Question

Suppose the equation \(a x^2+b x+c=0\) has no real solution and a graph of the related function has a vertex that lies in the second quadrant. a. Is the value of \(a\) positive or negative? Explain your reasoning. b. Suppose the graph is translated so the vertex is in the fourth quadrant. Does the graph have any \(x\)-intercepts? Explain.

Short answer

a) The value of \(a\) is negative because the vertex of the parabola is in the second quadrant, which means the parabola opens downwards. b) No, the graph does not have any x-intercepts. This is because after the graph is translated to the fourth quadrant, the parabola still opens downwards and thus remains below the x-axis.

Step-by-step solution

Identify the condition for imaginary roots

The quadratic equation \(a x^2+b x+c=0\) is said to have no real solution when the discriminant, \(b^2 - 4ac\), is less than zero. This is because the square root of a negative number is not a real number.

Determine the sign of \(a\)

For a quadratic equation, when the vertex of the graph lies in the second quadrant, the parabola opens downwards indicating \(a\) is negative. This is because in the second quadrant, \(x\) is negative but \(y\) is positive, and for the maximum point to be obtained, the parabola has to face downwards.

Understand the effect of translation

When the parabola is translated so that its vertex is in the fourth quadrant, it still retains its original shape and direction (downwards). This is because the translation only affects the position of the graph, not its shape or direction.

Analyze the existence of \(x\)-intercepts

In the fourth quadrant, the graph of the parabola remains below the x-axis as it still opens downwards. Thus, the graph doesn't cross the x-axis, which means it doesn't have any x-intercepts.

Key concepts

Understanding the Discriminant

The discriminant is a crucial part of a quadratic equation, helping us determine the nature of the roots. In a quadratic equation like \( ax^2 + bx + c = 0 \), the discriminant is given by \( b^2 - 4ac \). This value tells us whether the roots of the equation are real or imaginary.

• If the discriminant is positive, the equation has two distinct real roots.
• If it is zero, there is exactly one real root, also known as a repeated or double root.
• When the discriminant is less than zero, the equation has no real roots; instead, it has two complex or imaginary roots.

Imaginary roots occur because it is not possible to take the square root of a negative number in the set of real numbers. Therefore, a negative discriminant indicates that the quadratic equation's parabola graph, if plotted, does not cross the x-axis.

Exploring Imaginary Roots

Imaginary roots are the solutions of a quadratic equation when the discriminant is less than zero. These roots aren't found on the real number line but as part of complex numbers involving the imaginary unit \( i \), where \( i^2 = -1 \).

For the quadratic equation \( ax^2 + bx + c = 0 \) with a negative discriminant, the roots can be calculated using the quadratic formula:
\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\]
Since \( b^2 - 4ac \) is negative, the square root part renders the solution imaginary. These roots show up as \( A \pm Bi \), where \( A \) and \( B \) are real numbers. Though they don't mark x-intercepts, imaginary roots are vital for understanding the full scope of a parabola's behavior.

Parabola Translation and Its Effects

Translation of a parabola involves shifting its position in the Cartesian plane without altering its shape or direction. When we translate a parabola, we are effectively moving the entire graph horizontally, vertically, or both, by changing its vertex's position.

In this context, moving the vertex from the second to the fourth quadrant requires shifting the graph both downwards and possibly to the right. However, it's important to remember:

• The direction of the parabola (upward or downward opening) remains unchanged.
• The general shape stays the same without any stretching or compression.

These translations only relocate the vertex, altering where the parabola appears on the graph. Even with a new position, its interaction with the x-axis or lack thereof is dictated by the original characteristics of the quadratic equation, including whether the roots were real or imaginary.

Vertex of a Parabola in Quadrants

The vertex of a parabola, which is its highest or lowest point, provides pivotal information about the parabola's placement and orientation. This point can lie in any of the four quadrants on a graph, based on the values of the quadratic equation’s coefficients.

When a vertex is in the second quadrant, this typically means:

• The parabola opens downwards, since the vertex is a maximum point, signifying a negative \( a \).
• The vertex has negative x-coordinates and positive y-coordinates.

On translating this vertex to the fourth quadrant, the coordinates shift such that the x-value is positive and the y-value is negative. Despite this move, if the parabola already had no real x-intercepts because of a negative discriminant, it will remain so. The translation affects position, but not how the graph behaves in relation to the x-axis.