Problem 33 - Intermediate Algebra

Question

Graph the system \(\left(\begin{array}{l}y=x^{2}+2 \\ 6 x-4 y=-5\end{array}\right)\) and use the TRACE and ZOOM features of your calculator to demonstrate clearly that this system has no real number solutions.

Step-by-Step Solution

Answer

The system has no real number solutions as the graphs do not intersect.

Step 1: Identify the Equations

The given system of equations consists of a quadratic equation and a linear equation. They are: \( y = x^2 + 2 \) and \( 6x - 4y = -5 \). We need to graph these equations.

Step 2: Rearrange the Linear Equation

Rearrange the linear equation for \( y \). Start from: \( 6x - 4y = -5 \). Solve for \( y \): \( 4y = 6x + 5 \). Therefore, \( y = \frac{3}{2}x + \frac{5}{4} \).

Step 3: Graph the Equations

Use a graphing calculator to input the equations: \( y = x^2 + 2 \) (a parabola opening upwards) and \( y = \frac{3}{2}x + \frac{5}{4} \) (a straight line). Enter these into the y= editor of the calculator.

Step 4: Use ZOOM Feature

Use the 'ZOOM' feature on your calculator to adjust the view so that you can see the general behavior of both graphs. A good initial choice is the 'ZOOM 6:ZStandard' option to have a standard viewing window.

Step 5: Use TRACE Feature

Use the 'TRACE' feature to move along both curves. Investigate their paths to see if they intersect at any point within the visible window.

Step 6: Analyze Intersection

Observe that as you use the 'TRACE' feature, the parabola and the line do not intersect. This suggests that the graphs of the equations do not meet within the real number plane.

Key Concepts

Quadratic Equations

Quadratic equations are fundamental elements in algebra, forming U-shaped graphs known as parabolas. These equations take the form \( y = ax^2 + bx + c \), where \( a, b, \) and \( c \) are constants. The coefficient \( a \) determines the parabola's direction:

• If \( a > 0 \), the parabola opens upwards.
• If \( a < 0 \), the parabola opens downwards.

Parabolas have specific characteristics, such as a vertex, which is the highest or lowest point. Additionally, the axis of symmetry divides the parabola into two mirror-image halves, making analysis easier.
For our system, \( y = x^2 + 2 \) is in the standard quadratic form. The parabola opens upward because the coefficient of \( x^2 \) is positive. There is no linear \( x \) term, meaning it is centered along the y-axis at \( (0, 2) \). Understanding these basic properties helps assess the graph's behavior and relationships.

Linear Equations

Linear equations create straight lines on a graph. Their general form is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

• The slope \( m \) indicates the steepness and direction of the line.
• The y-intercept \( b \) is the point where the line crosses the y-axis.

For the given system, the linear equation \( 6x - 4y = -5 \) rearranges to the slope-intercept form \( y = \frac{3}{2}x + \frac{5}{4} \). Here, \( \frac{3}{2} \) is the slope, hinting at a relatively steep slope, and \( \frac{5}{4} \) tells us that the line crosses the y-axis just above 1.
Graphically, this line will extend infinitely, maintaining the same direction defined by its slope. Linear equations often serve as a comparison or solution method when they intersect with parabolas or other functions.

Graphing Calculator Techniques

Mastering a graphing calculator is crucial for visualizing complex systems involving multiple equations. For graphing the system of equations, several techniques come into play:

• Entering Equations: Input each equation separately through the 'y=' function on the calculator.
• Using the ZOOM Feature: This feature adjusts the view to better capture the entirety of both graphs. 'ZOOM 6:ZStandard' is an excellent starting point as it provides a general view.
• Employing the TRACE Feature: This allows closer examination of specific graph points, letting you move along the graph's path. It is essential for investigating intersections.

In our case, using TRACE showed that the quadratic and linear graphs did not intersect, indicating no real solutions for the system. Understanding these features enables more accurate and efficient problem-solving in exploring graph behaviors.