I was out on Thursday at a leadership meeting so on Friday, we graphed piecewise functions that didn't look like straight lines, using graphs from the quadratic, cubic, radical, and exponential function families. We use exactly the same process as we do for graphing linear functions: graph both graphs, graph the asymptote, and determine which part of which graph goes on the right and left side of the asymptote.
There are only two things that we need to keep in mind when graphing nonlinear piecewise functions:
1) Unlike linear functions, both nonlinear functions don't have to touch the asymptote
2) Unlike linear functions, both nonlinear functions don't have to show up on the graph. Sometimes, only one graph will be present.
Sunday, February 2, 2014
Wednesday, January 29th: Graphing Linear Piecewise Functions
Piecewise functions are strange because they are written in a very specific notation. They look something like this:
y = 2x + 1, x > 3
-x - 1, x <= 3
Basically, piecewise functions contain two functions that are on the same coordinate plane. The first part of the function gives the equation graphed, and the second part of the equation (with the greater than or less than) tells you where to put an asymptote on the graph. The number at the end will always be the same number because it tells where the asymptote is.
Here are three easy steps for graphing piecewise functions:
1) Graph both equations on a coordinate plane
2) Graph the asymptote, and determine which equation goes on which side of the asymptote
3) Re-graph the equation, with one line on one side of the asymptote and the other equation on the other side of the asymptote
y = 2x + 1, x > 3
-x - 1, x <= 3
Basically, piecewise functions contain two functions that are on the same coordinate plane. The first part of the function gives the equation graphed, and the second part of the equation (with the greater than or less than) tells you where to put an asymptote on the graph. The number at the end will always be the same number because it tells where the asymptote is.
Here are three easy steps for graphing piecewise functions:
1) Graph both equations on a coordinate plane
2) Graph the asymptote, and determine which equation goes on which side of the asymptote
3) Re-graph the equation, with one line on one side of the asymptote and the other equation on the other side of the asymptote
Tuesday, January 28th: Graphing Project Day
We made creative-looking projects today with your favorite graph families. For each family, you had to graph a vertical translation, horizontal translation, a reflection, and a dilation using your own functions. You also had to give me the equation that you used for each graph.
But most importantly, you had to provide evidence on how you KNEW that you were graphing the type of translation that was described. This tells me that you understand how the equations were formed and how the graphs visually show what is stated in the equation.
The projects that I have are very impressive - great job! Mrs. Long says that they are great-looking, and she loves that they are in the hallway.
But most importantly, you had to provide evidence on how you KNEW that you were graphing the type of translation that was described. This tells me that you understand how the equations were formed and how the graphs visually show what is stated in the equation.
The projects that I have are very impressive - great job! Mrs. Long says that they are great-looking, and she loves that they are in the hallway.
Monday, January 27th: Reflections and Dilations of Function Families
A reflection is when the graph is flipped across the x-axis. We see a reflection in an equation whenever there is a negative sign in front of the highest exponent, or in front of x if there isn't an exponent. The equations below all have reflections:
y = -x^2 + 4
y = -x^3 + 2x
y = -4^x
A dilation makes the graph bigger or smaller. The graph becomes bigger when there is a number in front of the x. If the number is a fraction, the graph becomes bigger/wider. If the number is a whole number, the graph becomes smaller/thinner. Examples:
y = 3x^2
y = 1/3x^3
y = -x^2 + 4
y = -x^3 + 2x
y = -4^x
A dilation makes the graph bigger or smaller. The graph becomes bigger when there is a number in front of the x. If the number is a fraction, the graph becomes bigger/wider. If the number is a whole number, the graph becomes smaller/thinner. Examples:
y = 3x^2
y = 1/3x^3
Thursday, January 23rd: Horizontal and Vertical Translations for Function Families
We have already done horizontal and vertical translations for trig families, and algebraic families are very similar. Vertical translations are found at the end of the equation, not next to the x, and tell us which way to go on the y-axis. The three functions below are examples of vertical translations:
y = x - 3
y = x^2 + 4
y = 3^x - 1
Horizontal translations are found either inside the parenthesis or, in radical equations, underneath the radical. They tell us where to move on the x-axis, BUT we have to write it backwards in the equation.
For example, if we wanted to do a horizontal translation of +4, we could write:
y = (x - 4)^2 because we would have to write it BACKWARDS.
Note that when we actually graph the equation, we still want to move +4 on the x-axis. We only write it backwards in the equation.
y = x - 3
y = x^2 + 4
y = 3^x - 1
Horizontal translations are found either inside the parenthesis or, in radical equations, underneath the radical. They tell us where to move on the x-axis, BUT we have to write it backwards in the equation.
For example, if we wanted to do a horizontal translation of +4, we could write:
y = (x - 4)^2 because we would have to write it BACKWARDS.
Note that when we actually graph the equation, we still want to move +4 on the x-axis. We only write it backwards in the equation.
Wednesday, January 22nd: Function Families
We explored five different function families in class today. You can tell if a function belongs to a certain family by either
1) the shape it makes when you graph it on a coordinate plane or 2) what it looks like as an equation
1) The linear function family are straight-line graphs. They look like y = 4x + 2 in an equation, where the x is next to the slope and there's usually (but not always!) a number behind it, which represents the y-intercept.
2) The quadratic function family are parabolas. They have x to the second power in their equations.
3) The cubic function family look like tangent graphs. They have x to the third power in their equations.
4) The radical function family looks like an arc going in one direction on the graph. They have x underneath the radical sign in their equations.
and finally...
5) The exponential function family looks like a giant curve on the graph. X is to a power (so the x is little) in a function.
1) the shape it makes when you graph it on a coordinate plane or 2) what it looks like as an equation
1) The linear function family are straight-line graphs. They look like y = 4x + 2 in an equation, where the x is next to the slope and there's usually (but not always!) a number behind it, which represents the y-intercept.
2) The quadratic function family are parabolas. They have x to the second power in their equations.
3) The cubic function family look like tangent graphs. They have x to the third power in their equations.
4) The radical function family looks like an arc going in one direction on the graph. They have x underneath the radical sign in their equations.
and finally...
5) The exponential function family looks like a giant curve on the graph. X is to a power (so the x is little) in a function.
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