key term - Quadratic Graphs

Definition

Quadratic graphs are a type of mathematical function that are represented by a parabolic curve. They are characterized by a U-shaped or inverted U-shaped curve and are described by the general equation $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

5 Must Know Facts For Your Next Test

1. Quadratic graphs are used to model a wide range of physical quantities, such as projectile motion, the relationship between force and acceleration, and the behavior of electrical circuits.
2. The shape of a quadratic graph can be determined by the sign of the coefficient $a$: if $a$ is positive, the graph opens upwards (a U-shape), and if $a$ is negative, the graph opens downwards (an inverted U-shape).
3. The vertex of a quadratic graph represents the point of maximum or minimum value, which can be found by using the formula $x = -b/(2a)$.
4. Quadratic graphs can be transformed by changing the values of the coefficients $a$, $b$, and $c$, which affects the position, orientation, and shape of the parabola.
5. The roots or solutions of a quadratic equation are the $x$-intercepts of the corresponding quadratic graph, which can be found using the quadratic formula: $x = (-b \pm \sqrt{b^2 - 4ac})/(2a)$.

Review Questions

• Explain how the coefficient $a$ in the equation $y = ax^2 + bx + c$ affects the shape and orientation of a quadratic graph.
  • The coefficient $a$ in the equation $y = ax^2 + bx + c$ determines the shape and orientation of the quadratic graph. If $a$ is positive, the graph opens upwards, forming a U-shape. If $a$ is negative, the graph opens downwards, forming an inverted U-shape. The magnitude of $a$ also affects the steepness of the parabola, with larger absolute values of $a$ resulting in a more sharply curved graph.
• Describe the relationship between the vertex of a quadratic graph and the coefficients $a$, $b$, and $c$.
  • The vertex of a quadratic graph represents the point of maximum or minimum value of the function. The coordinates of the vertex can be found using the formula $x = -b/(2a)$ and $y = a(-b)^2/(4a) + c$. The value of $a$ determines the orientation of the vertex (upwards or downwards), while the values of $b$ and $c$ affect the position of the vertex along the $x$- and $y$-axes, respectively. Understanding the relationship between the vertex and the coefficients is crucial for analyzing and interpreting the behavior of quadratic functions.
• Explain how the roots or solutions of a quadratic equation are related to the $x$-intercepts of the corresponding quadratic graph.
  • The roots or solutions of a quadratic equation are the values of $x$ that make the equation equal to zero. These roots correspond to the $x$-intercepts of the quadratic graph, which are the points where the graph crosses the $x$-axis. The $x$-intercepts can be found by solving the quadratic equation using the quadratic formula: $x = (-b \pm \sqrt{b^2 - 4ac})/(2a)$. The number and nature of the roots (real, complex, or repeated) determine the number and location of the $x$-intercepts on the graph, which is crucial for understanding the behavior and properties of quadratic functions.

Related terms

Parabola: A parabola is the curved path traced by a quadratic function, which has the characteristic U-shape or inverted U-shape.

Vertex: The vertex is the point on the quadratic graph where the curve changes direction, representing the minimum or maximum value of the function.

Axis of Symmetry: The axis of symmetry is the vertical line that passes through the vertex of a quadratic graph, dividing the parabola into two equal halves.