Concave Down

Definition

Concave down is a term used to describe the shape of a quadratic function when its graph curves downward, forming a bowl-like shape. This type of function has a negative leading coefficient, indicating that the function decreases as the input variable increases.

5 Must Know Facts For Your Next Test

1. A quadratic function with a negative leading coefficient will have a concave down parabolic graph.
2. The vertex of a concave down parabola is the point where the function reaches its maximum value.
3. Concave down quadratic functions have a minimum value, which occurs at the vertex of the parabola.
4. The shape of a concave down parabola is determined by the sign and magnitude of the leading coefficient, $a$, in the quadratic function $f(x) = ax^2 + bx + c$.
5. Concave down quadratic functions are often used to model situations where a quantity reaches a maximum and then decreases, such as the trajectory of a projectile or the cost of producing a good.

Review Questions

• Explain how the sign of the leading coefficient in a quadratic function determines whether the graph is concave up or concave down.
  • The sign of the leading coefficient $a$ in the quadratic function $f(x) = ax^2 + bx + c$ determines the orientation of the parabolic graph. If $a$ is positive, the graph is concave up, opening upward. If $a$ is negative, the graph is concave down, opening downward. This is because the term $ax^2$ dominates the function as $x$ increases, causing the graph to curve in the direction of the sign of $a$.
• Describe the key features of a concave down quadratic function, including the location of the vertex and the minimum value of the function.
  • A concave down quadratic function has a negative leading coefficient, $a < 0$, which means the graph opens downward, forming a bowl-like shape. The vertex of the parabola is the point where the function reaches its maximum value, and it is the point where the graph changes direction from decreasing to increasing. The vertex is the point of symmetry for the parabola, and the minimum value of the function occurs at the vertex. The shape and properties of a concave down quadratic function are determined by the values of the coefficients $a$, $b$, and $c$ in the function $f(x) = ax^2 + bx + c$.
• Explain how concave down quadratic functions can be used to model real-world situations, and provide an example of such a situation.
  • Concave down quadratic functions are often used to model situations where a quantity reaches a maximum and then decreases. For example, the trajectory of a projectile, such as a ball thrown in the air, can be modeled using a concave down quadratic function. The function $h(t) = -16t^2 + vt + h_0$, where $h$ is the height of the projectile, $t$ is the time, $v$ is the initial velocity, and $h_0$ is the initial height, represents a concave down parabola. This model can be used to predict the maximum height reached by the projectile, as well as the time at which it reaches that maximum height, which corresponds to the vertex of the parabola.

Related terms

Quadratic Function: A quadratic function is a polynomial function of the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are real numbers, and $a \neq 0$.

Parabola: A parabola is the graph of a quadratic function, which can be either concave up or concave down depending on the sign of the leading coefficient.

Vertex: The vertex of a parabola is the point where the graph changes direction, either from decreasing to increasing or from increasing to decreasing.