Problem 6 - Intermediate Algebra

Question

The equation \(y=5 x^{2}-6 x+1\) is written in the form \(y=a x^{2}+b x+c .\) What are \(a, b,\) and \(c ?\)

Step-by-Step Solution

Answer

The values are \(a = 5\), \(b = -6\), and \(c = 1\).

Step 1: Identify the Standard Form

The standard form of a quadratic equation is given by \( y = ax^2 + bx + c \). Our task is to compare this form with the given equation to identify the coefficients \(a\), \(b\), and \(c\).

Step 2: Match the Coefficients

The given equation is \( y = 5x^2 - 6x + 1 \). By comparing it with the standard form \( y = ax^2 + bx + c \), we can directly read off the values: \( a = 5 \), \( b = -6 \), and \( c = 1 \).

Step 3: Verify the Coefficients

To ensure accuracy, verify the identified values by checking if substituting them back into the standard form recovers the original equation. Substituting \( a = 5 \), \( b = -6 \), and \( c = 1 \) yields \( y = 5x^2 - 6x + 1 \), which matches the given equation.

Key Concepts

Coefficients Identification

In a quadratic equation of the form \( y = ax^2 + bx + c \), the letters \(a\), \(b\), and \(c\) are known as coefficients. Understanding these coefficients is crucial because they determine the shape and position of the parabola represented by the equation.

• The coefficient \(a\) is tied directly to the \(x^2\) term and it influences the direction the parabola opens. If \(a\) is positive, the parabola opens upwards. Conversely, if \(a\) is negative, it opens downwards.
• Coefficient \(b\), associated with the \(x\) term, affects the line of symmetry and position of the vertex horizontally along the x-axis.
• The constant term \(c\) identifies the y-intercept, the point where the parabola crosses the y-axis.

For the given equation \(y=5x^2-6x+1\), identifying these coefficients involves linking each term in the equation with its counterpart in the standard form. By isolating the terms, you find\(a = 5\), \(b = -6\), and \(c = 1\). It is essential to correctly recognize these coefficients to understand the parabola's properties.

Standard Form

The standard form of a quadratic equation is a crucial representation and follows the structure \( y = ax^2 + bx + c \). It's essential to comprehend this form because it provides a straightforward method to identify and work with quadratic equations.

• The \(x^2\) term is always accompanied by its coefficient \(a\). Regardless of its value, \(a\) should not be zero, or else the equation will not remain quadratic.
• The \(x\) term is followed by its coefficient \(b\), which is essential for understanding how the quadratic function shifts horizontally.
• Finally, \(c\) is a constant that provides a vertical translation of the graph along the y-axis.

To transform an equation into this standard form, rearrange or simplify its terms to fit \( y = ax^2 + bx + c \). In the exercise, the equation \(y=5x^2-6x+1\) fits seamlessly into this form, demonstrating it is already in standard structure. This allows easy extraction of coefficients and further mathematical analysis.

Comparing Equations

When solving quadratic equations, comparing them to the standard form \( y = ax^2 + bx + c \) can simplify the complex process of identifying key components. This approach involves aligning each term from your equation to the respective term in the standard form.

• Start by writing both equations in a similar format. For the problem at hand, both are written as \(y = 5x^2 - 6x + 1\) and \(y = ax^2 + bx + c\) respectively.
• Compare each position in both equations specifically: the \(x^2\) terms, the \(x\) terms, and the constant terms. This comparison directly tells you what \(a\), \(b\), and \(c\) are equivalent to.
• Ensuring proper alignment allows you to see at a glance, like in our exercise, that \(a = 5\), \(b = -6\), and \(c = 1\). Using this method aids in quick identification and verification of the coefficients.

This comparison strategy is a practical way to ensure calculations are correct and can serve as a conversational checkpoint when explaining the rationale behind identified values.