Concave Down

Definition

Concave down refers to a curve or function that is curved downward, with the ends of the curve pointing downward. This shape is the opposite of a concave up function, which curves upward.

5 Must Know Facts For Your Next Test

1. A concave down quadratic function has a negative leading coefficient ($a < 0$), which means the function opens downward.
2. The vertex of a concave down quadratic function represents the maximum value of the function.
3. Concave down quadratic functions have a parabolic shape that is symmetric about the vertical line passing through the vertex.
4. The graph of a concave down quadratic function is a downward-facing parabola.
5. Concave down quadratic functions can be used to model real-world situations, such as the path of a projectile or the cost of producing a good.

Review Questions

• Explain how the sign of the leading coefficient ($a$) in a quadratic function determines whether the function is concave up or concave down.
  • The sign of the leading coefficient $a$ in a quadratic function $f(x) = ax^2 + bx + c$ determines the orientation of the parabolic curve. If $a > 0$, the function is concave up, meaning the curve opens upward. If $a < 0$, the function is concave down, meaning the curve opens downward. This is because the sign of $a$ determines the direction of the curvature of the parabola, with a positive $a$ resulting in an upward-facing parabola and a negative $a$ resulting in a downward-facing parabola.
• Describe the relationship between the vertex of a concave down quadratic function and the function's maximum value.
  • For a concave down quadratic function, the vertex represents the maximum value of the function. This is because the vertex is the point where the function changes from increasing to decreasing, and for a concave down function, this point corresponds to the highest point on the parabolic curve. The $x$-coordinate of the vertex indicates the input value that produces the maximum output value of the function, while the $y$-coordinate of the vertex represents the maximum value itself.
• Discuss how the symmetry of a concave down quadratic function relates to its graph and the location of the vertex.
  • Concave down quadratic functions exhibit symmetry about the vertical line passing through the vertex. This means that the left and right halves of the parabolic curve are mirror images of each other. The vertex, being the point of symmetry, represents the midpoint of the parabola. This symmetry is a consequence of the quadratic function's mathematical properties, specifically the fact that the function can be written in the form $f(x) = a(x - h)^2 + k$, where $(h, k)$ represents the coordinates of the vertex. The symmetry of the graph allows for the easy identification of the vertex and the function's maximum value.

Related terms

Quadratic Function: A quadratic function is a polynomial function of degree two, often written in the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

Vertex:

The vertex of a quadratic function is the point where the function changes from increasing to decreasing or vice versa. It represents the maximum or minimum value of the function.

Parabola:

A parabola is the graph of a quadratic function, which has a characteristic U-shaped curve.