Problem 54 - Algebra and Trigonometry

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 negative slope and a parabola whose equation has a negative leading coefficient, has one solution.

Step-by-Step Solution

Answer

The graphs of the given equations can be represented by a downward-sloping line and an upside-down parabola. Their intersection shows the one solution to the system.

Step 1: Sketch the linear function

First, plot the straight line given with a negative slope. Without a set equation, we cannot determine exact points on our line. However, knowing that the slope is negative means that the line will decrease as we move from left to right along the x-axis. Draw a line reflecting this.

Step 2: Sketch the quadratic function

Next, sketch a parabola to represent the quadratic function. As the leading coefficient is negative, the parabola will open downwards. This means it has a maximum turning point and its graph will look like an upside-down 'U'.

Step 3: Identify the solution

Finally, the solution is the intersection point of the two graphs (the line and the parabola). As given, there is one solution to this system, meaning the line and parabola should intersect at one point.