ax2 + bx + c, where a, b, and c are constants. The moment you see a squared term, you can be sure that the graph of the equation will not be a straight line. Equations that represent straight lines are always of degree 1. They do not have any squared terms.
The
graph of a quadratic equation will be a parabola.
In general, a parabola looks like the picture at left, which I made
using Microsoft Math 3.0. Depending on the equation, the
parabola might be wider or narrower than this one here, it might
have its turning point at a different coordinate,
or it might be upside down or sideways.
The study of quadratic equations is much easier if you have a graphing calculator to work with. You can alter the values of a, b, and c in the equation above, and see how it affects the parabola. In this lesson, we'll learn some tips that will tells us what the parabola will look like, based on the values of a, b, and c.
Note that an easy way of graphing a parabola is to pick values of x, and see what the corresponding y values are when you substitute them into the equation. It's usually good to pick x values that are positive, negative, and 0. Plot the corresponding points, and then connect them with a smooth curve like you see above. Don't connect the points using straight lines. Don't worry if your curve is perfect for now.
Every parabola has an axis of symmetry.
This is the vertical line which divides the parabola in half such
that one side is the mirror image of the other. The formula
for finding the x value of the axis of symmetry is: x = -b/2a.
For example, in the equation
y = 2x2 - 4x + 5, the axis of symmetry is -(-4)/4, which
is 1. Be very careful with signs. Once you know this
value of x, you can substitute it into the equation to find the
value of y. The coordinate that you find is the
turning point of the parabola. That is the point
where the parabola stops sloping down, and starts sloping up.
If there is no x term in the equation, then b will equal 0. That will mean that the x value of the axis of symmetry is 0. Stated another way, the axis of symmetry is actually the y axis.
Note that if value of a in the equation is positive, the parabola will open "upwards," like the one in the diagram. If the value of a is negative, the parabola will be upside-down, and open downwards. In this case, the turning point will actually be the highest point on the parabola, instead of the lowest point.
If there is no x term in the equation, then the value of c tells us something special. We know that the parabola's axis of symmetry is actually the y-axis. The value of c will tell us how much to translate the parabola up or down. For example, if c = 3, the turning point of the parabola will be (0, 3). If it is -2, it will be (0, -2).
Working with parabolas takes practice, and some experimentation. Try to use your graphing calculator as much as you can to practice. Remember, even if you forget all of the tips in this lesson, you can always graph your parabola by picking values of x to substitute in the equation, finding the corresponding values of y, and then connecting the points with a smooth curve.
Remember that you can ask a math question if you have additional questions about a topic, or you can contact me if you have any comments or suggestions for this site.