Problem 57 - College Algebra and Calculus: An Applied Approach

Question

Use a graphing utility to find the point(s) of intersection of the graphs. Then confirm your solution algebraically. $$\left\\{\begin{array}{l}y=x^{2}+3 x-1 \\ y=-x^{2}-2 x+2\end{array}\right.$$

Step-by-Step Solution

Answer

The points of intersection of the two graphs are (-3, 11) and (0.5, -0.25).

Step 1: Graph the functions

Use a graphing calculator or an online graphing utility to plot the graphs of both functions \(y=x^2+3x-1\) and \(y=-x^2-2x+2\). Note down the points where the graphs intersect.

Step 2: Solve for x

Setting the two functions equal to each other will give a quadratic equation: \(x^2+3x-1=-x^2-2x+2\), which simplifies to \(2x^2+5x-3=0\). Solve this equation for x using the quadratic formula, \(x=(-b± \sqrt{b^2-4ac}) / 2a\). Substitute \(a=2\), \(b=5\), and \(c=-3\) into the formula, you will get \(x\) as -3 and 0.5.

Step 3: Solve for y

Substitute the \(x\) values into the original equation to find the corresponding \(y\) values. You will get the values of \(y\) for \(x=-3\) and \(x=0.5\) to be 11 and -0.25 respectively, which are the y-coordinates of the points of intersection.

Step 4: Confirm Solution

Verify that the intersection points (-3, 11) and (0.5, -0.25) that were obtained algebraically match with the intersection points found graphically. This confirms the solution.

Key Concepts

Quadratic Equations

Quadratic equations play a fundamental role in mathematics, often coming up in problems relating to the intersection of graphs. A quadratic equation is generally expressed in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. Quadratics form parabolas when graphed, which can open either upward or downward, depending on the sign of \(a\). In our exercise, we are dealing with two such equations:

• \(y = x^2 + 3x - 1\)
• \(y = -x^2 - 2x + 2\)

To find their intersection points, we set the equations equal. This process combines them into a single quadratic equation, allowing us to find values for \(x\) at which both functions have the same \(y\) value. Quadratic equations often have two solutions, as seen in our scenario, due to the squared term, which can result in up to two intersection points on a graph. Understanding how to handle and solve these equations is crucial for algebraic problem-solving.

Graphing Utilities

Modern graphing utilities, such as graphing calculators or online tools, make it easy to visualize functions and their intersections. By entering the equations \(y = x^2 + 3x - 1\) and \(y = -x^2 - 2x + 2\) into a graphing utility, students can instantly see where the graphs intersect. Using graphing utilities offers several advantages:

• They provide a visual representation that makes understanding concepts easier.
• They allow checking solutions by matching graphical data with algebraic calculations.
• They enable exploration by quickly altering equations and observing changes in the graph.

This visualization is a powerful tool that supports learning and engagement, letting students explore the shaping and intersection dynamics of quadratic functions. Seeing the intersection graphically helps to confirm algebraic solutions and deepen understanding.

Quadratic Formula

The quadratic formula is a reliable method for solving any quadratic equation. It is particularly useful when equations are not easily factorable. The formula is given by:\[ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \]In our case, after equating the two functions, we arrived at a new quadratic equation: \(2x^2 + 5x - 3 = 0\). Here, the coefficients are \(a = 2\), \(b = 5\), and \(c = -3\). By substituting these values into the quadratic formula, we calculate the potential intersection points:

• For the first solution, use the \( + \) sign.
• For the second solution, use the \( - \) sign.

Completing these calculations provided the \(x\)-values of -3 and 0.5. These values pinpoint where the two parabolas intersect in the \(x\)-direction. Calculating using the quadratic formula is often a step that unlocks the solution to complex intersection problems, making it an essential algebraic tool.