Problem 53 - College Algebra Essentials

Question

make a rough sketch in a rectangular coordinate system of the graphs representing the equations in each system. The system, whose graphs are a line with positive slope and a parabola whose equation has a positive leading coefficient, has two solutions.

Step-by-Step Solution

Answer

The graph will depict a line with positive slope and an upward-opening parabola. The system will have two solutions, which are the intersection points of the line and the parabola.

Step 1: Determining the Line

Identify the line with positive slope in the Cartesian coordinate system. Suppose the equation of the line is \(y = mx + b\), where \(m > 0\). Draw the line starting from the y-intercept \(b\) and rising as you move to the right since the slope is positive.

Step 2: Determining the Parabola

Identify the parabola with a positive leading coefficient in the same Cartesian coordinate system. Suppose the equation of the parabola is \(y = ax^2 + bx + c\), where \(a > 0\). This indicates an upwards-opening parabola. The vertex of the parabola will serve as the starting point, and it will open upwards because \(a > 0\). Draw this on the same set of axes as the line.

Step 3: Identifying the Solutions

Look for the points of intersection between the line and the parabola. These intersection points represent the solutions to the system of equations. Our initial assumption was that the system has two solutions so we should see two crossing points.

Key Concepts

Rectangular Coordinate System

The rectangular coordinate system is more commonly known as the Cartesian coordinate system. It consists of two perpendicular lines or axes: the horizontal axis, called the x-axis, and the vertical axis, known as the y-axis. Together, these axes divide the plane into four quadrants, which we use to locate points by their coordinates \(x, y\). Each point on this plane has a unique pair of numbers, defining its position.

• Quadrants: The four regions created by the axes are numbered counterclockwise starting from the upper right, as Quadrants I to IV.
• Origin: The point where the x and y axes intersect is called the origin, designated as \(0, 0\).
• Coordinates: Every point’s position is determined by its distance along the x and y axes.

Understanding this system is essential for graphing equations like lines and parabolas because it allows us to visualize mathematical relationships.

Line with Positive Slope

A line with a positive slope rises as it moves from left to right across the graph. In the slope-intercept form of a line, given by the equation \(y = mx + b\), the slope is represented by the variable \(m\). If \(m > 0\), the line will ascend.

• Slope (m): Indicates how steeply the line rises or falls. A positive value means the line ascends.
• Y-intercept (b): The point where the line crosses the y-axis. The line moves upward starting from this point if the slope is positive.

By plotting the y-intercept on the coordinate system and using the slope to determine the line’s path, you can draw a line with a positive slope effectively. This is crucial when solving systems of equations involving a line.

Parabola with Positive Leading Coefficient

A parabola with a positive leading coefficient is a curve on the graph that opens upward. This shape is a hallmark of quadratic equations in the form \(y = ax^2 + bx + c\) where \(a > 0\).

• Vertex: The highest or lowest point of the parabola. For positive \(a\), it is the lowest point.
• Axis of symmetry: A vertical line that divides the parabola into two mirror-image halves, passing through the vertex.

Drawing this parabola involves identifying the vertex and then plotting points on either side following the general curve. Since \(a > 0\), focus on sketching the parabola as a U-shape that opens upwards.

Intersection Points

Intersection points are key in graphing systems of equations as they signal where two graphs meet. When dealing with a line and a parabola in a coordinate plane, these points represent the solutions to the system of equations.

• Graphical Solution: Intersection points can be seen where the line and the parabola cross each other on the graph.
• Coordinates of Intersection: Each intersection point has coordinates \(x, y\) that satisfy both the line's and parabola's equations.

In our example, you should expect to see two such intersection points, which indicate the existence of two solutions for the system. Checking these graphically helps confirm the validity of mathematical solutions.