Problem 25 - Algebra and Trigonometry

Question

23-26 \(\mathbf{}\) Do the graphs intersect in the given viewing rectangle? If they do, how many points of intersection are there? $$ y=6-4 x-x^{2}, y=3 x+18 ;[-6,2] \text { by }[-5,20] $$

Step-by-Step Solution

Answer

The graphs intersect at two points within the viewing rectangle.

Step 1: Understand the Problem

We need to determine whether the graphs of the equations \(y = 6 - 4x - x^2\) and \(y = 3x + 18\) intersect in the specified viewing rectangle \([-6, 2]\) by \([-5, 20]\). If they intersect, we should find the number of intersection points.

Step 2: Set the Equations Equal to Each Other

To find the points of intersection, set the equations equal: \(6 - 4x - x^2 = 3x + 18\), and solve for \(x\).

Step 3: Rearrange the Equation

Rearrange the equation to form a quadratic equation: \[- x^2 - 7x - 12 = 0.\]

Step 4: Solve the Quadratic Equation

Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = -1\), \(b = -7\), and \(c = -12\) to solve for \(x\).

Step 5: Calculate the Discriminant

The discriminant \(b^2 - 4ac\) helps determine the nature of the solutions. Calculate it:\[(-7)^2 - 4(-1)(-12) = 49 - 48 = 1.\]

Step 6: Compute the Roots

Since the discriminant is positive, compute the roots:\[x = \frac{-(-7) \pm \sqrt{1}}{2(-1)} = \frac{7 \pm 1}{-2}.\]The roots are \(x = -4\) and \(x = -3\).

Step 7: Verify the Points of Intersection

Substitute \(x = -4\) and \(x = -3\) back into either original equation to verify the \(y\)-coordinates:- For \(x = -4\): - Using the first equation: \(y = 6 - 4(-4) - (-4)^2 = 6 + 16 - 16 = 6\).- For \(x = -3\): - Using the first equation: \(y = 6 - 4(-3) - (-3)^2 = 6 + 12 - 9 = 9\).Both points fall within the given viewing rectangle.

Key Concepts

Quadratic Equations

A quadratic equation is a type of polynomial equation that follows the standard form: \[ ax^2 + bx + c = 0, \]where \(a, b,\) and \(c\) are constants, and \(a eq 0\). The graph of a quadratic equation is a parabola. Depending on the sign of \(a\), the parabola opens upwards (\(a > 0\)) or downwards (\(a < 0\)). For example, in the problem, the equation \(6 - 4x - x^2 = y\) represents a downward-opening parabola, due to the coefficient \(a = -1\) being negative.
Quadratic equations often arise in various contexts, such as projectile motion or finding the area of certain geometric shapes. They are crucial when determining the points of intersection between curves, as they provide values where two graphs meet.

• To solve a quadratic manually, we can factorize it, complete the square, or use the quadratic formula.

Understanding how to manipulate and solve quadratic equations is a key skill in algebra, necessary for exploring more complex mathematical concepts.

Discriminant

The discriminant is a powerful tool in determining the nature and number of solutions for a quadratic equation. It's derived from the quadratic formula itself and is given by:\[ b^2 - 4ac. \] In our case, with the equation \(-x^2 - 7x - 12 = 0\), the values are \(a = -1\), \(b = -7\), and \(c = -12\). Calculating the discriminant gives:\[ (-7)^2 - 4(-1)(-12) = 49 - 48 = 1. \]This value tells us a lot:

• If the discriminant is positive, there are two distinct real roots.
• If the discriminant is zero, there is one real root (a double root).
• If the discriminant is negative, there are no real roots, but two complex roots.

For the given problem, a positive discriminant of 1 indicates two points where the graphs intersect, meaning the lines and parabola meet at these distinct points within the specified viewing rectangle.

Finding Roots

Finding the roots or solutions of a quadratic equation means figuring out the values of \(x\) that satisfy the equation. This is where the quadratic formula comes into play:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. \]In the example, we first rearrange the intersection equation into quadratic form, then apply the formula with \(a = -1, b = -7, \) and \(c = -12\):\[x = \frac{-(-7) \pm \sqrt{1}}{2(-1)}.\]Solving this gives the roots:

• \(x = -4\)
• \(x = -3\)

These roots represent the \(x\)-coordinates where the two graphs intersect. To verify the intersection, we substitute them back into either original equation to check the corresponding \(y\)-coordinates, ensuring these points fall within the specified viewing rectangle. This process solidifies our understanding of how graphical intersections relate to solving equations algebraically.