Quadratic Functions

Definition

Quadratic functions are a type of polynomial function where the highest exponent of the independent variable is 2. They are characterized by a U-shaped graph and can be used to model a variety of real-world phenomena, such as the motion of projectiles, the growth of populations, and the optimization of economic processes.

5 Must Know Facts For Your Next Test

1. Quadratic functions can be represented in the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.
2. The constant $a$ determines the direction and steepness of the parabola, with $a > 0$ resulting in an upward-opening parabola and $a < 0$ resulting in a downward-opening parabola.
3. The vertex of a quadratic function can be found using the formula $x = -b/(2a)$, and the y-coordinate of the vertex can be found by substituting this value into the original function.
4. Quadratic functions can have up to two real roots, which are the x-intercepts of the graph. These roots can be found using the quadratic formula: $x = (-b \pm \sqrt{b^2 - 4ac})/(2a)$.
5. Quadratic functions are often used to model optimization problems, where the goal is to find the maximum or minimum value of the function.

Review Questions

• Explain how the value of the constant $a$ in the equation $f(x) = ax^2 + bx + c$ affects the shape and orientation of the parabolic graph.
  • The value of the constant $a$ in the equation $f(x) = ax^2 + bx + c$ determines the direction and steepness of the parabolic graph. When $a > 0$, the parabola opens upward, and when $a < 0$, the parabola opens downward. The magnitude of $a$ affects the steepness of the parabola, with larger absolute values of $a$ resulting in a more narrow and steep parabola.
• Describe the process of finding the vertex of a quadratic function and explain its significance in understanding the behavior of the function.
  • The vertex of a quadratic function is the point on the graph where the function changes from decreasing to increasing, or vice versa. To find the vertex, we can use the formula $x = -b/(2a)$, where $a$ and $b$ are the coefficients in the equation $f(x) = ax^2 + bx + c$. The x-coordinate of the vertex represents the point of maximum or minimum value of the function, which is crucial for understanding the function's behavior and potential applications, such as in optimization problems.
• Explain how the quadratic formula can be used to determine the roots of a quadratic function and discuss the significance of these roots in the context of the function's graph.
  • The roots of a quadratic function are the x-intercepts of the parabolic graph, which can be found using the quadratic formula: $x = (-b \pm \sqrt{b^2 - 4ac})/(2a)$. These roots represent the values of $x$ where the function equals zero, and they are important for understanding the behavior of the function. The number and nature of the roots (real, complex, or repeated) can provide insights into the function's shape, symmetry, and potential applications. For example, the number of real roots can determine whether the function has a maximum, minimum, or neither, which is crucial in optimization problems.

Related terms

Parabola: The graph of a quadratic function, which has a characteristic U-shape.

Vertex: The point on the graph of a quadratic function where the function changes from decreasing to increasing, or vice versa.

Axis of Symmetry: The vertical line that passes through the vertex of a parabola, dividing the graph into two symmetric halves.