Problem 1 - Algebra 1

Question

Identify the values of \(a, b,\) and \(c\) for the quadratic function in standard form \(y=-5 x^{2}+7 x-4\)

Step-by-Step Solution

Answer

For the quadratic function \(y=-5 x^{2}+7 x-4\), the values of \(a, b,\) and \(c\) are -5, 7, and -4 respectively.

Step 1: Identify the Coefficient of \(x^{2}\)

The coefficient of \(x^{2}\) in the quadratic function \(y=-5 x^{2}+7 x-4\) is -5. This is the value of \(a\), the first variable in the standard form \(y = ax^{2} + bx + c\). So, \(a = -5\).

Step 2: Identify the Coefficient of \(x\)

The coefficient of \(x\) in the quadratic function \(y=-5 x^{2}+7 x-4\) is 7. This is the value of \(b\), the second variable in the standard form \(y = ax^{2} + bx + c\). Hence, \(b = 7\).

Step 3: Identify the Constant Term

The constant term in the quadratic function \(y=-5 x^{2}+7 x-4\) is -4. This corresponds to \(c\), the third variable in the standard form \(y = ax^{2} + bx + c\). Thus, \(c = -4\).

Key Concepts

Standard Form

The standard form of a quadratic function is a very useful way to write quadratic equations. It looks like this: \(y = ax^2 + bx + c\). In this form, you can easily identify the different parts of the quadratic equation:

• The term \(ax^2\) is the quadratic term, which determines the parabola's direction—whether it opens upwards or downwards.
• The term \(bx\) is the linear term, influencing the slope and position of the symmetry axis of the parabola.
• The term \(c\) is the constant term, which shifts the graph vertically, changing the y-intercept.

By writing functions in this standard form, you can easily read off the important values that influence the shape and position of a parabola in a graph. For example, in the equation \(y = -5x^2 + 7x - 4\), it is straightforward to identify that \(a = -5\), \(b = 7\), and \(c = -4\). These constants are key to graphing and shifting quadratic functions.

Coefficients

Coefficients are numbers used to multiply variables. In a quadratic function like \(y = ax^2 + bx + c\), the coefficients are the numbers that are attached to the variable terms. Understanding these coefficients is crucial because they give you a lot of information about the function's graph.

• Coefficient \(a\): This coefficient affects the direction and width of the parabola. If \(a\) is positive, the parabola opens upwards. If \(a\) is negative, it opens downwards. The larger the absolute value of \(a\), the narrower the parabola; the smaller it is, the wider the parabola.
• Coefficient \(b\): This coefficient affects the position of the vertex and determines how shifted the parabola is along the x-axis.
• Coefficient \(c\): This number is the constant term and represents the point where the parabola intersects the y-axis, called the y-intercept.

Thus, identifying coefficients helps to translate the equation into real-world geometry, which is essential for graphing and solving quadratic equations accurately.

Quadratic Equation

A quadratic equation is a second-degree polynomial equation and is typically in the form \(ax^2 + bx + c = 0\). These equations are fundamental because they represent parabolas in mathematics and can be used to model various real-world situations, such as projectile motion or optimizing areas. Here's what you need to know about them:

• The "quadratic" term comes from "quad," meaning square. This term \(x^2\) indicates that you will find squares in the equation.
• Quadratic equations can have 0, 1, or 2 real solutions. Finding these solutions often involve methods like factoring, completing the square, or using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
• They graph as parabolic shapes, with their vertex representing the maximum or minimum value depending on whether the parabola opens upwards or downwards.

Understanding quadratic equations goes beyond just solving for \(x\). It's about analyzing how changing \(a\), \(b\), and \(c\) can alter its graph and behavior in applied contexts.