Nonlinear Systems of Equations

Front

What is a nonlinear system of equations?

Back

A nonlinear system of equations is a set of equations in which at least one equation is not linear, meaning it cannot be represented as a straight line on a graph.

Front

What is the general form of a quadratic equation?

Back

The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0.

Front

How do you determine the number of solutions in a linear-quadratic system?

Back

The number of solutions can be determined by finding the points of intersection between the linear and quadratic equations, which can be 0, 1, or 2 solutions.

Front

What does it mean if a system has no solutions?

Back

A system has no solutions when the graphs of the equations do not intersect, indicating that there are no common points.

Front

What is the vertex of a parabola?

Back

The vertex of a parabola is the highest or lowest point on the graph, depending on whether it opens upwards or downwards.

Front

How can you find the intersection points of a linear and a quadratic equation?

Back

To find the intersection points, set the equations equal to each other and solve for the variable.

Front

What is the significance of the discriminant in a quadratic equation?

Back

The discriminant (b² - 4ac) determines the nature of the roots: if it's positive, there are 2 real solutions; if zero, 1 real solution; if negative, no real solutions.

Front

What is the standard form of a quadratic function?

Back

The standard form of a quadratic function is f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola.

Front

What does it mean for a quadratic function to be concave up or concave down?

Back

A quadratic function is concave up if it opens upwards (a > 0) and concave down if it opens downwards (a < 0).

Front

How do you graph a quadratic function?

Back

To graph a quadratic function, find the vertex, axis of symmetry, and intercepts, then plot points and draw a smooth curve.