Question

REASONING A nonlinear system contains the equations of a constant function and a quadratic function. The system has one solution. Describe the relationship between the graphs.

Short answer

When a system has a constant function and a quadratic function and they share one solution, it means the graphs intersect at exactly one point: the vertex of the parabolic graph of the quadratic function. The line of the constant function lies either completely above or below the parabola depending on whether the parabola opens upwards or downwards.

Step-by-step solution

Understanding the functions

A constant function has the form \(y = c\), where \(c\) is a constant and a quadratic function has the form \(y = ax^2 + bx + c\). For there to be one solution, it means that there is only one point where the constant function and the quadratic function intersect.

Visualizing the relationship

The graph of a quadratic function is parabolic. When a system of equations containing a quadratic function and a constant function has only one solution, it means that the vertex of the parabola lies on the line of the constant function. This is because the parabola either opens upwards or downwards while a constant function is a straight line. If the parabolic graph opens upwards, the vertex is the point on the parabola that is the lowest, so it must touch the constant function line at this point only. Similarly, if the parabolic graph opens downwards, the vertex is the point on the parabola that is the highest.

Describing the relationship

In conclusion, when the system of equations containing a constant function and a quadratic function has one solution, the parabolic graph of the quadratic function must touch the line of the constant function exactly at the vertex of the parabola. If the parabola opens upwards, the line of the constant function lies above the parabola except for the one point that is the vertex. If the parabola opens downwards, the line of the constant function lies below the parabola except for the one point that is the vertex.

Key concepts

Constant Function

A constant function is one of the simplest types of functions in mathematics. It is represented by the equation \(y = c\), where \(c\) is a constant number. This means, no matter what value \(x\) takes, \(y\) will always be equal to \(c\). The graph of a constant function is a horizontal line. It stretches infinitely in both directions along the \(x\)-axis but stays at the same value on the \(y\)-axis.

• For example, if the constant function is \(y = 3\), it is a line crossing the \(y\)-axis at 3, remaining parallel to the \(x\)-axis.
• This simple nature makes it crucial when considering its intersection with other types of functions like quadratic functions.

Quadratic Function

A quadratic function is a type of polynomial function with the general form \(y = ax^2 + bx + c\). It is characterized by its highest exponent being 2, often forming a parabolic curve when graphed. Here are the critical features of quadratic functions:

• The coefficient \(a\) determines the direction of the parabola. If \(a > 0\), the parabola opens upwards. If \(a < 0\), it opens downwards.
• The parabolic shape arises due to the presence of the square term \(x^2\).
• The graph of a quadratic function is symmetrical around its vertex, which is the highest or lowest point depending on the direction in which the parabola opens.

Graph Intersection

The point where two graphs meet or touch is known as the intersection. For a nonlinear system containing a quadratic and a constant function, the intersection point contributes to the solution of the system. Key elements about graph intersection:

• If there is one point of intersection between these functions, it means the line representing the constant function tangentially touches the parabola at that point.
• This single solution occurs precisely when the vertex of the parabola is at the same level as the constant function, making them intersect at one point.
• If there are zero or two points of intersection, it suggests a different number of solutions for the system, but in this scenario, there is exactly one.

Vertex of a Parabola

The vertex of a parabola is a crucial point on its graph. It represents the peak or trough, depending on the direction the parabola opens. The vertex will have different characteristics based on the parabola's direction:

• If the parabola opens upwards, the vertex is at the lowest point.
• If it opens downwards, the vertex is at the highest point.

Given that a parabola opens according to its leading coefficient, the vertex formula is a useful tool. The vertex \((h, k)\) can be found using the formula: \[h = \frac{-b}{2a} \k = f(h) = a(h)^2 + bh + c\]In the context of intersecting with a constant function, the vertex's \(y\)-value would match the constant value \(c\) for the intersection to be possible.