For what values of a does y = ax^2 - 8x + 4 have no x-intercepts?

Verified step by step guidance

1. To determine when the quadratic equation \( y = ax^2 - 8x + 4 \) has no x-intercepts, we need to analyze the discriminant of the quadratic formula.
2. The quadratic formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). The discriminant is the part under the square root: \( b^2 - 4ac \).
3. For the quadratic to have no x-intercepts, the discriminant must be less than zero: \( b^2 - 4ac < 0 \).
4. In our equation, \( a = a \), \( b = -8 \), and \( c = 4 \). Substitute these into the discriminant: \( (-8)^2 - 4(a)(4) < 0 \).
5. Simplify the inequality: \( 64 - 16a < 0 \). Solve for \( a \) to find the range of values where the quadratic has no x-intercepts.

Key Concepts

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form y = ax^2 + bx + c. The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of the coefficient 'a'. Understanding the general shape and properties of parabolas is essential for analyzing their intercepts.

Discriminant

The discriminant of a quadratic equation, given by the formula D = b^2 - 4ac, determines the nature of the roots of the equation. If D > 0, there are two distinct real roots; if D = 0, there is one real root (the vertex touches the x-axis); and if D < 0, there are no real roots, indicating that the parabola does not intersect the x-axis.

X-Intercepts

X-intercepts are the points where a graph crosses the x-axis, which occur when y = 0. For a quadratic function, finding the x-intercepts involves solving the equation ax^2 + bx + c = 0. The absence of x-intercepts indicates that the quadratic function remains entirely above or below the x-axis, which is directly related to the sign of the discriminant.