Problem 32 - A Graphical Approach to College Algebra

Question

It is not apparent from the standard viewing window whether the graph of the quadratic function intersects the \(x\) -axis once, twice, or not at all. Experiment with various windows to find the number of \(x\) -intercepts. If there are \(x\) -intercepts, give their coordinates to the nearest hundredth. (GRAPH CANT COPY) $$y=-2 x^{2}+0.2 x-0.15$$

Step-by-Step Solution

Answer

No real x-intercepts.

Step 1: Analyze the Quadratic Function

The quadratic function provided is \( y = -2x^2 + 0.2x - 0.15 \). It is a downward opening parabola because the coefficient of \( x^2 \) is negative.

Step 2: Calculate the Discriminant

To determine the number of \( x \)-intercepts, calculate the discriminant \( \Delta \) using the formula \( \Delta = b^2 - 4ac \), where \( a = -2 \), \( b = 0.2 \), and \( c = -0.15 \).

Step 3: Perform the Discriminant Calculation

Substitute the values into the discriminant formula: \[ \Delta = (0.2)^2 - 4(-2)(-0.15) = 0.04 - 1.2 = -1.16 \]

Step 4: Interpret the Discriminant

Since \( \Delta < 0 \), this means that the quadratic function has no real \( x \)-intercepts. A negative discriminant indicates that the parabola does not intersect the \( x \)-axis.

Key Concepts

Understanding the Discriminant

In the context of quadratic functions, the discriminant plays a crucial role in determining the nature of the roots. For a given quadratic equation of the form \( ax^2 + bx + c = 0 \), the discriminant is calculated using the formula \( \Delta = b^2 - 4ac \). This expression helps us understand how the quadratic equation will behave in relation to the \( x \)-axis.

Here's what the value of the discriminant tells us about the quadratic function:

• If \( \Delta > 0 \), there are two distinct real solutions or \( x \)-intercepts. This means the parabola intersects the \( x \)-axis at two points.
• If \( \Delta = 0 \), there is exactly one real solution or \( x \)-intercept. Here, the parabola touches the \( x \)-axis at exactly one point, known as a double root.
• If \( \Delta < 0 \), as in our exercise, there are no real solutions, indicating that the parabola does not intersect the \( x \)-axis at all.

The discriminant is a powerful tool as it quickly tells us about the intersection between the parabola and the \( x \)-axis without having to graph the function.

Finding the X-intercepts

The \( x \)-intercepts of a quadratic function are points where the parabola crosses the \( x \)-axis. These are also known as the roots or solutions of the quadratic equation. To find these \( x \)-intercepts, we typically solve the equation \( ax^2 + bx + c = 0 \).

However, calculation of the \( x \)-intercepts involves evaluating the discriminant first. Depending on the discriminant's value:

• If \( \Delta > 0 \), apply the quadratic formula \( x = \frac{-b \pm \sqrt{\Delta}}{2a} \) to find the two real and distinct \( x \)-intercepts.
• If \( \Delta = 0 \), use the quadratic formula to find the single repeated \( x \)-intercept, where the square root term becomes zero.
• If \( \Delta < 0 \), as in our example \( y = -2x^2 + 0.2x - 0.15 \), no real solutions mean there are no \( x \)-intercepts to find on the real number line.

Analyzing the \( x \)-intercepts is essential for understanding the function's behavior and visualizing its graph in the coordinate plane.

Characteristics of a Parabola

A parabola is the graphical representation of a quadratic function. Understanding its features provides insight into the function’s behavior and characteristics. The standard form of a quadratic function is \( y = ax^2 + bx + c \).

The parabola opens either upward or downward depending on the coefficient \( a \):

• If \( a > 0 \), the parabola opens upwards, resembling a U-shape.
• If \( a < 0 \), as in the given equation \( y = -2x^2 + 0.2x - 0.15 \), the parabola opens downwards, forming an upside-down U-shape.

Some key features of a parabola include:

• Vertex: The highest or lowest point, depending on the direction of opening, offers a way to find the axis of symmetry.
• Axis of Symmetry: A vertical line passing through the vertex that divides the parabola into two symmetrical halves.
• X-intercepts and Y-intercept: Points where the parabola crosses the \( x \)-axis and \( y \)-axis respectively.

In this particular exercise, the negative discriminant indicates the parabola does not cross the \( x \)-axis, confirming its nature of having no real \( x \)-intercepts.