Problem 32 - Algebra 2

Question

Rewrite each equation in vertex form. $$ y=2 x^{2}-8 x+1 $$

Step-by-Step Solution

Answer

\( y = 2(x - 2)^2 - 7 \)

Step 1: Identify the coefficients in standard form

In the standard form of a quadratic equation, which is given by \( y = ax^2 + bx + c \), identify the coefficients. For this equation, \( a = 2 \), \( b = -8 \), and \( c = 1 \).

Step 2: Find the vertex \( x \)-coordinate

Use the formula \( h = -\frac{b}{2a} \) to calculate the x-coordinate of the vertex. Substitute \( a = 2 \) and \( b = -8 \) into the formula to get \( h = -\frac{-8}{2 \cdot 2} = \frac{8}{4} = 2 \).

Step 3: Find the vertex \( y \)-coordinate

Plug the x-coordinate of the vertex back into the original equation to find the y-coordinate of the vertex. Thus, \( k = y(2) = 2(2)^2 - 8(2) + 1 = 8 - 16 + 1 = -7 \).

Step 4: Rewrite in vertex form

With the vertex found as \( (h, k) = (2, -7) \), write the equation in vertex form, which is \( y = a(x - h)^2 + k \). Substituting \( a = 2 \), \( h = 2 \), and \( k = -7 \) gives \( y = 2(x - 2)^2 - 7 \).

Key Concepts

Quadratic Equations

Quadratic equations hold a fundamental place in algebra, serving as a bridge to more advanced topics in mathematics. Characteristically, a quadratic equation is any equation that can be rearranged into the standard form y = ax^2 + bx + c, where a, b, and c are constants, and a ≠ 0. These equations are represented graphically by parabolas, which open upwards if a > 0 and downwards if a < 0.

The roots or solutions to these equations are the x-values where the parabola intersects the x-axis. These points are critical in understanding the behavior of the quadratic function. Moreover, the y-intercept, which is the value of c, tells us where the parabola crosses the y-axis. Quadratic equations are used in various areas of science and engineering to describe curves, trajectories, and optimization problems.

Completing the Square

The method of completing the square transforms a quadratic equation into vertex form to easily identify the coordinates of the parabola's vertex. This process involves creating a perfect square trinomial from the quadratic expression, allowing us to express it as (x-h)^2 + k, where (h,k) is the vertex of the parabola. The general steps include:

• Dividing the equation by a if it’s not 1.
• Rearranging the equation to leave space for completing the square.
• Adding and subtracting (inside the parentheses) the square of half the coefficient of x, which turns the expression into a perfect square trinomial.
• Simplifying the equation to achieve the vertex form.

For example, to complete the square for 2x^2 - 8x, we take half of -8, which is -4, and square it to get 16. We add and subtract 16 within the equation to maintain equality and obtain the form (x-4)^2 -16. With some adjustments, this gives us the vertex form.

Standard Form of a Quadratic

The standard form of a quadratic equation, represented as y = ax^2 + bx + c, provides a useful structure for quadratic functions, giving immediate visibility into certain characteristics. Here, a determines the width and direction of the parabola, b affects the location of the vertex and axis of symmetry, and c establishes the y-intercept. While this form is convenient for quickly identifying these properties and solving the equation by factoring or using the quadratic formula, it doesn’t immediately reveal the vertex. That’s where converting to vertex form comes into play, offering insight into the parabola's peak and aiding in graphing and solving problems related to maxima and minima.

Vertex of a Parabola

The vertex of a parabola is a paramount feature that represents the highest or lowest point on the graph, depending on whether the parabola opens upward or downward. In the context of a quadratic function, the vertex form y = a(x-h)^2 + k explicitly provides the vertex as the point (h, k).

Calculating the vertex coordinates directly from standard form involves using the formula h = -b/(2a) for the x-coordinate and substituting this back into the original equation to find the y-coordinate. This vertex acts as a turning point for the parabola and signifies the axis of symmetry, which is a vertical line that passes through the vertex and divides the parabola into two symmetrical halves. Understanding the vertex helps in analyzing the graph and forecasting the behavior of the quadratic function for real-world applications.