key term - Vertex Form

Definition

The vertex form of a quadratic function is a way of expressing the function that highlights the vertex, which is the point on the graph where the function changes direction. This form is particularly useful for understanding the behavior and properties of parabolic curves.

5 Must Know Facts For Your Next Test

1. The vertex form of a quadratic function is $y = a(x - h)^2 + k$, where $(h, k)$ is the coordinates of the vertex.
2. The vertex form directly provides the coordinates of the vertex, which is useful for analyzing the behavior of the function.
3. The vertex form can be used to easily determine the axis of symmetry, which is the vertical line $x = h$.
4. The vertex form can be used to identify the maximum or minimum value of the function, which occurs at the vertex.
5. Transforming a quadratic function from standard form to vertex form can be done by completing the square.

Review Questions

• Explain how the vertex form of a quadratic function is different from the standard form, and why the vertex form is particularly useful.
  • The vertex form of a quadratic function, $y = a(x - h)^2 + k$, is different from the standard form, $y = ax^2 + bx + c$, in that it directly provides the coordinates of the vertex, $(h, k)$. This is particularly useful because the vertex represents the point where the function changes direction, either from decreasing to increasing or from increasing to decreasing. The vertex form allows you to easily identify the maximum or minimum value of the function, as well as the axis of symmetry, which is the vertical line $x = h$. This information is crucial for understanding the behavior and properties of parabolic curves.
• Describe the process of transforming a quadratic function from standard form to vertex form by completing the square.
  • To transform a quadratic function from standard form, $y = ax^2 + bx + c$, to vertex form, $y = a(x - h)^2 + k$, you can use the process of completing the square. First, you need to isolate the $x^2$ term by dividing the entire equation by the coefficient $a$. Then, you add the square of half the coefficient of the $x$ term, $(b/2)^2$, to both sides of the equation. This completes the square for the $x$ terms, allowing you to factor the left side into the form $a(x - h)^2$. Finally, you can identify the coordinates of the vertex, $(h, k)$, where $h = -b/(2a)$ and $k = c - b^2/(4a)$.
• Explain how the vertex form of a quadratic function can be used to analyze the behavior and properties of a parabolic curve, such as the maximum or minimum value, the axis of symmetry, and the direction of the curve.
  • The vertex form of a quadratic function, $y = a(x - h)^2 + k$, provides a wealth of information about the behavior and properties of the corresponding parabolic curve. The vertex, $(h, k)$, represents the point where the function changes direction, either from decreasing to increasing or from increasing to decreasing. This allows you to easily identify the maximum or minimum value of the function, which occurs at the vertex. The axis of symmetry, the vertical line $x = h$, can also be readily determined from the vertex form. Additionally, the sign of the coefficient $a$ indicates the direction of the parabolic curve, with $a > 0$ resulting in an upward-opening parabola and $a < 0$ resulting in a downward-opening parabola. By understanding these key features of the vertex form, you can gain valuable insights into the properties and behavior of the parabolic curve.

Related terms

Standard Form:

The standard form of a quadratic function is $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

Axis of Symmetry:

The axis of symmetry is the vertical line that passes through the vertex of a parabolic curve, dividing the curve into two equal halves.

Vertex:

The vertex is the point on the graph of a parabolic curve where the function changes direction, either from decreasing to increasing or from increasing to decreasing.