Problem 33 - Algebra 2

Question

Write each function in vertex form. $$ y=2 x^{2}-5 x+12 $$

Step-by-Step Solution

Answer

\(y = 2(x - \frac{5}{4})^2 + \frac{71}{8}\)

Step 1: Understand Vertex Form

Vertex form of a quadratic equation is given by: \(y = a(x-h)^2 + k\) where \((h, k)\) is the vertex of the parabola. The coefficient \(a\) gives the direction and width of the parabola.

Step 2: Identify Coefficient 'a'

Here, the given function is \(y = 2x^{2}-5x+12\). The coefficient of \(x^{2}\) is 2, so \(a = 2\).

Step 3: Complete the Square

To write the function in vertex form, we need to complete the square for the \(x\) terms. First, factor out the coefficient \(a\) from the \(x^2\) and \(x\) terms: \(y = 2(x^{2} - \frac{5}{2}x) + 12\). Then, add and subtract the square of half the coefficient of \(x\) inside the parentheses: \(y = 2(x^{2} - \frac{5}{2}x + (\frac{5}{4})^2) - 2(\frac{5}{4})^2 + 12\).

Step 4: Simplify and Write in Vertex Form

Simplify by adding the constants outside the parentheses and complete the binomial square: \(y = 2(x - \frac{5}{4})^2 + 12 - 2(\frac{5}{4})^2\). Calculate the constant term inside \((...)\) which results in: \(y = 2(x - \frac{5}{4})^2 + 12 - \frac{25}{8}\), and finally simplify to get the vertex form: \(y = 2(x - \frac{5}{4})^2 + \frac{71}{8}\).

Key Concepts

Completing the Square

Completing the square is a method used to rewrite quadratic equations in a form that reveals the vertex of the corresponding parabola. It involves creating a perfect square trinomial from the quadratic terms, which allows for easier analysis and graphing of the quadratic function.

Here's how you can complete the square:

• Start with a quadratic in the form of \(y = ax^2 + bx + c\).
• If \(a\) is not 1, factor it out from the \(x\)-terms.
• Take half the coefficient of \(x\) (which is \(b\) when \(a = 1\)), square it, and add it inside the parentheses.
• Add the same number outside the parentheses but with a negative sign, to keep the equation balanced.
• Rewrite the equation now with a perfect square trinomial and combine like terms if necessary.

Through this process, a quadratic equation such as \(y = 2x^2 - 5x + 12\) is transformed into its vertex form, which not only identifies the peak or trough of the graph but also simplifies the process of graphing the quadratic relation.

Quadratic Equations

Quadratic equations are mathematical expressions of the second degree, generally presented in the standard form \(y = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). The solutions to a quadratic equation are the values of \(x\) that make the equation equal to zero. These solutions are also known as the 'roots' or 'zeroes' of the equation.

There are various methods to solve quadratic equations:

• Factoring the quadratic into two binomials, if possible.
• Using the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
• Completing the square to solve for \(x\) directly or to put the equation in vertex form, which can highlight the solutions.

Understanding quadratic equations is crucial not just for solving for \(x\) but also for graphing the parabola and finding its properties, such as the vertex, axis of symmetry, and intercepts.

Parabola Transformation

Parabola transformation refers to shifting the graph of a quadratic function, which is a parabola, horizontally and/or vertically, due to changes in its equation. A parabola can be described in vertex form as \(y = a(x-h)^2 + k\), where the vertex is located at the point \((h, k)\).

Here are some key transformations:

• When \(h\) is positive, the graph shifts \(h\) units to the right; when it's negative, the graph shifts to the left.
• When \(k\) is positive, the graph shifts \(k\) units up; when it's negative, the graph shifts down.
• The coefficient \(a\) affects the width and direction of the parabola: if \(a > 0\), the parabola opens upwards, and if \(a < 0\), it opens downwards. The greater the absolute value of \(a\), the narrower the parabola.

By manipulating these parameters, you can easily move and reshape the parabola on a coordinate plane, promoting a profound understanding of how the quadratic function behaves in various situations.