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A polynomial function is an expression involving a sum of powers in one or more variables multiplied by coefficients. The highest power of the variable is called the degree of the polynomial. A quadratic polynomial function specifically has the form \( f(x) = ax^2 + bx + c \). This means it is a second-degree polynomial because the highest power is 2.

In the expression \( ax^2 + bx + c \), each term involves a constant coefficient, like \( a \), \( b \), and \( c \). These coefficients are essential because they dictate the shape and position of the parabola that represents the quadratic function.

• \( a \), \( b \), and \( c \) are real numbers.
• If \( a eq 0 \), the function is truly quadratic.
• If \( a = 0 \), the function becomes linear.

Understanding quadratic polynomial functions is crucial in algebra as they appear in various scenarios, from physics to engineering.

The graph of a polynomial function provides a visual representation of all its possible solutions. This is essentially plotting the function on a coordinate plane.
The most important aspect of the graph of a quadratic function, such as the one in our exercise, is that it forms a parabola. Parabolas have unique features:

• A symmetric axis, often vertical along the line \( x = -\frac{b}{2a} \).
• A vertex, which is the highest or lowest point, depending on whether the parabola opens upwards (\( a > 0 \)) or downwards (\( a < 0 \)).

For the given quadratic function \( f(x) = 3x^2 + 2x - 1 \), the graph passes through the points (1,4), (2,15), and (3,32). Plotting these points shows how the function increases or decreases, illustrating the relationship between the x-values and the y-values determined by the function. Knowing how to read and interpret the graph is invaluable for solving equations and analyzing functions.

A system of equations is a set of equations with multiple variables. In solving these, the goal is to find a common solution that satisfies all of the equations in the system.
In our exercise, from using the points (1, 4), (2, 15), and (3, 32), we form a system of three equations:

• \( a + b + c = 4 \)
• \( 4a + 2b + c = 15 \)
• \( 9a + 3b + c = 32 \)

The coefficients of \( a \), \( b \), and \( c \) in these equations come from substituting the known x-values into the general quadratic equation form. These equations must be solved concurrently, and common methods such as substitution or elimination simplify the process. A system of equations allows us to solve for more than one unknown variable, providing us with the necessary coefficients \( a \), \( b \), and \( c \) for our quadratic function.

The substitution method is a reliable strategy for solving a system of equations. It involves solving one of the equations for one variable and then substituting that expression into another equation.
Here's how it works with a system of equations from the exercise:

• Solve one equation for one variable. For instance, solving \( 2a = 6 \) gives \( a = 3 \).
• Substitute \( a = 3 \) into another equation such as \( 3a + b = 11 \), resulting in \( b = 2 \).
• Use these values to solve for the remaining variable using another equation, for example, \( a + b + c = 4 \) gives \( c = -1 \).

By substituting known values back into other equations, you systematically reduce the unknowns and reach a solution. The substitution method is particularly effective when one equation is already solved for a variable or can be easily manipulated. This method provides a step-by-step approach that builds upon each substitution, making it easier to follow and understand.