1.
a. The growth factor in an exponential line is how steep the curve is. The y-intercept is where the line starts on the y-axis. In a table, you divide the y of any x by the y of the x before it, and you get the growth factor. In a graph, you do the same, only you have to find the points, and divide the y values. In the equation y= a(b^x), b is the growth factor. To find the y-intercept in a table, you just look to where the x=o. On a graph, you just look at where the line crosses the y-axis. In the equation y=a(b^x), the y-intercept is a.
b. If you know what the y-intercept and growth factor are, you can plug them into the standard equation y= a(b^x).
2.
a. The a is the y-intercept, the b is the growth factor.
b. a is the y when x=o, and where the line crosses the y-axis
c. b is the steepness of the line.
Monday, March 22, 2010
Yesterday, our class worked on graphing Exponential Functions in order to learn how certain variables effect the function's graph. In the equation y=a(b)*+c the variables a, b and c all change the way the graph looks. The a is the y-intercept or initial value, unless there is a c in the equation. If there is a c, the two variables values are added to create the y-intercept. The b in the equation is the growth factor. It represents the rate that the line is increasing or decreasing much like the slope does in a linear equation. For example, if the b is two, then the y is doubling every interval. Next, the c is, again, the added or subtracted variable that is able to change the y-intercept and the a. The c sometimes is the y-intercept when you add one to it, because in many Exponential Functions the initial factor (a) is one. These three important variable can reveal a lot of information about an Exponential Function's graph.
Mathematical reflections 2
1. a. You can find the y-intercept in an exponential relationship by using a table, a graph, and an equation. You can find the y-intercept from a table by looking at the y when the x is zero on the table. Whatever the y is will be the y-intercept. you can find the y-intercept from a graph by looking at where the line passes the y axis and that will be the y-intercept. You can find the y-intrcept from an equation by looking at the number that is being multiplied by a number with an exponent and that will be the y-intercept.
You can also find the growth factor in an exponential relationship by using a table, a graph, and an equation. you can find the growth factor from a table by looking at ho much y is being multiplied by for each x and that will be the growth factor. You can find the growth factor from a graph by looking at how much y is multiplied on the graph for every x and that will be the growth factor. You can find the growth factor of an equation by seeing what number has an exponent and that will be the growth factor.
1. b. You can use the y-intercept and growth factor to write an equation for an exponential relationship. You can put the y-intercept first and have it multiplied by the growth factor with an exponent of whatever number x the problem is at.
2. a. In the equation y=a(b^x), a is the initial value of the exponetial relationship and b is the growth factor of the exponetial relationship.
2. b. In a graph of y=a(b^x), a is the y-intercept.
2. c. In a graph of y=a(b^x), b is the amount y goes up for every time x goes up one interval or the steepness of the line.
You can also find the growth factor in an exponential relationship by using a table, a graph, and an equation. you can find the growth factor from a table by looking at ho much y is being multiplied by for each x and that will be the growth factor. You can find the growth factor from a graph by looking at how much y is multiplied on the graph for every x and that will be the growth factor. You can find the growth factor of an equation by seeing what number has an exponent and that will be the growth factor.
1. b. You can use the y-intercept and growth factor to write an equation for an exponential relationship. You can put the y-intercept first and have it multiplied by the growth factor with an exponent of whatever number x the problem is at.
2. a. In the equation y=a(b^x), a is the initial value of the exponetial relationship and b is the growth factor of the exponetial relationship.
2. b. In a graph of y=a(b^x), a is the y-intercept.
2. c. In a graph of y=a(b^x), b is the amount y goes up for every time x goes up one interval or the steepness of the line.
Math Reflection 2
1a. The growth factor of an expenentionally grown line is the steepness. It shows how fast it goes up for example doubling or tripleing. To find it in in a table you divide y when x equals 2 by y when x equals 1. To get it from a graph you do the same thing from the table exept you have to find the points. From the equation: Y= a(b to the x) the growth factor is the b. To find the y intercept in a table or graph you take the number when x is zero. To find it from the equation you fill in 0for x or take the a value.
1b. You fill in the Y intercept as the a value and the growth factor for the b value.
2a. A is the y intercept b is the growth factor.
2b. A is the spot where x equals 0.
2c. B is how rapidly the line is going up.
1b. You fill in the Y intercept as the a value and the growth factor for the b value.
2a. A is the y intercept b is the growth factor.
2b. A is the spot where x equals 0.
2c. B is how rapidly the line is going up.
Math Reflection
1.a.The growth factor is how fast the slope grows and the steepness of the graph. You can get it by dividing y of any given x value by the y of the x value below that. For example, for a graph with the points (0,1) and (1,4) you would divide 4 by 1 to get a growth factor of 4. To find the y-intercept you take the a in and equation or whatever the y is when x=0 in a graph or table.
1.b.You plug the y-intercept in as a and the growth factor as b.
2.a.A represents the original value or the y-intercept. B represents the growth rate.
2.b.A is the y-intercept of the graph.
2.c.B is how fast the slope grows and how curved the line is.
1.b.You plug the y-intercept in as a and the growth factor as b.
2.a.A represents the original value or the y-intercept. B represents the growth rate.
2.b.A is the y-intercept of the graph.
2.c.B is how fast the slope grows and how curved the line is.
Exploring Exponential Functions
In class on Friday we did a graphing activity called exploring exponential functions. In the activity we graphed different sets of equations that all were the same except 1 or 2 factors were changed to figure out what the different elements did to the graph.
In set 1, a and b were the same but c varied. From this we figured out that c moves the graph up and down. To figure out the y-intercept you can use the equation (when y is the intercept) c+a=y.
In set 2, b and c stayed the same but b varied. The higher b was, the higher the y-intercept was. The graphs look similar except that the graph accelerates faster to start out when the a is higher.
In set 3, the only thing that changed was the growth factor which effected how fast the slope grew and moreover the steepness of the graph. The higher the growth factor the steeper the graph.
Finally, in set four we explored a fraction as a growth factor and a negative. When the growth factor is a fraction the slope starts out large and gets smaller and smaller as the x continues. When the growth factor is negative the graph goes down instead of up and mirrors whatever the graph would be if the growth factor were not a negative.
In set 1, a and b were the same but c varied. From this we figured out that c moves the graph up and down. To figure out the y-intercept you can use the equation (when y is the intercept) c+a=y.
In set 2, b and c stayed the same but b varied. The higher b was, the higher the y-intercept was. The graphs look similar except that the graph accelerates faster to start out when the a is higher.
In set 3, the only thing that changed was the growth factor which effected how fast the slope grew and moreover the steepness of the graph. The higher the growth factor the steeper the graph.
Finally, in set four we explored a fraction as a growth factor and a negative. When the growth factor is a fraction the slope starts out large and gets smaller and smaller as the x continues. When the growth factor is negative the graph goes down instead of up and mirrors whatever the graph would be if the growth factor were not a negative.
Mathematical Reflecations 2
1. a. To find the y-intercept for an exponential equation using the graph, you must extend the graph to the point where the line in question crosses the y-axis. To find the growth factor of an exponential relationship using a graph, you need to compare different values. The easiest way to do this is to divide a y-value by the y-value preceding it, and the quotient is the growth factor. If you are not sure if it is an exponential relationship, it is always good to try this a few times with different numbers.
The y-intercept is most easily found using a table. To do this, you simply find where the value of x is 0 and look at the corresponding value of y. The growth factor is found with a table the same way it is found with a graph. By dividing a value of y by the one before it you find the growth factor of that particular exponential equation.
The third way to find the y-intercept and growth factor of an exponential equation is to look at the equation itself.
The y-intercept is most easily found using a table. To do this, you simply find where the value of x is 0 and look at the corresponding value of y. The growth factor is found with a table the same way it is found with a graph. By dividing a value of y by the one before it you find the growth factor of that particular exponential equation.
The third way to find the y-intercept and growth factor of an exponential equation is to look at the equation itself.
The equation above is the standard form for exponential equations. The a is the initial valuue, also known as the y-intercept. The b is the growth factor.
b. Writing and exponential equation is very simple. Since a is the initial value, or y-intercept, you just have to substitute your y-intercept into the equation given in 1.a. The growth factor is represented by b, and so all that is left to do is substitute your second variable in and you have your equation.
2.a. In the standard form of the exponential equation shown in 1.a., the a represents the intial values, or y-intercept. The b, in turn, represents the growth factor of the relationship.
b. On a graph, the a is represented by where the line crosses the y-axis.
c. On a graph, the b is represented by the slope or "shape" of the line.
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