Explain Linear and Exponential Functions
Linear: The graph of a linear equation can either run uphill or downhill. The slope is positive when the line is uphill, while the downhill line represents a negative slope. To find the slope, find two coordinates along the line. Then take the y-value of the second coordinate and subtract the y-value of the first coordinate. Do the same with the x-coordinates, Take the difference of the y-values and divide it by the difference of the x-values. The remaining value should give you the slope of the line. If you want to find the y-intercept of the graph, let x=0 in your equation. Then find the y-coordinate. Do the same for the x-intercept, except let y=0 and find the x-coordinate.
Slope= (Y2 - Y1)/(X2 - X1)
Equation of linear line: y=mx+b
Exponential: The graph of an exponential function can increase or decrease at different rates. Unlike the linear equation graph, the shape of an exponential function starts out as a straight line, then begins to rise up and ends up looking like a straight vertical line. Between the horizontal and vertical points, there is a curve in the graph.
How to find the slope, y-intercept, and solve problems using graphs and equations:
Linear - With Equation: In this example, we are using the equation y=3x+2
The equation for a line is y=mx+b. In this equation, we already know the m which is 3 (slope). The y-intercept is found by equaling the equation to 0 (0=3x+2). We would subtract the 2 from both sides (-2=3x) and then divide the equation by 3 leaving us with x=-2/3 (y-intercept).
Linear - With Coordinates: In this example, we are using the coordinates (0,0) and (2,4)
We don't have the slope, so we will have to find it using the equation (Y2 - Y1)/(X2 - X1) and plug the coordinates into it. This will leave us with (4-0)/(2-0) = 2. The slope is 2. We will plug that into the equation y=mx+b (y=2x+b). Then we will plug on of the sets of coordinates into the equation (0=2(0)+b). If we solve this equation, we will be left with b=0. So the y-intercept would be 0. We would expect this because the line goes through the origin.
Exponential: In this example we are using the numbers: 200,000 people, 25%, and 5 years.
We are trying to find here the amount of something at a particular point. Typically this would be a population growth because of the extreme change. We already have all of the appropriate numbers. Now all we have to do is write an equation and solve it. Using the form y=m*p^x, we plug 200,000 in as m,.25+1 as p, and 5 as x. The reason we add 1 to .25 in our p, is because the population is increasing, if we didn't add 1, the population would be decreasing, because you are using a number less than 1. You can then solve the problem and find the right answer.
EXPLAIN!
A) Consider the table of votes, over the past 3 years cast for ASB president.
Year |
2008 | 2009 | 2010 |
Total Votes |
74 | 78 | 82 |
Show that the relationship between variables is linear
B) Below is a table of population growing in a small rural town in Eastern Washington.
End of Year |
Number of people |
| 0 | 500 |
| 1 | 550 |
| 2 | 605 |
| 3 | 665 |
| 4 | 732 |
Show the relationship is exponential.
After writing your response, move on to ELABORATE!