Questions: Rachel sketches the graph of a quadratic function that never crosses the x-axis. Which statement is TRUE concerning the roots of the function? A. The function has two real roots. B. The function has two imaginary roots. C. The function has no real or imaginary roots. D. The function has one imaginary root

Rachel sketches the graph of a quadratic function that never crosses the x-axis. Which statement is TRUE concerning the roots of the function? A. The function has two real roots. B. The function has two imaginary roots. C. The function has no real or imaginary roots. D. The function has one imaginary root

Transcript text: Rachel sketches the graph of a quadratic function that never crosses the $x$-axis. Which statement is TRUE concerning the foots of the function? A. The function has two real roots. B. The function has two imaginary roots. C. The function has no real or imaginary roots. D. The function has one imaginary root

Solution

Solution Steps

To determine the nature of the roots of a quadratic function that never crosses the x-axis, we need to consider the discriminant of the quadratic equation. The discriminant is part of the quadratic formula and is given by $b^2 - 4ac$. If the discriminant is less than zero, the quadratic has two complex (imaginary) roots. Since the graph never crosses the x-axis, the discriminant must be negative.

Step 1: Identify the Quadratic Function

We are given a quadratic function of the form $ f(x) = ax^2 + bx + c $. For our analysis, we can use the coefficients $ a = 1 $, $ b = 0 $, and $ c = 1 $.

Step 2: Calculate the Discriminant

The discriminant $ D $ of a quadratic function is calculated using the formula: \[ D = b^2 - 4ac \] Substituting the values: \[ D = 0^2 - 4 \cdot 1 \cdot 1 = -4 \]

Step 3: Determine the Nature of the Roots

Since the discriminant $ D = -4 < 0 $, this indicates that the quadratic function has two complex (imaginary) roots. Therefore, the function does not intersect the x-axis.