key term - Vertex Form

Definition

The vertex form of a quadratic equation is a way of expressing the equation in a specific format that highlights the vertex of the parabolic graph. The vertex form emphasizes the coordinates of the vertex, which are the point where the parabolic curve changes direction from increasing to decreasing or vice versa.

5 Must Know Facts For Your Next Test

1. The vertex form of a quadratic equation is $y = a(x - h)^2 + k$, where $(h, k)$ represents the coordinates of the vertex.
2. The vertex form is useful for identifying the vertex, which is the point where the parabolic graph changes direction from increasing to decreasing or vice versa.
3. Completing the square is a technique used to convert a quadratic equation from standard form to vertex form.
4. The quadratic formula can also be used to find the vertex of a parabolic graph by calculating the $x$-coordinate of the vertex and then substituting it into the equation to find the $y$-coordinate.
5. Transformations, such as translations and reflections, can be easily identified and applied to a quadratic function in vertex form.

Review Questions

• Explain how the vertex form of a quadratic equation is related to solving quadratic equations by completing the square.
  • The vertex form of a quadratic equation, $y = a(x - h)^2 + k$, is directly related to the process of solving quadratic equations by completing the square. To convert a quadratic equation from standard form to vertex form, you must complete the square by adding and subtracting a constant to isolate the $(x - h)^2$ term. This allows you to identify the coordinates of the vertex, $(h, k)$, which is the point where the parabolic graph changes direction.
• Describe how the vertex form of a quadratic equation can be used to graph the function using transformations.
  • The vertex form of a quadratic equation, $y = a(x - h)^2 + k$, provides the necessary information to graph the function using transformations. The $(h, k)$ coordinates represent the vertex of the parabolic graph, which can be used as the starting point for the transformation. The value of $a$ determines the stretch or compression of the graph, while the signs of $a$ and $k$ indicate the orientation of the parabola (opening upward or downward). By understanding how to apply these transformations to the basic parabolic graph, you can accurately sketch the graph of any quadratic function in vertex form.
• Analyze how the vertex form of a quadratic equation can be used to solve quadratic equations using the quadratic formula, and explain the connection between the vertex and the solutions.
  • The vertex form of a quadratic equation, $y = a(x - h)^2 + k$, can be used in conjunction with the quadratic formula to solve quadratic equations. The quadratic formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, can be used to find the $x$-coordinates of the solutions, which are the points where the parabolic graph intersects the $x$-axis. The $x$-coordinates of the solutions are directly related to the $x$-coordinate of the vertex, $h$, as the solutions are equidistant from the vertex. Furthermore, the $y$-coordinate of the vertex, $k$, represents the minimum or maximum value of the parabolic function, depending on the sign of $a$, which is also related to the solutions. By understanding the connections between the vertex form, the quadratic formula, and the solutions, you can more effectively solve and analyze quadratic equations.

Related terms

Standard Form:

The standard form of a quadratic equation is $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are coefficients that determine the shape and position of the parabolic graph.

Axis of Symmetry:

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

Transformations:

Transformations are changes made to the graph of a function, such as translations, reflections, and dilations, that can alter the shape, size, and position of the graph.