B. Direction of Opening and y-intercept

Glossary Checklist FAQ

B. Direction of Opening and y-intercept

Variables a, b, and c

It is useful to examine the impact the coefficient of each term has on the graph of a quadratic function.

Open the Quadratic Function applet. Move the sliders to change the \(a\), \(b\), and \(c\) values. Try changing each value, while leaving the other two constant to determine the effect of each individual parameter. The table below may help you to organize your thinking.

Parameter	Effect on the Graph of \(f(x) = ax^2 + bx + c, a \ne 0 \)
Changing \(a\)
Changing \(b\)
Changing \(c\)

Using the Quadratic Function applet, you may have noticed that parameters \(a\), \(b\), and \(c \) all affected the graph differently. You may have noticed the following patterns:

Parameter	Effect on the Graph of \(f(x) = ax^2 + bx + c, a \ne 0 \)
Changing \(a\)	Changes how wide or narrow the parabola is. When \(a > 0\), the parabola opens up, and when \(a < 0\), the parabola opens down.
Changing \(b\)	Changes the location of the graph both vertically and horizontally.
Changing \(c\)	Changes the location of the graph vertically. Changes the \(y\)-intercept of the graph.

Conclusion

The direction of opening can be determined by simply looking at the sign of \(a\). The \(y\)-intercept can also be easily determined by simply looking at \(c\). Unfortunately, most other characteristics of the graph are difficult to determine directly from the standard form of a quadratic function. The following table summarizes the effect of parameters \(a\) and \(c\) on the graph of a quadratic function.