Problem 84 - A Graphical Approach to College Algebra

Question

Sketch a graph of \(f(x)=a x^{2}+b x+c\) that satisfies each set of conditions. $$a>0, b^{2}-4 a c<0$$

Step-by-Step Solution

Answer

The graph of \(f(x)=ax^2+bx+c\) is an upward-opening parabola that lies entirely above the x-axis.

Step 1: Understanding the Quadratic Function Structure

The function given is a quadratic function of the form \[ f(x) = ax^2 + bx + c \].In this expression, \(a\), \(b\), and \(c\) are constants. The coefficient \(a\) determines the direction of the parabola, with it opening upwards when \(a > 0\). The quadratic formula \(b^2 - 4ac\) is known as the discriminant and helps determine the nature of the roots of the quadratic equation.

Step 2: Analyzing the Condition \(a > 0\)

The condition \(a > 0\) indicates that the parabola opens upwards. Therefore, the vertex of the parabola is the lowest point on the graph. This implies that the graph will start from the top and curve downwards on either side of the vertex, rising infinitely as \(x\) moves away from the vertex in both directions.

Step 3: Analyzing the Condition \(b^2 - 4ac < 0\)

The discriminant \(b^2 - 4ac < 0\) implies that the quadratic equation has no real roots. In terms of the graph, this means the parabola does not intersect the x-axis at any point. The entire graph of the parabola lies either above or below the x-axis, with the vertex being above it since \(a > 0\).

Step 4: Combining the Conditions to Sketch the Graph

Given that \(a > 0\) and \(b^2 - 4ac < 0\), sketch the graph starting with an upward-opening parabola that does not intersect the x-axis.1. Identify the vertex of the parabola using the vertex formula \[ x = -\frac{b}{2a} \] and substitute in \(f(x)\) to find the vertex's y-coordinate.2. Plot the vertex and draw the symmetry axis.3. Ensure the parabola stays entirely above the x-axis without intersecting it.

Key Concepts

Parabola Graphing

Graphing a parabola involves plotting a specific kind of curve formed by a quadratic function. The general form of a quadratic function is \( f(x) = ax^2 + bx + c \). Here, the coefficient \(a\) influences the shape and direction of the parabola:

• If \( a > 0 \), the parabola opens upwards, resembling a smile.
• If \( a < 0 \), it opens downwards, resembling a frown.

The graph's symmetry is always about the vertical line passing through the vertex, making it essential to determine the axis of symmetry. This axis is where you find both the minimum or maximum point of the parabola, depending on its direction. In the specific scenario given, where \(a > 0\) and \( b^2 - 4ac < 0 \), the parabola opens upwards and does not cross the x-axis at all. Instead of intersecting the axis, the entire graph sits either completely above it when viewed on a typical Cartesian plane.

Quadratic Discriminant

The quadratic discriminant is key to understanding the nature of the roots of a quadratic equation. It is found using the expression \( b^2 - 4ac \). This value can help predict how the parabola associated with a quadratic equation interacts with the x-axis:

• If \( b^2 - 4ac > 0 \), there are two distinct real roots, meaning the parabola intersects the x-axis at two points.
• If \( b^2 - 4ac = 0 \), there is exactly one real root, meaning the parabola touches the x-axis at one point, called a double root.
• If \( b^2 - 4ac < 0 \), there are no real roots, implying the parabola does not touch or cross the x-axis at all.

For the exercise at hand, with \( b^2 - 4ac < 0 \), the discriminant tells us the parabola is entirely above the x-axis since \(a > 0\). This is a crucial aspect when sketching the graph because it directly indicates the absence of x-intercepts.

Vertex of a Parabola

The vertex of a parabola is either the lowest or highest point on the graph, depending on how the parabola opens. Finding the vertex is fundamental for sketching the graph accurately. To locate the vertex, use the formula:\[ x = -\frac{b}{2a} \]This formula finds the x-coordinate of the vertex, and substituting it back in the original quadratic equation gives the corresponding y-coordinate:\[ y = f\left(-\frac{b}{2a}\right) \]

• For an upward-opening parabola (\(a > 0\)), the vertex will be the minimum point.
• For a downward-opening parabola (\(a < 0\)), the vertex will be the maximum point.

In our exercise, with \( a > 0 \) and no real roots, the vertex is not just the minimum point but also the lowest point entirely above the x-axis. Knowing the vertex helps to accurately sketch the parabola, ensuring it correctly represents its mathematical behavior and context in the scenario.