Question

Write each function in vertex form. $$ y=-2 x^{2}+2 x+5 $$

   Write each function in vertex form.
$$
y=-2 x^{2}+2 x+5
$$

Algebra 2

Prentice Hall, Basia… 1st Edition

Chapter 5, Problem 86

Instant Answer

Solved by Expert Amrita Bhasin

08/08/2019

Step 1

Step 1: First, we factor out the coefficient of the x^2 term from the first two terms: $$ y=-2(x^{2}-x)+5 $$ Show more…

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Write each function in vertex form. $$ y=-2 x^{2}+2 x+5 $$

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Key Concepts

Vertex Form

Vertex form represents a quadratic function in a way that clearly shows the coordinates of the vertex (h, k) and the vertical stretch/compression factor a. Writing the function in this form simplifies the process of graphing the quadratic function, as the vertex of the parabola can be directly read from the expression.

Quadratic Functions

A quadratic function is a type of polynomial function of degree two, typically expressed in a standard form as ax² + bx + c. It produces a U-shaped graph known as a parabola, which can open upward or downward depending on the sign of the coefficient a. Understanding quadratic functions is crucial for analyzing the behavior of parabolic graphs, including determining their maximum or minimum values.

Standard Form and Vertex Form

The standard form (ax² + bx + c) and the vertex form (a(x - h)² + k) are two ways of expressing quadratic functions. While the standard form is useful for identifying the coefficients affecting the shape of the parabola, the vertex form makes it easier to determine the vertex of the parabola, which represents the maximum or minimum point of the function.

Completing the Square

Completing the square is a mathematical technique used to transform a quadratic function from its standard form to vertex form. By manipulating the quadratic and linear terms into a perfect square trinomial, this method reveals the vertex and allows for easier analysis and graphing of the parabola.

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