Problem 79 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the equation and graphically approximate the values of \(x\) that satisfy the specified inequalities. Then solve each inequality algebraically. Equation \(y=-x^{2}+2 x+3\) Inequalities (a) \(y \leq 0\) (b) \(y \geq 3\)

Step-by-Step Solution

Answer

After completing the above steps, one would have graphically approximated the values of \(x\) satisfying the given inequalities and also solved inequalities algebraically. Exact solutions would depend on the results of algebraic calculations in steps 3 and 4.

Step 1: Plotting the quadratic equation

Start by plotting the equation \(y=-x^{2}+2 x+3\) using a graphing utility. Pay attention to the shape and the key points of the graph (such as vertex and x-intercepts). From this graph, one can have an initial approximation of the values of \(x\) that will satisfy the given inequalities.

Step 2: Graphical approximation of solution within given inequalities

Visually approximate the values of \(x\) that make \(y \leq 0\) on the graph. This would be the range of \(x\) when the graph of the equation lies on or below the x-axis. Similarly, approximate values of \(x\) that make \(y \geq 3\). The goal is to find the range of \(x\) when the graph of the equation lies on or above the line \(y = 3\).

Step 3: Algebraic solution of first inequality

While visual approximation gives us an idea of the solution, finding exact solution will require solving the inequalities algebraically. Begin by setting our quadratic equation equal to 0, \(-x^{2}+2 x+3=0\). Solve this equation for \(x\), which gives possible boundary points for our interval. Then, use these boundary points to determine our solution set. Solve the inequality \(-x^{2}+2 x+3 \leq 0\) by using the boundary points to sign test the intervals.

Step 4: Algebraic solution of second inequality

The process is repeated to solve the second inequality algebraically. Start by setting our equation equal to 3, \(-x^{2}+2 x+3=3\), and solve for \(x\). Again, these solutions will be the boundary points for our interval. Then solve the inequality \(-x^{2}+2 x+3 \geq 3\), by sign testing the intervals using the boundary points.

Key Concepts

Graphing Quadratic Functions

Graphing quadratic functions involves plotting a curve that represents the equation of the form \( y = ax^2 + bx + c \). In our specific example, we have the quadratic equation \( y = -x^2 + 2x + 3 \). This is a downward-facing parabola because the coefficient of \( x^2 \) is negative.

When graphing this function, the key features to identify include:

• Vertex: The highest point on a downward-facing parabola. Use the formula \( x = -\frac{b}{2a} \) to find the x-coordinate of the vertex.
• Axis of Symmetry: A vertical line through the vertex, where the parabola is symmetric.
• X-intercepts: The points where the graph crosses the x-axis, found by solving the equation \( -x^2 + 2x + 3 = 0 \).
• Y-intercept: The point where the graph crosses the y-axis, which in our equation is \( y = 3 \).

To accurately graph the function, use these details to plot both intercepts and the vertex, and then sketch the parabola with these points as guides. Observing this graph will help visually approximate where specific inequalities are true, such as \( y \leq 0 \) or \( y \geq 3 \).

Solving Equations Algebraically

To solve quadratic inequalities algebraically, we start by solving the associated quadratic equation. For instance, to solve \( -x^2 + 2x + 3 = 0 \), employ methods such as factoring, completing the square, or the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

This provides the x-values where the equation equals zero, known as the boundary points. In our case, these solutions might include irrational numbers, which should be computed or simplified as needed.

To solve the inequality \( -x^2 + 2x + 3 \leq 0 \), examine intervals created by the boundary points and test points within those intervals to determine the sign of the inequality.

• If the value is negative, the inequality holds true for that interval.
• For \( -x^2 + 2x + 3 \geq 3 \), set the equation to 3 and solve \( -x^2 + 2x + 3 = 3 \), forming new boundary points. Again, check intervals to find where the inequality is satisfied.

Reassessing each inequality within these intervals provides the sets of x-values that satisfy them, allowing for a fully exact algebraic solution.

Using Graphing Utilities

Graphing utilities, such as graphing calculators or software, are powerful tools for visualizing functions. In solving quadratic inequalities, they can significantly ease the process by providing a visual approximation of solutions.

To use a graphing utility for the function \( y = -x^2 + 2x + 3 \):

• Input the equation into the utility.
• Observe the plotted parabola for how it interacts with the x-axis or horizontal lines, such as \( y = 3 \).
• Estimate the values of \( x \) where \( y \leq 0 \) or \( y \geq 3 \).

The graph provides a clear picture of where the function dips below the x-axis and where it rises above the line \( y = 3 \). While precise calculations must still be done algebraically, visual insights from the graphing utility often speed up understanding, provide confirmations, and facilitate a more intuitive grasp of the problem.