Graphing Problems for Functions
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1. (a) - (c) Given the graph, identify the graph of the function (line, parabola, hyperbola, or exponential), explain your choice, and give the domain and range as shown in the graph, and also the domain and range of the entire function.
1. (d) Given a line containing the points (1,4), (2,7), and (3,10) determine the slope-intercept form of the equation, provide one additional point on this line, and graph the function.
2. In most businesses, increasing prices of products can negatively impact the number of customers. A bus company in a small town has an average number of riders of 800 per day. The bus company charges $2.25 for a ride. They conducted a survey of their customers and found that they will lose approximately 40 customers per day for each $.25 increase in fare.
(a) Let the number of riders be a function of the fare charged. Graph the function, identify the graph of the function (line, parabola, hyperbola, or exponential), find the slope of the graph, find the price at which there will be no more riders, and the maximum number of riders possible.
(b) The bus company has determined that even if they set the price very low, there is a maximum number of riders permitted each day. If the price is $0 (free), how many riders are permitted each day?
(c) If the bus company sets the price too high, no one will be willing to ride the bus. Beginning at what ticket price will no one be willing to ride the bus?
3. It is approximately 480 miles from Los Angeles, California, to San Francisco, California. Allowing for various traffic conditions, a driver can average approximately 60 miles per hour.
(a) How far have you traveled after 3 hours?
(b) How far have you traveled after 4 hours?
(c) How far have you traveled after t hours (i.e., write a linear function that expresses the distance traveled, d, as a function of time, t).
(d) How far will you HAVE LEFT to travel to reach San Francisco after you have traveled 3 hours?
(e) How far will you HAVE LEFT to travel to reach San Francisco after you have traveled 4 hours?
(f) How far will you HAVE LEFT to travel to reach San Francisco after you have traveled t hours (i.e., write a linear function that expresses the distance to be traveled to reach San Francisco, s, as a function of time, t).
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Solution Summary
Step by step instructions are given to demonstrate the graphing problems and to arrive at the answers.
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1) a) Given the above graph, identify the graph of the function (line, parabola, hyperbola, or exponential), explain your choice, and give the domain and range as shown in the graph, and also the domain and range of the entire function.
This is a parabola. It is curved, not linear, so it can't be a line; it is symmetrical, so it can't be an exponential function; and it is continuous, so it's not a hyperbola.
Parabolas are curved in such a way that they are symmetric about a vertical line (the left side is the mirror image of the right) and the slope of the curve gets steeper and steeper as the curve moves away from the vertex. This is exactly what this graph does.
The domain (x values) is all real numbers.
From looking at the graph, the minimum value seems to be at y = -1 (this happens at x = 2, so the vertex of the parabola is at the point (2, -1)). The graph never goes lower than -2, so the range (y values) is all real numbers greater than or equal to -2.
b) Given the above graph, identify the graph of the function (line, parabola, hyperbola, or exponential), explain your choice, and give the domain and range as shown in the graph, and also the domain and range of the entire function.
This is an exponential function. Exponential functions are curved, and their slope gets steeper and steeper as values of x increase (or decrease). This particular graph has a slope of almost 0 when x is small and the slope increases as x increases.
The domain is all real numbers.
The range is all real numbers greater than 0. If you look at negative values of x, the graph will get close to the x-axis (i.e. the value of y will get close to 0), but the graph will never actually touch the x-axis.
c) Given the graph above, identify the graph of the function (line, parabola, hyperbola, or exponential) and explain ...
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