Problem 82 - College Algebra with Modeling and Visualization

Question

Give an example of a quadratic function that has only real zeros and an example of one that has only imaginary zeros. How do their graphs compare? Explain how to determine from a graph whether a quadratic function has real zeros.

Step-by-Step Solution

Answer

Quadratic with real zeros: \( x^2 - 4 \); imaginary zeros: \( x^2 + 4 \). Real zeros touch/intersect x-axis.

Step 1: Quadratic Function with Real Zeros

A quadratic function with real zeros has roots that can be found on the x-axis of the graph. An example is \( f(x) = x^2 - 4 \). This function can be factored as \( (x - 2)(x + 2) = 0 \), which gives real zeros at \( x = 2 \) and \( x = -2 \). The graph of this function is a parabola that intersects the x-axis at these points.

Step 2: Quadratic Function with Imaginary Zeros

A quadratic function with imaginary zeros does not intersect the x-axis. An example is \( g(x) = x^2 + 4 \). This function cannot be factored into real numbers, and solving \( x^2 + 4 = 0 \) gives imaginary zeros \( x = 2i \) and \( x = -2i \). The graph of this function is a parabola that lies entirely above the x-axis.

Step 3: Comparing Graphs

The graph of a quadratic function with real zeros crosses the x-axis at its zeros. In contrast, the graph of a quadratic function with imaginary zeros does not touch the x-axis at all. The vertex of a parabola with imaginary roots lies either completely above or below the x-axis, depending on the sign of the quadratic coefficient when the parabola opens upwards (or downwards).

Step 4: Determining Type of Zeros from a Graph

To determine the type of zeros a quadratic function has by looking at its graph, observe whether the parabola touches or intersects the x-axis. If it intersects the x-axis at one or two points, the function has real zeros. If it does not touch the x-axis, the zeros are imaginary.

Key Concepts

Real Zeros

In mathematics, real zeros of a quadratic function are the points where the graph of the function intersects the x-axis. This usually means that the solutions to the quadratic equation are real numbers. For instance, consider the quadratic function \( f(x) = x^2 - 4 \). In this case, the real zeros are found by setting the function equal to zero: \( x^2 - 4 = 0 \). Solving this, we get \( (x - 2)(x + 2) = 0 \), which gives the real zeros at \( x = 2 \) and \( x = -2 \). These are the points where the graph hits the x-axis, making it clear that the function has real zeros.

When a quadratic function has real zeros:

• The parabola of the function will intersect the x-axis at these zero points.
• The solutions to the equation are real numbers, possibly distinct or the same (if the vertex lies on the x-axis).
• The discriminant (\(b^2 - 4ac\)) of the quadratic formula will be positive or zero.

Imaginary Zeros

Imaginary zeros occur when the solutions to a quadratic equation involve imaginary numbers. This means the graph of the quadratic function does not cross or touch the x-axis at any point. For example, consider \( g(x) = x^2 + 4 \). Set this equal to zero: \( x^2 + 4 = 0 \). The equation cannot be factored over the real numbers as there are no real solutions for this equation. Solving for \( x \), we find \( x = 2i \) and \( x = -2i \)—these are imaginary zeros because they contain the imaginary unit \( i \).

In scenarios with imaginary zeros:

• The graph of the parabola lies completely above or below the x-axis, depending on its orientation.
• There is no intersection with the x-axis.
• The discriminant (\(b^2 - 4ac\)) is negative, indicating complex solutions.

Graph Interpretation

Visually interpreting the graph of a quadratic function helps determine the nature of its zeros. The key aspect to focus on is whether the parabola, which is the graphical representation of the quadratic function, makes contact with the x-axis.

Here's how to interpret the graph:

• If the parabola crosses the x-axis at one or more points, it has real zeros. The crossing points are the zeros.
• If the parabola does not touch the x-axis at any point, the quadratic function has imaginary zeros.
• The position of the vertex also gives insight; if the vertex is above or below the x-axis and the parabola does not intersect, then the zeros are imaginary.

Through graph interpretation, it becomes incredibly intuitive to distinguish between real and imaginary zeros. Just remember: x-axis interaction implies real zeros; none implies imaginary zeros. This approach provides a quick and efficient visual understanding of the nature of the quadratic's solutions.