Problem 24 - 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? $$ y=\sqrt{49-x^{2}}, y=\frac{1}{5}(41-3 x) ; \quad[-8,8] \text { by }[-1,8] $$

Step-by-Step Solution

Answer

The graphs intersect at one point within the viewing rectangle.

Step 1: Understand the Functions

The graphs we need to consider are those of the functions \( y = \sqrt{49-x^2} \) and \( y = \frac{1}{5}(41-3x) \). The first function is a semi-circle with radius 7 centered at the origin, and the second function is a line with a slope of \(-\frac{3}{5}\) and a y-intercept of \( \frac{41}{5} \). We need to find points where the semi-circle and the line intersect.

Step 2: Set the Functions Equal

To find the points of intersection, set the equations equal to each other: \( \sqrt{49-x^2} = \frac{1}{5}(41 - 3x) \). This equation will help us identify potential intersection points by solving for \(x\).

Step 3: Square Both Sides to Eliminate the Square Root

Square both sides of the equation to remove the square root: \( 49 - x^2 = \left( \frac{1}{5}(41 - 3x) \right)^2 \). This results in a polynomial equation to solve.

Step 4: Simplify and Solve the Polynomial Equation

Expand and simplify the right side: \( \left( \frac{41}{5} - \frac{3}{5}x \right)^2 = \frac{1}{25}(1681 - 246x + 9x^2) \), leading to \( 49 - x^2 = \frac{1}{25}(1681 - 246x + 9x^2) \). Solve for \(x\) by multiplying through by 25 to remove the fraction and simplify the expression, resulting in a standard quadratic equation.

Step 5: Identify Real Solutions

Once the polynomial is simplified, solve for \(x\) by using the quadratic formula. Check for real solutions by calculating the discriminant (\(b^2 - 4ac\)) to ensure it is non-negative. This tells us whether the functions actually intersect.

Step 6: Verify Solutions on Intervals

Once solutions for \(x\) have been found, substitute back into either of the original equations to find \(y\). Ensure that the \((x, y)\) points lie within the viewing rectangle \([-8, 8] \text{ by } [-1, 8]\).

Key Concepts

Polynomial Equation

A polynomial equation is an expression involving a sum of powers in one or more variables multiplied by constant coefficients. In our exercise, the transformed equation from Step 4 reads as a quadratic polynomial equation once modified.

Key components of polynomial equations include:

• Degree: Indicates the highest power of the variable. For instance, a quadratic equation has a degree of 2.
• Coefficients: These are the constants that multiply the variable(s), influencing the shape and position of the polynomial on the graph.
• Terms: These are the segments separated by addition or subtraction within the polynomial.

For the given exercise, we derived a quadratic polynomial equation by clearing out the square root through squaring and simplifying the terms. The equation was then suitable for applying formulas like the quadratic formula to find its solutions.

Quadratic Formula

The quadratic formula is a tool used to find the solutions of a quadratic equation, which is a polynomial equation of the form: \[ ax^2 + bx + c = 0 \].

The solutions for \(x\) can be found using:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]where \(a\), \(b\), and \(c\) are coefficients from the equation.

Using the quadratic formula involves:

• Identifying the coefficients \(a\), \(b\), and \(c\).
• Substituting these into the quadratic formula.
• Solving the formula considering the plus-minus operation, which may result in two solutions.

The plus-minus sign indicates two potential solutions, which are essential because a quadratic equation can have up to two real roots. By applying the quadratic formula to the reduced polynomial from the exercise, we can determine the exact points of intersections.

Real Solutions

Real solutions to a polynomial equation indicate the real-number values of \(x\) that satisfy the equation—where the graph intersects the \(x\)-axis or, in our context, where two functions intersect.

To identify real solutions in polynomial contexts:

• Calculate the discriminant: \(b^2 - 4ac\) of the quadratic equation. This tells us about the nature of the roots:
  • If it's positive, there are two distinct real solutions.
  • If it's zero, there is exactly one real solution.
  • If it's negative, there are no real solutions in terms of real numbers (but instead complex solutions).
• Verify these solutions by plugging them back into the original equations to find corresponding \(y\) values and ensure they fall within given constraints like viewing rectangles.

Understanding whether solutions are real helps in determining the actual points of intersection between different functions such as the semi-circle and the line in the original problem.