Question

State the circumstances that cause the graph of a quadratic function \(f(x)=a x^{2}+b x+c\) to have no \(x\) -intercepts.

Short answer

The graph of the quadratic function has no x-intercepts when its discriminant is negative ( )

Step-by-step solution

Title - Identify the Quadratic Formula

The quadratic formula is used to determine the roots (or x-intercepts) of the quadratic function. The formula is:

Title - Understand Discriminant

The discriminant of a quadratic function, given by

Title - Know the Conditions

To have no x-intercepts, the roots of the quadratic equation do not exist within the real number system. This occurs when the discriminant is negative. Thus, if

Conclusion

Therefore, the graph of a quadratic function

Key concepts

quadratic formula

The quadratic formula is a powerful tool for solving quadratic equations. It helps to find the roots or x-intercepts of a quadratic function. The general form of a quadratic equation is: \(ax^2 + bx + c = 0\).
To find the roots, we use the quadratic formula: \x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\. This equation provides the solutions for x in terms of the coefficients a, b, and c.
The \(\pm\) symbol means there are usually two roots: one with a positive square root and one with a negative square root.
These roots represent the points where the parabola (graph of the quadratic function) crosses the x-axis.

discriminant

The discriminant is a key part of the quadratic formula. It is found under the square root sign in the quadratic formula: \(b^2 - 4ac\).
The value of the discriminant helps us determine the nature of the roots of the quadratic equation.
The discriminant can be positive, zero, or negative:

• If the discriminant is positive (\(b^2 - 4ac > 0\)), the quadratic equation has two distinct real roots. This means the graph intersects the x-axis at two different points.
• If the discriminant is zero (\(b^2 - 4ac = 0\)), the quadratic equation has exactly one real root. In this case, the graph touches the x-axis at exactly one point (also called a repeated root or double root).
• If the discriminant is negative (\(b^2 - 4ac < 0\)), the quadratic equation has no real roots. This indicates that the graph does not intersect the x-axis at all; the parabola is either entirely above or entirely below the x-axis.

roots of quadratic equation

The roots of a quadratic equation are the solutions obtained through the quadratic formula. These roots, often referred to as x-intercepts or zeros, represent the points where the quadratic function crosses or touches the x-axis.
In context of the discriminant:

• When \(b^2 - 4ac > 0\), the quadratic equation has two distinct real roots, and thus the graph has two x-intercepts.
• When \(b^2 - 4ac = 0\), the quadratic equation has one real root, and the graph has one x-intercept (vertex touches the x-axis).
• When \(b^2 - 4ac < 0\), there are no real roots, which means the graph of the quadratic function has no x-intercepts.

Understanding these roots' nature helps in graphing and analyzing quadratic functions effectively. The values of a, b, and c define the precise location and number of these intercepts on the x-axis.