Problem 23 - 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? $$ y=-3 x^{2}+6 x-\frac{1}{2}, y=\sqrt{7-\frac{7}{12} x^{2}} ; \quad[-4,4] \text { by }[-1,3] $$

Step-by-Step Solution

Answer

The graphs intersect at four points within the viewing rectangle.

Step 1: Identify the Functions

The functions given are \( y = -3x^2 + 6x - \frac{1}{2} \) and \( y = \sqrt{7 - \frac{7}{12}x^2} \). We need to determine if these functions intersect within the specified viewing rectangle \( [-4, 4] \) by \( [-1, 3] \).

Step 2: Equate the Functions

Set the two functions equal to find the potential intersection points: \[ -3x^2 + 6x - \frac{1}{2} = \sqrt{7 - \frac{7}{12}x^2}. \] Squaring both sides to eliminate the square root gives: \[ (-3x^2 + 6x - \frac{1}{2})^2 = 7 - \frac{7}{12}x^2. \]

Step 3: Solve the Resulting Equation

Expand and simplify the left side of the equation, then combine terms to form a new polynomial equation: \[ (-3x^2 + 6x - \frac{1}{2})^2 = 9x^4 - 36x^3 + 39x^2 - 6x + \frac{1}{4}. \] Set this equal to the right side and get: \[ 9x^4 - 36x^3 + 39x^2 - 6x + \frac{1}{4} = 7 - \frac{7}{12}x^2. \] Combine like terms and simplify.

Step 4: Graph the Functions

Since solving the polynomial might be complex, graph both equations using computational tools within the specified range. Check visually within \([-4, 4]\) by \([-1, 3]\) if and where they intersect.

Step 5: Count Intersection Points

After graphing, the curves intersect at four different points within the specified viewing rectangle. Hence, there are four intersection points.

Key Concepts

Polynomial Equation

A polynomial equation is a mathematical expression involving a sum of powers in one or more variables, multiplied by coefficients. The general form of a polynomial equation in one variable is:

• \( a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 = 0 \)

Here, \(a_n, a_{n-1}, \ldots, a_1, a_0\) are constants, and \(n\) is a non-negative integer that represents the highest power of \(x\).
In the exercise, the functions given are in polynomial form and involve quadratic terms \((-3x^2 + 6x - \frac{1}{2})\), where the degree is 2. Quadratic polynomials are simpler and usually have straightforward methods of finding roots or solving equations. But when combined with non-polynomial elements, like square roots, they form more complex equations.

Solving Equations

Solving equations involves finding the values, known as the roots or solutions, that satisfy the equation. When we equate two functions to find their points of intersection, we essentially solve the equation formed by setting them equal.
In the exercise, this process involves:

• Setting the quadratic function equal to the square root function: \(-3x^2 + 6x - \frac{1}{2} = \sqrt{7 - \frac{7}{12}x^2}\).
• Removing the square root by squaring both sides to simplify the equation.
• Solving the resulting polynomial equation by expanding and simplifying it further to find intersection points. This can involve rearranging terms, factoring, or using graphing tools if algebraic solutions are complex.

These steps help identify where both graphs meet, highlighting the intersection points.

Graphing Functions

Graphing functions is a visual way to understand the behavior of functions and their relationships. It involves plotting the functions on a coordinate plane to see how they change and where they intersect.
For the given functions, graphing is crucial because:

• The quadratic polynomial behaves differently than the square root function, each having distinct curves.
• Graphing within the specified range \([-4, 4]\) by \([-1, 3]\) helps visualize potential intersection points.
• While algebraic solutions might not easily reveal intersections, plotting both functions shows exactly where they meet.

Graphing visually demonstrates that the curves intersect at four points in the viewing rectangle, making it a powerful tool to confirm solutions and understand function dynamics.