Problem 25 - Precalculus Mathematics for Calculus

Question

Do the graphs intersect in the given viewing rectangle? If they do, how many points of intersection are there? $$y=6-4 x-x^{2}, y=3 x+18 ; \quad[-6,2] \text { by }[-5,20]$$

Step-by-Step Solution

Answer

The graphs intersect at two points.

Step 1: Set the Equations Equal

To find the points where the graphs intersect, we need to set the two equations equal to each other:\[6 - 4x - x^2 = 3x + 18\] and solve for \(x\).

Step 2: Rearrange the Equation

Rearrange the equation into a standard quadratic form. Subtract \(3x + 18\) from both sides to get:\[-x^2 - 7x - 12 = 0\]

Step 3: Solve the Quadratic Equation

Multiply the entire equation by -1 to simplify:\[x^2 + 7x + 12 = 0\]Factor the quadratic:\[(x + 3)(x + 4) = 0\]Setting each factor equal to zero gives solutions:\[x = -3\] and \[x = -4\]

Step 4: Check Solutions in the Interval

We need to ensure the solutions are within the given interval of \([-6, 2]\). Both solutions \(x = -3\) and \(x = -4\) are within this interval.

Step 5: Determine the Intersection Points

Substitute \(x = -3\) and \(x = -4\) back into either of the original equations to find the corresponding \(y\) values. Using \(y = 3x + 18\):- For \(x = -3\), \(y = 3(-3) + 18 = 9\)- For \(x = -4\), \(y = 3(-4) + 18 = 6\)Thus, the points of intersection are \((-3, 9)\) and \((-4, 6)\).

Step 6: Conclusion about Intersection

Since we have found two valid intersection points within the given viewing rectangle, the graphs intersect at two points.

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation where the highest degree of the variable is squared. It usually follows the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable we are solving for. Quadratic equations, like the one encountered in this exercise, appear in many mathematical problems, especially in algebra and calculus.

To solve a quadratic, you can use several methods:

• Factoring: This involves expressing the quadratic as a product of two binomials.
• Quadratic formula: A general formula that can solve any quadratic equation, given by \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
• Completing the square: This method transforms the quadratic into a perfect square trinomial.

In this problem, after setting the two equations equal, \(-x^2 - 7x - 12 = 0\), we simplified it to \(x^2 + 7x + 12 = 0\). The equation was then solved by factoring.

Factoring Polynomials

Factoring polynomials means expressing a polynomial as the product of other polynomials. It is a crucial step in solving quadratic equations, as seen in the current problem, \(x^2 + 7x + 12 = 0\). Factoring makes it easier to find the roots of the equation.

Here are some steps for factoring quadratic polynomials:

• Identify the components: Look for two numbers that multiply to give the constant term (\(c\)) and add to give the linear coefficient (\(b\)).
• Rewrite the middle term: Split the middle term based on the two numbers identified, and group the terms.
• Factor by grouping: Factor out the greatest common factor from each group.

In the given exercise, the polynomial \(x^2 + 7x + 12\) was factored as \((x + 3)(x + 4)\) because \(3\) and \(4\) added equal \(7\) and multiplied equal \(12\). This factorization revealed the solutions \(x = -3\) and \(x = -4\).

Coordinate System

The coordinate system is a fundamental concept in mathematics that allows us to locate points in a plane using pairs of numbers (coordinates). The most common system is the Cartesian coordinate system, which consists of two perpendicular axes: the horizontal \(x\)-axis and the vertical \(y\)-axis.

Each point on the plane can be identified by an \((x, y)\) pair, where \(x\) is the position along the horizontal axis, and \(y\) is the position along the vertical axis. For our exercise, finding the coordinates \((-3, 9)\) and \((-4, 6)\) was crucial for identifying the intersection points within the given viewing rectangle \([-6, 2] \text{ by } [-5, 20]\).

When plotting graphs, the coordinate system helps visualize the intersection of functions by looking at where their graphs meet or cross over each other.

Points of Intersection

Points of intersection between graphs are where two or more graphs meet on a coordinate plane. They are crucial for solving simultaneous equations and can provide solutions to many mathematical problems. In the context of the given exercise, we set the two functions equal to find where they intersect:

\[6 - 4x - x^2 = 3x + 18\]

The solutions we found, \(x = -3\) and \(x = -4\), were then used to find corresponding \(y\)-values to determine the exact points: \((-3, 9)\) and \((-4, 6)\). These are the points of intersection within the viewing interval.

To find points of intersection:

• Set the equations of the functions equal to each other.
• Solve for \(x\) to find where the graphs intersect.
• Substitute the \(x\) values back into either original equation to find corresponding \(y\) values.

By checking the interval \([-6, 2]\), we confirmed that both points lie within the appropriate section of the graph, ensuring our results are accurate.