In Exercises 19–22, find the quadratic function y = ax^2+bx+c whose graph passes through the given points. (−1, 6), (1, 4), (2, 9)

Verified step by step guidance

1. Step 1: Start by substituting each point into the quadratic equation y = ax^2 + bx + c to create a system of equations. For the point (-1, 6), substitute x = -1 and y = 6 to get the equation: 6 = a(-1)^2 + b(-1) + c.
2. Step 2: Substitute the second point (1, 4) into the equation y = ax^2 + bx + c. This gives you the equation: 4 = a(1)^2 + b(1) + c.
3. Step 3: Substitute the third point (2, 9) into the equation y = ax^2 + bx + c. This results in the equation: 9 = a(2)^2 + b(2) + c.
4. Step 4: You now have a system of three equations: 6 = a - b + c, 4 = a + b + c, and 9 = 4a + 2b + c. Solve this system of equations using methods such as substitution or elimination to find the values of a, b, and c.
5. Step 5: Once you have the values of a, b, and c, substitute them back into the quadratic equation y = ax^2 + bx + c to get the specific quadratic function that passes through the given points.

Key Concepts

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form y = ax^2 + bx + c, where a, b, and c are constants. The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of 'a'. Understanding the general form is essential for identifying the coefficients that will satisfy the conditions given by specific points.

Systems of Equations

To find the specific quadratic function that passes through given points, we set up a system of equations based on the coordinates of those points. Each point (x, y) provides an equation when substituted into the quadratic formula. Solving this system allows us to determine the values of a, b, and c that define the unique quadratic function.

Substitution Method

The substitution method is a technique used to solve systems of equations by expressing one variable in terms of another and substituting it into the other equations. In the context of finding a quadratic function, this method can simplify the process of solving for the coefficients by reducing the number of variables in the equations, making it easier to isolate and calculate the values of a, b, and c.