Concavity of Graphs
Concavity of graphs is a Grade 7 math skill from Yoshiwara Intermediate Algebra describing whether a curve bends upward (concave up) or downward (concave down). For quadratics and parabolas, the sign of the leading coefficient a determines concavity, while in calculus this is linked to the second derivative.
Key Concepts
Property A graph that bends upward is called concave up , and one that bends down is concave down .
If a graph is concave up, its slopes are increasing. This describes a rate of change that is speeding up.
If a graph is concave down, its slopes are decreasing. This describes a rate of change that is slowing down.
Common Questions
What is concavity of a graph?
Concavity describes the direction a curve bends. A graph is concave up if it bends like a bowl (opens up) and concave down if it bends like an arch (opens down).
How do you determine concavity from a quadratic equation?
In y = ax^2 + bx + c, if a > 0 the parabola is concave up (minimum), and if a < 0 it is concave down (maximum).
What is an inflection point?
An inflection point is where a curve changes from concave up to concave down (or vice versa). Parabolas do not have inflection points.
Why does concavity matter?
Concavity tells you whether a critical point is a minimum or maximum, which is crucial for optimization problems.