Problem 70 - College Algebra and Calculus: An Applied Approach

Question

Solve the quadratic equation and then use a graphing utility to graph the related quadratic function in the standard viewing window. Discuss how the graph of the quadratic function relates to the solutions of the quadratic equation. Function \(y=-x^{2}-3 x+4\) Equation $$-x^{2}-3 x+4=0$$

Step-by-Step Solution

Answer

The solutions of the quadratic equation are the x-values where the graph of the function intercepts the x-axis. The solutions to \( -x^{2}-3x+4=0 \) are the x-coordinates of the intercept points of the graph of \( y=-x^{2}-3x+4 \).

Step 1: Solve the Quadratic Equation

The quadratic equation to solve is \(-x^{2}-3 x+4=0\). This can be addressed by using the quadratic formula: \(x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}\) where \(a=-1\), \(b=-3\), \(c=4\). Plugging values and solving will yield the roots of the equation.

Step 2: Graph the Quadratic Function

The quadratic function to be graphed is \(y=-x^{2}-3x+4\). This can be done by using a graphical calculator or an online graphing utility to get an idea of the shape and salient points of the function.

Step 3: Analyze the Relationship

Discussion on how the graph of the quadratic function relates to the solutions of the quadratic equation. The roots calculated in the first step should correspond to the points where the graph intersects the x-axis (these are known as 'zeros' or 'x-intercepts' of the function).

Key Concepts

Quadratic Formula

The quadratic formula is a fundamental tool for solving quadratic equations of the form \( ax^2 + bx + c = 0 \). It is expressed as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula allows you to find the roots or solutions of any quadratic equation. It involves three coefficients:

• \( a \): the coefficient of \( x^2 \)
• \( b \): the coefficient of \( x \)
• \( c \): the constant term

In the context of the equation \(-x^2 - 3x + 4 = 0\), the values are \( a = -1 \), \( b = -3 \), and \( c = 4 \). Plugging these into the quadratic formula simplifies the process of finding the roots of the equation.Let's break it down further:

• Calculate the discriminant \( b^2 - 4ac \). If it's positive, there are two real solutions. If zero, one real solution. If negative, there are no real solutions (the solutions are complex).
• Substitute \( a \), \( b \), and \( c \) into the formula to find the exact roots.

These roots indicate where the function \(-x^2 - 3x + 4\) crosses the x-axis, helping us understand the underlying graph.

Graphing Quadratic Functions

Graphing quadratic functions projects the equation onto a coordinate plane, giving a visual representation of its behavior and solutions.Quadratics generally form a parabola, a U-shaped curve, which can open upwards or downwards depending on the coefficient \( a \). If \( a < 0 \), the parabola opens downward, as in our function \(-x^2 - 3x + 4\).Here's how to graph this quadratic function:

• Identify the vertex, the highest or lowest point on the graph. Use the formula \( x = -\frac{b}{2a} \) to find the x-coordinate of the vertex, then solve for y.
• Plot the y-intercept, where \( x = 0 \). It's the constant term \( c \) in the equation.
• Use symmetrical points around the vertex to complete your parabola. The vertex axis helps in mirroring points across it.

Knowing how to graph it offers insight into the function's structure and roots, setting the stage for deeper analysis.

Solutions and Graph Analysis

Understanding the relationship between the solutions of a quadratic equation and its graph is crucial for a complete analysis.The roots calculated from the quadratic formula are the x-intercepts of the function's graph, confirming where the parabola crosses the x-axis.Here’s what you should consider:

• X-intercepts: The points where the parabola intersects the x-axis. In our problem, these are the solutions from the quadratic formula.
• Vertex: This gives the maximum or minimum value of the function, offering a deeper insight into how the function behaves.
• Axis of Symmetry: A vertical line through the vertex, showing the parabola's symmetrical nature. It’s calculated as \( x = -\frac{b}{2a} \).

The graph not only visualizes the equation’s solutions but also provides a comprehensive view of the function's overall behavior. This helps in predicting future trends and understanding the real-world phenomena described by the equation.